Enumeration of Frobenius Local Rings of Order p^6r via Bilinear Forms

Abstract

Let p be a prime number and r a positive integer. This paper investigates the construction and classification of finite commutative rings of order $p^{6r}$ in which the set of zero-divisors J forms an ideal satisfying the conditions $J^3=0$, $J^2\ne 0$ with $J^2$ being principal. Under these conditions, the rings considered are precisely the Frobenius local rings. A Frobenius local (completely primary) ring R with these properties is referred to as a ring with property (P). These rings naturally divide into three classes according to their characteristic: p, $p^2$, or $p^3$. In the case of characteristic $p^2$, a further distinction is made depending on whether p lies in $J^2$ or in $J\setminus J^2$, where J denotes the Jacobson radical of R. The classification is achieved by associating to each ring a canonical matrix corresponding to a bilinear form and then applying matrix congruence techniques to reduce the problem to linear algebra over finite fields. This yields a complete and explicit description of all Frobenius local rings with property (P) of order $p^{6r}$, including their algebraic structure and enumeration.

1. Introduction

Let p be a prime number, and let m, n, k, and r denote positive integers. Throughout this work, all rings are finite, commutative, and associative with identity. Homomorphisms are assumed to be unital, subrings share the same identity, and modules are unital. This paper investigates the classification of finite local rings whose Jacobson radical J has nilpotency index three and whose square $J^2$ is principal. Rings of this kind, hereafter referred to as having property (P), form a significant subclass of Frobenius local rings and play a central role in algebraic coding theory, see [1], where Frobenius structures govern duality and weight properties of linear codes.

Let R be a finite local ring with Jacobson radical J and residue field $R/J\cong {\mathbb{F}}_{p^r}$. If $\left|R\right|=p^{mr}$, then J coincides with the set of zero-divisors of R, and the characteristic of R is $p^n$ for some $1\le n\le m$. The nilpotency index of J determines much of the structure of R. When $n=m$, the ring is a Galois ring and contains a maximal Galois subring $R_0\cong \mathrm{GR}(p^n,r)$. Moreover, there exist elements $x_1,\dots ,x_k\in J$ such that

$$R=R_0\oplus x_1{R}_0\oplus \cdots \oplus x_k{R}_0,J=p{R}_0\oplus x_1{R}_0\oplus \cdots \oplus x_k{R}_0,$$

(1)

and if $x\in J$ has additive order $p^{t_0}$, then $x{R}_0$ has order $p^{t_0}$, since $x{R}_0\cong R_0/p^{t_0}{R}_0$. Every element of $R_0$ admits a unique p-adic expansion ${\sum }_{i=0}^{n-1}{p}^i{a}_i$ with $a_i\in \Gamma \left(r\right)=\{0,1,\alpha ,{\alpha }^2,\dots ,{\alpha }^{p^r-2}\}$, the Teichmüller system of $R_0$.

The classification of finite local rings has a long and rich history. Raghavendran [2] provided foundational results by describing all local rings of order $p^{mr}$ and characteristic $p^n$ in the extremal cases $n=1$ and $n=m$, obtaining complete classifications when $J^2=0$ and when the radical reaches maximal nilpotency. Corbas [3,4] extended this work to general local rings with $J^2=0$ and analyzed their dependence on the characteristic, while Krull [5] introduced the notion of Galois rings and developed their fundamental properties. Classical references such as Wilkerson [6], Zariski and Samuel [7], Matsumura [8], Wirt [9], McDonald [10], and Honold [11] provide essential background on the structure of local and Frobenius rings, including the role of maximal ideals, coefficient subrings, and module decompositions. Subsequent works by Alkhamees [12] investigated the case $J^2\subseteq R_0$, while Alabiad and Alkhamees [13,14,15] classified local principal ideal (chain) rings and singleton local rings, and completed the enumeration of all local rings of order $p^{5r}$ with nilpotency index three, laying the groundwork for the present study.

In contrast to the case of order $p^{5r}$, where ${dim}_F(J/J^2)=3$, the present setting $\left|R\right|=p^{6r}$ forces ${dim}_F(J/J^2)=4$ under property (P). Consequently, the multiplication on the radical is governed by symmetric bilinear forms on a 4-dimensional vector space over the residue field, rather than on a 3-dimensional one. This leads to a fundamentally different congruence classification problem, namely the determination of orbits in $S_4\left(F\right)$ instead of $S_3\left(F\right)$. The change in dimension alters the canonical forms, the orbit structure, and the resulting enumeration, and therefore the case $p^{6r}$ cannot be obtained from the $p^{5r}$ case by a simple parameter shift.

The aim of this paper is to extend these classical results to the next nontrivial case, namely Frobenius local rings of order $p^{6r}$ satisfying property (P). To each such ring, we associate a canonical matrix that encodes a bilinear form over the residue field ${\mathbb{F}}_{p^r}$, and we show that the classification problem is equivalent to determining the congruence classes of these matrices under the action of ${\mathrm{GL}}_t\left({\mathbb{F}}_{p^r}\right)$, field automorphisms, and scalar multiplication by ${\Gamma }^{\times }\left(r\right)$. This matrix-based approach provides a transparent, constructive, and computationally effective method for describing all isomorphism classes of Frobenius local rings with property (P). While a uniform classification for arbitrary exponents q remains a challenging open problem, the present paper provides a complete solution for the first nontrivial extension beyond previously known cases within the Frobenius and property (P) framework.

The classification of Frobenius local rings of a fixed order is motivated by the fact that the order determines the dimensions of the successive radical layers $J/J^2$ and $J^2$, which are the key invariants controlling the ring multiplication under property (P). Fixing the order therefore fixes the size of the associated bilinear forms and makes a complete and explicit classification feasible. Although the techniques employed are classical, the case $m=6$ is the first instance in which the associated bilinear-form classification occurs in dimension four, yielding new canonical types and enumeration behavior not present at lower orders, and it fits within the standard incremental methodology of finite local ring classification, revealing stable structural patterns such as the independence of the enumeration from the parameter r.

The main results of this work can be summarized as follows. For each characteristic $p^n$ with $n=1,2,3$, we explicitly construct all Frobenius local rings with $J^3=0$, $J^2\ne 0$, and $J^2$ principal, and we derive their canonical structure matrices. For $n=1$ and $n=3$, we obtain closed-form linear relations for the number of non-isomorphic classes as functions of the nilpotency index m. For $n=2$, we distinguish between the cases $p\in J^2$ and $p\in J\setminus J^2$, derive a unified enumeration formula, and highlight its dependence on both p and the parity of m. These results generalize the classifications in [2,3,4,5,6,7,8,9,10,11,12,13,14,15] and establish a coherent framework encompassing all Frobenius local rings of order $p^{6r}$.

The remainder of this paper is organized as follows. Section 2 treats the case of characteristic p, derives the corresponding bilinear forms, and presents the classification of the associated rings. Section 3 addresses the case of characteristic $p^2$, distinguishing the subcases $p\in J^2$ and $p\in J\setminus J^2$, while Section 4 completes the classification for characteristic $p^3$. Several corollaries extend these results to larger values of m, providing general formulas for the enumeration of isomorphism classes. The classification developed here deepens the structural understanding of Frobenius local rings and supports their applications in algebraic coding theory, see [1], where such rings serve as natural alphabets for the construction of linear codes with duality-preserving and distance-optimal properties.

2. Preliminaries

Throughout this paper, all rings are assumed to be finite, commutative, and associative with identity. All homomorphisms preserve the identity element, subrings share the same identity, and all modules are unital. This section collects fundamental structural facts about finite local rings that will be used throughout this paper. For a detailed exposition of these classical results, the reader is referred to [7,8,10,11,16,17,18].

Let R be a finite local ring with Jacobson radical J and residue field $R/J\cong \mathbb{F}$, where p is a prime and m, n, and r are positive integers. It is well known that $\left|R\right|=p^{mr}$ and that the Jacobson radical coincides with the set of zero-divisors of R. The radical J is nilpotent with $J^m=0$, and $\left|J\right|=p^{(m-1)r}$. The characteristic of R is $p^n$ for some $1\le n\le m$. The residue field $\mathbb{F}$ has order $p^r$, and its multiplicative group is cyclic of order $p^r-1$.

If $n=m$, then R contains an element $\alpha$ of multiplicative order $p^r-1$, and R is a Galois ring, denoted $\mathrm{GR}(p^n,r)$. More generally, every finite local ring of characteristic $p^n$ contains a distinguished Galois subring $R_0\cong \mathrm{GR}(p^n,r)$, called the coefficient subring. If u is a unit in R, then $R_1=u{R}_0{u}^{-1}$ is also a coefficient subring. Moreover, there exist elements $x_1,\dots ,x_k\in J$ such that

$$R=R_0\oplus x_1{R}_0\oplus \cdots \oplus x_k{R}_0(\mathrm{as}{R}_0\text{-}\mathrm{modules}).$$

(2)

This decomposition plays a key role in understanding the additive and multiplicative structure of R.

Let $x\in J$ have additive order $p^{t_0}$. Then, the principal $R_0$-submodule generated by x has size $|x{R}_0|=p^{t_0}$, since $x{R}_0\cong R_0/p^{t_0}{R}_0$.

Let $R_0={\mathbb{Z}}_{p^n}\left[\alpha \right]$ be the maximal Galois subring, where $\alpha$ generates the multiplicative group of ${\mathbb{F}}_{p^r}$. Denote by:

$$\begin{array}{cc}\hfill \Gamma \left(r\right)& =\langle \alpha \rangle \cup \left\{0\right\}=\{0,1,\alpha ,{\alpha }^2,\dots ,{\alpha }^{p^r-2}\},\hfill \\ \hfill {\Gamma }_1& =\left\{\mathrm{non}\text{-}\mathrm{squares}\mathrm{in}{\Gamma }^{\times }\left(r\right)\right\},\hfill \\ \hfill {\Gamma }_2& =\left\{\mathrm{squares}\mathrm{in}{\Gamma }^{\times }\left(r\right)\right\},\hfill \end{array}$$

(3)

where ${\Gamma }^{\times }\left(r\right)$ is the group of units of $\Gamma \left(r\right)$. Every element of $R_0$ admits a unique p-adic expansion

$$a=\sum _{i=0}^{n-1}{p}^i{a}_i,a_i\in \Gamma \left(r\right),$$

which reflects the isomorphism $R_0\cong {\mathbb{Z}}_{p^n}\otimes {\mathbb{F}}_{p^r}$.

Since R decomposes as in (2), the Jacobson radical has the form

$$J=p{R}_0\oplus x_1{R}_0\oplus \cdots \oplus x_k{R}_0.$$

This structural description will be fundamental in the classification of Frobenius local rings with property (P) developed in the next sections.

Remark 1.

An essential feature of our classification is the distinction between the cases $\mathrm{char}\left(\mathbb{F}\right)=2$ and $\mathrm{char}\left(\mathbb{F}\right)\ne 2$. This dichotomy arises from the fact that in characteristic 2, every element is a square, and alternating and symmetric bilinear forms coincide. This significantly reduces the number of non-isomorphic classes compared to the odd characteristic case, as shown in our enumeration results. This phenomenon is aligned with classical results in the theory of quadratic and bilinear forms, where the characteristic 2 case often requires separate treatment due to the degeneracy of the symmetric form.

Before proceeding to the case-by-case analysis, we briefly outline the common structural pattern underlying all classifications. Under property (P) with $J^3=0$ and $J^2$ principal, the multiplication on the radical is completely determined by a symmetric bilinear form

$$B:(J/J^2)\times (J/J^2)⟶J^2.$$

Thus, the classification of Frobenius local rings of order $p^{6r}$ reduces to the classification of congruence classes of symmetric matrices over the residue field. The subsequent case distinctions by characteristic reflect the known invariants of such bilinear forms, including rank, discriminant (for odd p), and parity phenomena (for $p=2$). The tables at the end of this section summarize the resulting canonical representatives.

3. Frobenius Local Rings of Characteristic p

In the following, we consider only Frobenius local rings R with $n=1,$ that is, of characteristic $p.$ The order of R is $p^{6r}$, implying that $k=5,$ and thus

$$R=R_0\oplus x_1{R}_0\oplus \cdots \oplus x_5{R}_0.$$

(4)

The ring R has a chain of ideals $J^3=0\subseteq J^2\subseteq J\subset R.$ Also, since R is Frobenius, $J^2$ is generated by a single element, say $x,$ over $\mathbb{F}$ as a vector space. As a result, we have the following products among the elements $x_1,x_2,x_3,x_4$ with respect to $x,$

$$x_i{x}_j={\alpha }_{ij}x,$$

(5)

where ${\alpha }_{ij}\in \mathbb{F}$ and $1\le i,j\le 4.$ The Formula (5) gives the following matrix:

$$M=\left(\begin{array}{cccc}{\alpha }_{11}& {\alpha }_{12}& {\alpha }_{13}& {\alpha }_{14}\\ {\alpha }_{12}& {\alpha }_{22}& {\alpha }_{23}& {\alpha }_{24}\\ {\alpha }_{13}& {\alpha }_{23}& {\alpha }_{33}& {\alpha }_{34}\\ {\alpha }_{14}& {\alpha }_{24}& {\alpha }_{34}& {\alpha }_{44}\end{array}\right).$$

(6)

The symmetry of M comes from the fact that R is commutative. We call M the structural matrix of $R.$

3.1. Construction

Let V and W be vector spaces over $\mathbb{F}$ with ${dim}_{\mathbb{F}}\left(V\right)=n_1$ and ${dim}_{\mathbb{F}}\left(W\right)=m_1$. Without loss of generality, we assume that $n_1\le m_1$.

A bilinear form is a mapping $B:V\times W\to \mathbb{F}$ that satisfies the following relation:

$$B\left(\sum _i{a}_i{x}_i,\sum _j{b}_j{x}_j^{\prime }\right)=\sum _i\sum _j{a}_i{b}_{j}B(x_i,x_j^{\prime }),$$

(7)

for all $a_i,b_j\in \mathbb{F}$, $x_i\in V$, and $x_j^{\prime }\in W$.

• Given the expansion formula above for bilinear forms, associativity of the multiplication in (11) follows directly. Consider arbitrary elements of R written in the form

$$X=\left(a_0+\sum _i{a}_i{x}_i+a_{x}x\right),Y=\left(b_0+\sum _j{b}_j{x}_j+b_{x}x\right),Z=\left(c_0+\sum _k{c}_k{x}_k+c_{x}x\right),$$

where all coefficients lie in $\mathbb{F}$. When forming the products $\left(XY\right)Z$ and $X\left(YZ\right)$, the scalar parts combine inside $\mathbb{F}$, whose multiplication is associative. The only additional contributions come from bilinear terms of the form $B\left({\sum }_i{a}_i{x}_i,{\sum }_j{b}_j{x}_j\right)$, and by bilinearity,

$$B\left(\sum _i{a}_i{x}_i,\sum _j{b}_j{x}_j\right)=\sum _{i,j}{a}_i{b}_{j}B(x_i,x_j).$$

Thus, both ways of expanding $\left(XY\right)Z$ and $X\left(YZ\right)$ yield the same $\mathbb{F}$–linear combination of basis elements. Since each $B(x_i,x_j)$ lies in $V=J^2$ and hence any triple product involving these elements lands in $J^3=0$, all higher-order terms vanish. Consequently, the expressions for $\left(XY\right)Z$ and $X\left(YZ\right)$ agree componentwise, and the multiplication defined in (11) is associative.

Consider the sets $\{x_1,x_2,\dots ,x_{n_1}\}$ and $\{x_1^{\prime },x_2^{\prime },\dots ,x_{m_1}^{\prime }\}$, which serve as bases for the vector spaces V and W over the field $\mathbb{F}$, respectively. In relation to these bases, the bilinear form B is uniquely determined by the matrix of dimensions $n_1\times m_1$ represented as

$$B={\left(b_{ij}\right)}_{1\le i\le n_1,1\le j\le m_1},\mathrm{where}{b}_{ij}=B(x_i,x_j^{\prime }).$$

(8)

The left radical of the bilinear form B is characterized as the collection of all $x\in V$ for which $B(x,y)=0$ holds for every $y\in W$. The rank of B is defined as

$$\mathrm{rank}\left(B\right)={dim}_{\mathbb{F}}\left(V\right)-{dim}_{\mathbb{F}}\left(\mathrm{rad}\left(B\right)\right).$$

It is important to observe that the rank of B is exactly equal to the rank of its corresponding matrix (8).

A symmetric bilinear form on V is defined as a bilinear form $B:V\times V\to \mathbb{F}$ that satisfies the symmetry property:

$$B(x,y)=B(y,x)\mathrm{for}\mathrm{every}x,y\in V.$$

By setting $x_i=x_i^{\prime }$ for $i=1,2,\dots ,e$, we derive from (8) that the matrix corresponding to a symmetric bilinear form is symmetric. Therefore, once a basis for V over $\mathbb{F}$ is established, a one-to-one correspondence can be observed between symmetric bilinear forms on V and $e\times e$ symmetric matrices over $\mathbb{F}$. The collection of symmetric bilinear forms on V is denoted by $S_e$. For future reference, we observe that

$$|S_e|={\left(p^r\right)}^{{\displaystyle \frac{e(e+1)}{2}}}.$$

(9)

The aim of this section is to demonstrate the correspondence between Frobenius local rings and symmetric bilinear forms on V with ${dim}_{\mathbb{F}}\left(V\right)=4$ through structural matrices $M.$

Let U and V be 4-dimensional and 1-dimensional $\mathbb{F}$-vector spaces, respectively. Furthermore, consider the additive sum

$$R=\mathbb{F}\oplus U\oplus V.$$

(10)

Let $\{x_1,x_2,x_3,x_4\}$ and $\left\{x\right\}$ be bases for U and $V,$ respectively. Insert a multiplication into R as

$$\begin{array}{c}\hfill (a_0+\sum _{i=1}^4{a}_i{x}_i+a_{x}x)(b_0+\sum _{i=1}^4{b}_i{x}_i+b_{x}x)=\hfill \\ \hfill a_0{b}_0+\sum _{i=0}^4[a_0{b}_i+b_0{a}_i]x_i+\sum _{i,j}{a}_i{b}_j{\alpha }_{ij}x+[a_0{b}_x+a_x{b}_0]x.\hfill \end{array}$$

(11)

Remark 2.

In the expression above, all coefficients $a_0,b_0,a_i,b_i$ lie in the finite field $\mathbb{F}$ and are interpreted inside R through their Teichmüller lifts. Since the Teichmüller map is multiplicative, all scalar coefficients occurring in the products $\left((a_0+{\sum }_{i=1}^4{a}_i{x}_i+a_{x}x)(b_0+{\sum }_{i=1}^4{b}_i{x}_i+b_{x}x)\right)(c_0+{\sum }_{i=1}^4{c}_i{x}_i+c_{x}x)$ and $(a_0+{\sum }_{i=1}^4{a}_i{x}_i+a_{x}x)\left((b_0+{\sum }_{i=1}^4{b}_i{x}_i+b_{x}x)(c_0+{\sum }_{i=1}^4{c}_i{x}_i+c_{x}x)\right)$ coincide in $\mathbb{F}$ before reduction into U or V. Moreover, the bilinear part

$$B(x_i,x_j)={\alpha }_{ij}x\in V=J^2$$

takes values in $J^2$, and because $J^3=0$, any term of degree three or higher in elements of J vanishes. Hence, every expression of the triple multiplication reduces to the same $\mathbb{F}$-linear combination of basis elements modulo $J^3=0$. This shows that the multiplication in Equation (11) is associative.

Theorem 1.

The additive group R in Equation (10) is a ring with multiplication given in Equation (11). Moreover, every ring with property (P) is of this form.

Proof. 

The product in (11) is clearly well-defined, and it is easy to see that it also satisfies the ring axioms. Since the sum is direct, we have

$$\left|R\right|=\left|\mathbb{F}\right|·\left|U\right|·\left|V\right|=p^r·p^{4r}·p^r=p^{6r}.$$

We now show that R is completely primary and satisfies property (P).

With the obvious identifications, we can think of $\mathbb{F}$, U, and V as subsets of $R.$ Put $J=U+V$. It follows immediately from the way multiplication was defined that $J^2=V$ and that $JV=VJ=0$. Hence, $J^3=0.$ Also, from the definition of multiplication, it follows that $RJ=JR\subseteq J$, so that J is an ideal.

Let now $a\in \mathbb{F}$ and $x\in J$. Since $x^m=0$ for some $m>0$, we have that $1+x$ is invertible for every $x\in J.$ Then, $a+x=a(1+a^{-1}x)$ is the product of two invertible elements, and hence it is invertible.

Because $\left|J\right|=p^{r(4+1)}$ and $|{\Gamma }^{\times }\left(r\right)+J|=(p^r-1)p^{5r}$, it follows that ${\Gamma }^{\times }\left(r\right)+J=R-J$, and hence all the elements outside J are invertible. Hence, ${\displaystyle \frac{R}{J}}\cong \mathbb{F}$; therefore, R is a local ring and satisfies property (P).

To demonstrate the converse, it is adequate to observe that the preceding arguments in the Construction confirm that all rings of characteristic p exhibiting property (P) resemble those presented in the Construction. □

3.2. Classification

Let R be a local ring of order $p^{6r}$ with characteristic p given in the Construction, and let M be its structural matrix given in (6). Now, assume T is another local ring with the same specifications and its structural matrix is $M^{\prime }=\left({\beta }_{ij}\right).$ Therefore, our objective is to identify the choices of M that result in rings that are not isomorphic to each other. The proposition below will illustrate the point. We denote the transpose of a matrix A by $A^t.$

Proposition 1.

Given the above notation, $R\cong T$ if and only if there is an invertible matrix D of size $4\times 4$ over $\mathbb{F}$ and $\mu \in \mathrm{Aut}\left(\mathbb{F}\right)$ such that

$$M^{\prime }=\beta D^t\mu \left(M\right)D,$$

(12)

for some $\beta \in {\Gamma }^{\times }\left(r\right).$

Proof. 

Since $J^2$ is principal and R is Frobenius, multiplication on $J/J^2$ is determined by the products of basis elements modulo $J^3$. Associativity and commutativity imply that these products define a symmetric bilinear form with values in $J^2$. Thus, we want to prove that two such rings are isomorphic if and only if the corresponding bilinear forms are congruent under a change in basis of $J/J^2$, which completes the reduction. Assume $R\cong T$ and let $\varphi$ be the isomorphism. As $\mathbb{F}$ is the coefficient subfield, so is $\varphi \left(\mathbb{F}\right)$ in T, and hence there is $u\in U\left(T\right)$ such that $u\varphi \left(\mathbb{F}\right)u^{-1}=\mathbb{F}$. Consider the map

$$\begin{array}{cc}\hfill \sigma :& R\to T\hfill \\ & a⟼u\varphi \left(a\right)u^{-1}.\hfill \end{array}$$

This is an isomorphism between R and T sending $\mathbb{F}$ to itself. Observe that

$$\begin{array}{cc}& \sigma \left(x_i\right)=\sum _{c=1}^4{\beta }_{ic}^{\prime }{y}_c,\hfill \\ & \sigma \left(x_i{x}_j\right)=\sigma \left(x_i\right)\sigma \left(x_j\right),\hfill \\ & \sigma \left(x_i{x}_j\right)=\sigma \left({\alpha }_{ij}x\right)=\beta \sigma \left({\alpha }_{ij}\right)y,\hfill \\ & \sigma \left(x_i\right)\sigma \left(x_j\right)=\left(\sum _{c=1}^4{\beta }_{ic}^{\prime }{y}_c\right)\left(\sum _{h=1}^4{\beta }_{jh}^{\prime }{y}_h\right).\hfill \end{array}$$

Thus,

$$\beta \sigma \left({\alpha }_{ij}\right)y=\sum _{h,c}{\beta }_{ic}^{\prime }{\beta }_{jh}^{\prime }{\beta }_{ch}y.$$

(13)

From the last equation, we get

$$\beta \sigma \left(M\right)=A{M}^{\prime }{A}^t,$$

where ${\mu =\sigma |}_{\mathbb{F}}$ and $A=\left({\beta }_{jc}^{\prime }\right).$ If we let $D=A^{-1}$, the result follows. For the converse, assume that we have $D=\left({\beta }_{ch}^{\prime }\right)$ and $\mu$; then, one can check the following correspondence:

$$\begin{array}{cc}\hfill \sigma :& R\to T\hfill \\ & x_i⟼\sum _j{\beta }_{ji}^{\prime }{y}_j,\mathrm{and}x⟼\beta y\hfill \end{array}$$

is an isomorphism. □

The last proposition builds a bridge between the classification of local rings of order $p^{6r}$ and symmetric bilinear forms in $\mathrm{GL}(4,\mathbb{F})$. In other words, the classification of bilinear forms B is enough. From now on, B is a bilinear form defined on U as a 4-dimensional $\mathbb{F}$-vector space. The associated matrix will be denoted by M as in (6). We say that M is congruent to $M^{\prime }$ if there is a unique invertible matrix $D\in \mathrm{GL}(4,\mathbb{F})$ such that

$$M^{\prime }=\beta D^{t}MD,$$

for some $\beta \in {\Gamma }^{\times }\left(r\right).$ Denote

$$\begin{array}{cccc}\hfill A& =\left(\begin{array}{cccc}{\alpha }_{11}& 0& 0& 0\\ 0& {\alpha }_{22}& 0& 0\\ 0& 0& {\alpha }_{33}& 0\\ 0& 0& 0& {\alpha }_{44}\end{array}\right),B^{\prime }\hfill & \hfill =\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),C& =\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right).\hfill \end{array}$$

(14)

Lemma 1.

Let $M\in S_4$ be the associated matrix of the bilinear form $B.$ Then, M is congruent to $A+\nu B^{\prime }+{\nu }_{1}C,$ where ν and ${\nu }_1$ may be $0.$

Proof. 

We may assume that the first row of M is nonzero. There are two cases, leading to two different reduction processes.

(I)

Suppose that some diagonal element of M is not 0. Then, by a suitable row permutation followed by a corresponding column permutation, this diagonal element may be brought into the $M_{11}$ position. Thus, we may assume that ${\alpha }_{11}\ne 0$. Now, subtract ${\displaystyle \frac{{\alpha }_{i1}}{{\alpha }_{11}}}$ times row 1 from row i and then subtract ${\displaystyle \frac{{\alpha }_{1j}}{{\alpha }_{11}}}$ times column 1 from column j, for $2\le j\le 4$. This produces a matrix congruent to M, which is the direct sum of $\left({\alpha }_{11}\right)$ with a $3\times 3$ matrix.

(II)

Suppose that every diagonal element of M is 0. Then, ${\alpha }_{ij}\ne 0$ for some j such that $2\le j\le 4$. Interchanging rows 2 and j and columns 2 and j brings this element into the $M_{12}$ position and does not change the fact that all the diagonal elements are 0. We may assume then that M has the form

$$\left(\begin{array}{cccc}0& {\alpha }_{12}& 0& 0\\ {\alpha }_{21}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$$

where ${\alpha }_{12}$ is a unit. Construct the matrix

$$D=\left(\begin{array}{cccc}0& {\alpha }_{12}^{-1}& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).$$

Note that

$$DM{D}^t=\left(\begin{array}{cccc}0& 1& {\alpha }_{13}^{\prime }& {\alpha }_{14}^{\prime }\\ 1& 0& {\alpha }_{23}^{\prime }& {\alpha }_{24}^{\prime }\\ {\alpha }_{31}^{\prime }& {\alpha }_{32}^{\prime }& {\alpha }_{33}^{\prime }& {\alpha }_{34}^{\prime }\\ {\alpha }_{41}^{\prime }& {\alpha }_{42}^{\prime }& {\alpha }_{43}^{\prime }& {\alpha }_{44}^{\prime }\end{array}\right).$$

Now, subtract ${\alpha }_{1j}^{\prime }$ times row 2 from row j and then ${\alpha }_{1j}^{\prime }$ times column 2 from column j, for $3\le j\le 4$, to produce the congruent matrix

$$\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& {\alpha }_{23}^{″}& {\alpha }_{24}^{″}\\ {\alpha }_{31}^{″}& {\alpha }_{32}^{″}& {\alpha }_{33}^{″}& {\alpha }_{34}^{″}\\ {\alpha }_{41}^{″}& {\alpha }_{42}^{″}& {\alpha }_{43}^{″}& {\alpha }_{44}^{″}\end{array}\right).$$

Finally, subtract ${\alpha }_{2j}^{″}$ times row 1 from row j and then ${\alpha }_{2j}^{″}$ times column 1 from column j, for $3\le j\le 4$. This produces a matrix congruent to M which is the direct sum of $B^{\prime }$ and a $2\times 2$ matrix. Repetition of these processes applied to the submatrices obtained by deleting the first row and column, or the first two rows and two columns, ultimately produces the desired form, up to order. Since the summands can be rearranged at will by simultaneous row and column permutations, the proof is complete. □

Before we go further, we mention some special cases which will be useful later. The block $2\times 2$ matrix

$$\left(\begin{array}{cc}\beta & 0\\ 0& \beta \end{array}\right)\mathrm{is}\mathrm{congruent}\mathrm{to}{I}_2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right).$$

(15)

This follows by using

$$D=\left(\begin{array}{cc}{\gamma }_1& -{\gamma }_2\\ {\gamma }_2& {\gamma }_1\end{array}\right),$$

where $\beta ={\gamma }_1^2+{\gamma }_2^2.$ In addition, when $\mathrm{char}\left(\mathbb{F}\right)\ne 2,$ the matrix

$$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\mathrm{is}\mathrm{congruent}\mathrm{to}\left(\begin{array}{cc}{2}^{-1}& 0\\ 0& 2^{-1}\end{array}\right).$$

(16)

In fact, (16) is true if and only if $\mathrm{char}\left(\mathbb{F}\right)\ne 2.$ Indeed, if $\mathrm{char}\left(\mathbb{F}\right)=2$ and $N=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ is congruent to a diagonal matrix $E,$ then there would exist an invertible matrix D such that $N=D^{t}ED.$ As every element of $\mathbb{F}$ is a square, that is, ${\mathbb{F}}^{\times }={\Gamma }_2,$ the equation ${\gamma }_1^2+{\gamma }_2^2=0$ implies ${\gamma }_1={\gamma }_2=0,$ and hence at least one row of D is zero. This contradicts the fact that $D\in \mathrm{GL}(2,\mathbb{F}).$

Remark 3.

Throughout this article, the parameter α is fixed as a generator of the cyclic group ${\mathbb{F}}^{\times }.$ Hence, the same α appears in all structural descriptions and normal forms. No assumption on α being a square or a nonsquare is required; only the condition $\alpha \in {\mathbb{F}}^{\times }$ is needed.

Theorem 2.

Suppose M is in $S_4$, and suppose also that $A_0$ is induced from A by taking ${\alpha }_{ii}\in \{\alpha ,1,0\}$ such that α appears only once on the diagonal.

(i) 

If $\mathrm{char}\left(\mathbb{F}\right)\ne 2,$ then M is congruent to $A_0$ with rank between 1 and 4

(ii) 

If $\mathrm{char}\left(\mathbb{F}\right)=2,$ then M is congruent to $A_0,$ $B^{\prime },$ or $C.$

Proof. 

(i)

We first suppose that $\mathrm{char}\left(\mathbb{F}\right)\ne 2.$ In light of Lemma 1, $M\cong A+\nu B^{\prime }+{\nu }_{1}C.$ Also, by (16), we must have $\nu ={\nu }_1=0.$ This means M is congruent to $A.$ Let the rank of A be $4.$ Write the elements ${\alpha }_{ii}$ as

$${\alpha }_{ii}={\alpha }^{e_i}{\alpha }_i^2\mathrm{where}1\le i\le 4,$$

where $e_i=0$ or 1 and ${\alpha }_i\in {\Gamma }^{\times }\left(r\right).$ Note that if $e_i=0,$ then ${\alpha }_{ii}\in {\Gamma }_2,$ and ${\alpha }_{ii}\in {\Gamma }_1$ otherwise. Define D to be a matrix of the form

$$\left(\begin{array}{cccc}{\alpha }_1^{-1}& 0& 0& 0\\ 0& {\alpha }_2^{-1}& 0& 0\\ 0& 0& {\alpha }_3^{-1}& 0\\ 0& 0& 0& {\alpha }_4^{-1}\end{array}\right)=D({\alpha }_1^{-1},{\alpha }_2^{-1},{\alpha }_3^{-1},{\alpha }_4^{-1}).$$

Thus, the matrix $D^{t}AD$ has the form $D({\alpha }^{e_1},{\alpha }^{e_2},{\alpha }^{e_3},{\alpha }^{e_4}).$ Depending on the values of $e_i,$ we get

$$A\cong \alpha I_{r_0}+I_{4-r_0},\mathrm{for}{r}_0\le 4.$$

Moreover, $\alpha I_{r_0}$ is congruent to either $I_{r_0}$, by (15), or $\alpha I_1+I_{r_0-1}$ when $r_0$ is even or odd, respectively. In any case, $I_{r_0}$ and $\alpha I_1+I_{r_0-1}$ are two non-congruent forms of $A_0.$ In conclusion, we have

$$M\cong A\cong A_0.$$

(ii)

When $\mathrm{char}\left(\mathbb{F}\right)=2,$ every element of $\mathbb{F}$ is a square. We encounter two cases. The first one is when M has at least one ${\alpha }_{ii}$ not equal to $0.$ Based on a similar technique as in (i), we must have $M\cong A_0.$ The second situation occurs when ${\alpha }_{ii}=0.$ Now, if the rank $r_0$ of M is odd, we have the following. When $r_0=1,$ then M is obviously congruent to a copy of $A_0,$ say $A_0^{\prime },$ where ${\alpha }_{11}=1$ and the rest are $0.$ In case $r_0=3,$ we obtain $M\cong A_0^{\prime }+B^{\prime }.$ The latter matrix is congruent to $A_0,$ where all ${\alpha }_{ii}=1$ except when $i=4$, ${\alpha }_{44}=0$, using the matrix

$$\left(\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 1& 0& 1\end{array}\right).$$

Finally, when M has even rank, M is congruent to $\nu B^{\prime }+{\nu }_{1}C,$ where $\nu$ and ${\nu }_1$ take the values 0 or 1 by Lemma 1. Assume $r_0=2,$ then based on the same lemma, $\nu =1$ and ${\nu }_1=0$, which means that $M\cong B^{\prime }.$ While if $r_0=4,$ then we must have $\nu =0$ and ${\nu }_1=1,$ and thus M is congruent to $C.$ The matrices $A_0,$ $B^{\prime },$ and C are not congruent. Since any matrix congruent to $B^{\prime }$ or C has diagonal elements equal to zero, the proof is complete. □

As the main result of this section, we construct a correspondence between rings which have property (P) and non-zero symmetric bilinear forms $S_4^{\times }.$ Furthermore, we have found the congruence classes of such bilinear forms. Thus, we introduce the following theorem.

Theorem 3.

Suppose that R is a ring with property (P) and structural matrix $M.$ Then, M is one and only one of the following:

(i) 

$\mathrm{char}\left(\mathbb{F}\right)\ne 2,$

$$\begin{array}{cc}\hfill & \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\hfill \\ \hfill & \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).\hfill \end{array}$$

(17)

(ii) 

$\mathrm{char}\left(\mathbb{F}\right)=2,$

$$\begin{gathered} \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right), \\ \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right). \end{gathered}$$

(18)

Corollary 1.

The number of Frobenius local rings of order $p^{6r}$ with characteristic p is $7$ classes if $p\ne 2$ and $6$ classes when $p=2.$

In general, one can deduce the following.

Corollary 2.

If $m\ge 3,$ then the number $N_F$ of non-isomorphic classes of such rings of order $p^{mr}$ is

$$N_F(p,n=1,m)=\left\{\begin{array}{cc}2m-5,\hfill & \mathit{if}p\ne 2,\hfill \\ \left\{\begin{array}{cc}{\displaystyle \frac{3m-6}{2}},\hfill & \mathit{if}m\mathit{is}\mathit{even},\hfill \\ m-2,\hfill & \mathit{if}m\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill & \mathit{if}p=2.\hfill \end{array}\right.$$

(19)

Remark 4.

If $N_F$ is considered as a function of the invariants $p,n,r,m,k,$ then $N_F$ is independent of the value of r as shown in the last corollary. Note that here $n=1$ and $k=m-1.$

4. Frobenius Local Rings of Characteristic p²

In this section, we classify Frobenius local rings with property (P) and characteristic $p^2$. Recall that property (P) means

$$J^3=0,J^2\ne 0,J^2\mathrm{is}\mathrm{principal}.$$

Since $\mathrm{char}\left(R\right)=p^2$, the coefficient subring is the Galois ring

$$R_0=\mathrm{GR}(p^2,r),\mathbb{F}\cong R_0/p{R}_0\cong {\mathbb{F}}_{p^r}.$$

Because $p\ne 0$ in $R_0$, the position of p inside the radical filtration $0\subset J^2\subset J\subset R$ splits the analysis into two disjoint cases:

$$p\in J^2\mathrm{or}p\in J\setminus J^2.$$

4.1. Case $p\in J^2$

Here, p lies entirely in $J^2$. Modulo p, products of radical generators land in $J^2/p{J}^2=0$, and the multiplicative structure is controlled by a symmetric bilinear form on $J/J^2$ with values in $pR$; equivalently, by a symmetric matrix over $\mathbb{F}$ that appears multiplied by p in R. This mirrors the $\mathrm{char}\left(R\right)=p$ case (Section 3), except that the image of $J^2$ is now scaled by p.

4.1.1. Construction and Verification

Let U be a free $R_0$–module with basis $x_1,x_2,x_3,x_4$ and set

$$R=R_0\oplus U.$$

(20)

Fix scalars ${\alpha }_{ij}\in \mathbb{F},$ and define a multiplication on R by

$$\begin{array}{ccc}\hfill (a_0+{\sum }_i{a}_i{x}_i)(b_0+{\sum }_j{b}_j{x}_j)& =& a_0{b}_0+\sum _i(a_0{b}_i+b_0{a}_i)x_i+\sum _{i,j}{a}_i{b}_j{\alpha }_{ij}p,\hfill \end{array}$$

(21)

where $a_0,b_0\in R_0$ and $a_i,b_j\in R_0$, and we identify ${\alpha }_{ij}\in \mathbb{F}$ with its Teichmüller lift in $R_0$ (so ${\alpha }_{ij}p\in p{R}_0\subset R$). The right-hand side of (21) lies in $R_0\oplus U$ because $a_0{b}_0,{\alpha }_{ij}p\in R_0$ and ${\sum }_i(a_0{b}_i+b_0{a}_i)x_i\in U$. Commutativity follows from ${\alpha }_{ij}={\alpha }_{ji}$ and commutativity of $R_0$. Bilinearity in each factor follows from bilinearity in all coordinates.

Radical, nilpotency, locality. Let $J=U+p{R}_0$. Then, J is an ideal:

$$R·J\subset U+p{R}_0,J·R\subset U+p{R}_0.$$

Moreover,

$$J^2={(U+p{R}_0)}^2=U^2+p{R}_{0}U+{\left(p{R}_0\right)}^2\subset p{R}_0+pU+p^2{R}_0=p{R}_0,$$

since $U^2\subset p{R}_0$ by definition of the multiplication, and $pU\subset p{R}_0$ because U is $R_0$-free. Hence, $J^2\subset p{R}_0$ and ${\left(J^2\right)}^2\subset p^2{R}_0=0$, so $J^3=0$. Because $R/J\cong R_0/\left(p{R}_0\right)\cong \mathbb{F}$ is a field, J is maximal and R is local.

For Frobenius property and socle, since $J^2=p{R}_0$ and $p{R}_0\cong R_0/p{R}_0$ as R-modules (the action of U on $p{R}_0$ is trivial and the action of $R_0$ factors through $R_0\to R_0/p{R}_0$), the socle $Soc\left(R\right)={Ann}_R\left(J\right)=J^2$ is cyclic. A finite local ring with cyclic socle is Frobenius. Hence, R is Frobenius with $Soc\left(R\right)=J^2$ principal.

Conversely, if R is Frobenius local with $J^3=0$, $J^2$ principal and $\mathrm{char}\left(R\right)=p^2$, then $R_0=GR(p^2,r)\subset R$ and $J/J^2$ is a 4-dimensional $\mathbb{F}$-space (because $\left|R\right|=p^{6r}$ and $|J^2|=p^r$). Choosing an $R_0$-basis $x_1,\dots ,x_4$ of a lift U of $J/J^2$ and setting $R=R_0\oplus U$, the multiplication must have the form (21) for some symmetric ${\alpha }_{ij}\in \mathbb{F}$. Hence, (21) gives a necessary and sufficient model.

Thus, the multiplication in (21) turns R into a ring. The construction of R given in (20) yields a ring with property (P) and characteristic $p^2$. The converse also holds: every Frobenius local ring with property (P) and characteristic $p^2$ can be obtained by this construction.

4.1.2. Classification up to Isomorphism

Define the structural matrix $M=\left({\alpha }_{ij}\right)\in S_4\left(\mathbb{F}\right)$ attached to R by

$$x_i{x}_j={\alpha }_{ij}p(1\le i,j\le 4).$$

Let T be another ring constructed as above with structural matrix $M^{\prime }=\left({\alpha }_{ij}^{\prime }\right)$ in the basis $y_1,\dots ,y_4$.

Proposition 2.

$R\simeq T$ if and only if there exist $D\in G{L}_4\left(\mathbb{F}\right),$ $\sigma \in \mathrm{Aut}\left(\mathbb{F}\right)$ and $\beta \in {\Gamma }^{\times }\left(r\right)$ such that

$$M^{\prime }=\beta D^t\sigma \left(M\right)D.$$

Proof. 

We recall that R and T share the same residue field $\mathbb{F}\simeq R/J\simeq T/J_T$, and that $J^3=0$ and $J^2$ is principal. Fix a generator p of $J^2$ (resp. p of $J_T^2$) and choose $\mathbb{F}$-bases $\{{\overline{x}}_1,\dots ,{\overline{x}}_4\}$ of $J/J^2$ and $\{{\overline{y}}_1,\dots ,{\overline{y}}_4\}$ of $J_T/J_T^2$. Based on the definition of the structural matrices $M=\left({\alpha }_{ij}\right)$ and $M^{\prime }=\left({\alpha }_{ij}^{\prime }\right)$, we have

$$x_i{x}_j={\alpha }_{ij}p,y_i{y}_j={\alpha }_{ij}^{\prime }p,$$

with ${\alpha }_{ij},{\alpha }_{ij}^{\prime }\in \mathbb{F}$, and symmetry follows from commutativity.

• (⇒) Assume $\varphi :R\to T$ is a ring isomorphism. Then, $\varphi \left(J\right)=J_T$ and $\varphi \left(J^2\right)=J_T^2$, so $\varphi$ induces an isomorphism of residue fields

$$\overline{\varphi }:R/J⟶T/J_T,$$

hence an automorphism $\sigma \in Aut\left(\mathbb{F}\right)$. Let $R_0\subset R$ and $T_0\subset T$ be fixed coefficient fields (Teichmüller systems/coefficient subrings) mapping isomorphically onto $\mathbb{F}$. Based on standard structure theory for finite local rings, any two coefficient fields in T are conjugate by a unit of T; therefore, composing $\varphi$ with an inner automorphism of T (which does not change the induced action on $J/J^2$ up to a basis change already accounted for below), we may assume that

$${\varphi |}_{R_0}=\sigma :R_0\stackrel{\sim }{⟶}{T}_0$$

lifts the given field automorphism $\sigma \in Aut\left(\mathbb{F}\right)$.

Next, since $J_T^2$ is principal generated by p and $\varphi \left(J^2\right)=J_T^2$, we must have

$$\varphi \left(p\right)=\beta p$$

for some unit $\beta \in T^{\times }$. Moreover, reducing modulo $J_T$ shows that $\beta$ is determined up to a unit coming from the coefficient field; equivalently, $\beta$ may be taken in the Teichmüller unit group, and we record this as $\beta \in {\Gamma }^{\times }\left(r\right)$.

Finally, $\varphi$ induces an $\mathbb{F}$-semilinear isomorphism on the radical layer:

$$\overline{\varphi }:J/J^2⟶J_T/J_T^2,\overline{\varphi }\left(a\overline{x}\right)=\sigma \left(a\right)\overline{\varphi }\left(\overline{x}\right)(a\in \mathbb{F}),$$

because $\varphi$ acts as $\sigma$ on $\mathbb{F}=R_0/(R_0\cap J)$. Choosing the fixed bases, we can write

$$\overline{\varphi }\left({\overline{x}}_i\right)=\sum _{c=1}^4{d}_{ci}{\overline{y}}_c,D=\left(d_{ci}\right)\in G{L}_4\left(\mathbb{F}\right),$$

and lifting to J we may choose representatives so that

$$\varphi \left(x_i\right)\equiv \sum _{c=1}^4{d}_{ci}{y}_c(mod{J}_T^2).$$

Multiplying and using $J_T^3=0$ (so terms involving $J_T^2·J_T$ vanish), we get

$$\varphi \left(x_i\right)\varphi \left(x_j\right)\equiv \left(\sum _c{d}_{ci}{y}_c\right)\left(\sum _h{d}_{hj}{y}_h\right)=\sum _{c,h}{d}_{ci}{d}_{hj}{y}_c{y}_h=\sum _{c,h}{d}_{ci}{d}_{hj}{\alpha }_{ch}^{\prime }p.$$

On the other hand,

$$\varphi \left(x_i{x}_j\right)=\varphi \left({\alpha }_{ij}p\right)=\varphi \left({\alpha }_{ij}\right)\varphi \left(p\right)=\sigma \left({\alpha }_{ij}\right)\beta p,$$

since ${\alpha }_{ij}\in \mathbb{F}$ and ${\varphi |}_{R_0}=\sigma$. Comparing p-coefficients yields

$$\sigma \left({\alpha }_{ij}\right)\beta =\sum _{c,h}{d}_{ci}{d}_{hj}{\alpha }_{ch}^{\prime }(1\le i,j\le 4).$$

In matrix form this is exactly

$$\beta \sigma \left(M\right)=D^t{M}^{\prime }D,$$

equivalently $M^{\prime }=\beta D^t\sigma \left(M\right)D$.

• (⇐) Conversely, assume there exist $\beta \in {\Gamma }^{\times }\left(r\right)$, $\sigma \in Aut\left(\mathbb{F}\right)$ and $D\in G{L}_4\left(\mathbb{F}\right)$ such that $M^{\prime }=\beta D^t\sigma \left(M\right)D$. Define $\varphi :R\to T$ on generators by

$${\varphi |}_{R_0}=\sigma ,\varphi \left(p\right)=\beta p,\varphi \left(x_i\right)=\sum _{c=1}^4{d}_{ci}{y}_c(1\le i\le 4),$$

and extend additively to $R=R_0\oplus J$ (as an $R_0$-module). It remains to check multiplicativity. For $a\in R_0$ and $x\in J$, we have $\varphi \left(ax\right)=\sigma \left(a\right)\varphi \left(x\right)$ by construction, so it suffices to verify products among the $x_i$’s. Using the defining relations in T,

$$\varphi \left(x_i\right)\varphi \left(x_j\right)=\sum _{c,h}{d}_{ci}{d}_{hj}{y}_c{y}_h=\sum _{c,h}{d}_{ci}{d}_{hj}{\alpha }_{ch}^{\prime }p.$$

By the assumed congruence relation, the coefficient of p equals $\sigma \left({\alpha }_{ij}\right)\beta$; hence,

$$\varphi \left(x_i\right)\varphi \left(x_j\right)=\sigma \left({\alpha }_{ij}\right)\beta p=\varphi \left({\alpha }_{ij}\right)\varphi \left(p\right)=\varphi \left({\alpha }_{ij}p\right)=\varphi \left(x_i{x}_j\right).$$

Therefore, $\varphi$ respects all defining products and is a ring homomorphism. Since $\sigma$ is bijective and $D\in G{L}_4\left(\mathbb{F}\right)$, $\varphi$ induces bijections on $R/J\simeq \mathbb{F}$ and on $J/J^2$; hence, $\varphi$ is bijective, i.e., a ring isomorphism. □

By the standard reduction of symmetric bilinear forms over finite fields, one obtains the following list of congruence classes (orbits) in $S_4\left(\mathbb{F}\right)$ modulo $D^t(-)D$, scalar multiplication by ${\Gamma }^{\times }\left(r\right)$, and field automorphisms. The outcome differs in characteristic 2 because symmetric and alternating forms coincide.

Theorem 4.

There are 8 and 6 non-isomorphic rings constructed in (20) when $\mathrm{char}\left(\mathbb{F}\right)\ne 2$ and $\mathrm{char}\left(\mathbb{F}\right)=2,$ respectively.

Proof. 

In the situation of (20), we have $J^3=0$, and $J^2=\langle p\rangle$ is a 1–dimensional $\mathbb{F}$-space. Moreover, the products of basis elements in $J/J^2$ land in $J^2$; hence, the multiplication on $J/J^2$ is completely encoded by

$$x_i{x}_j={\alpha }_{ij}p(1\le i,j\le 4),$$

(22)

with ${\alpha }_{ij}\in \mathbb{F}$. Since the ring is commutative, the matrix

$$M=\left({\alpha }_{ij}\right)\in S_4\left(\mathbb{F}\right)$$

is symmetric. Conversely, once M is fixed, the relations (22) determine the multiplication on $R_0\oplus J$ (and associativity has already been ensured by the construction in (20)).

Let T be another ring of the same type with structural matrix $M^{\prime }=\left({\alpha }_{ij}^{\prime }\right)$. Based on Proposition 2, the rings R and T are isomorphic if and only if there exist $\sigma \in Aut\left(\mathbb{F}\right)$, $D\in G{L}_4\left(\mathbb{F}\right)$ and $\beta \in {\Gamma }^{\times }\left(r\right)$ such that

$$M^{\prime }=\beta D^t\sigma \left(M\right)D.$$

(23)

Thus, the isomorphism types of rings arising from (20) are in bijection with the orbits of symmetric $4\times 4$ matrices under the action generated by: (i) congruence $M\mapsto D^{t}MD$, (ii) field automorphisms $M\mapsto \sigma \left(M\right)$, and (iii) scalar multiplication by $\beta \in {\Gamma }^{\times }\left(r\right)$.

Therefore, the classification reduces to the congruence classification of symmetric bilinear forms on a 4-dimensional $\mathbb{F}$-vector space, which is exactly the classification developed in Section 3. In particular:

• If $\mathrm{char}\left(\mathbb{F}\right)\ne 2$, symmetric forms are classified by rank together with a discriminant type (square vs. nonsquare) for each even rank, yielding the 7 standard congruence types in dimension 4 listed in Section 3. In our construction, there is in addition the orbit represented by the rank-1 diagonal matrix

  $$D(\alpha ,0,0,0),\alpha \in {\Gamma }_1,$$

  which does not merge with the other rank-1 cases under the restricted scaling in (23). Hence, we obtain $7+1=8$ non-isomorphic rings.
• If $\mathrm{char}\left(\mathbb{F}\right)=2$, there is no square/nonsquare discriminant distinction, and the congruence types in dimension 4 reduce accordingly (as summarized in Section 3), giving 5 standard types. As above, the additional class represented by $D(\alpha ,0,0,0)$ contributes one more orbit, so the total number is $5+1=6$.
  This proves the claimed numbers of isomorphism classes. □

Corollary 3.

For $m\ge 3,$ the number of isomorphism classes satisfies

$$N_F(p,n=2,m)=\left\{\begin{array}{cc}{N}_F(p,1,m)+1,\hfill & p\ne 2,\hfill \\ N_F(p,1,m),\hfill & p=2.\hfill \end{array}\right.$$

Remark 5.

The extra class when $p\ne 2$ corresponds to a rank 1 diagonal form with nonsquare discriminant, which survives after modding out by ${\Gamma }^{\times }\left(r\right)$; in characteristic 2, every element is a square, so this split disappears and the count drops to 6.

4.2. Case $p\in J\setminus J^2$

In the remaining characteristic $p^2$ case with $p\in J\setminus J^2$, the first step is to determine the $R_0$–module decomposition of R and to control the effect of multiplication by p on $J/J^2$. The Frobenius hypothesis (together with property (P)) forces $J^2$ to be 1-dimensional over the residue field, and this severely restricts how many generators of $J/J^2$ can have a nonzero p-multiple. The next proposition makes this precise.

Proposition 3.

Let R be a Frobenius local ring of order $p^{6r}$ and characteristic $p^2$ satisfying property (P), and assume that $p\in J\setminus J^2$. Then, ${dim}_{\mathbb{F}}(J/J^2)=4$ and, after a suitable choice of generators,

$$J=\langle p,x_1,x_2,x_3\rangle \mathit{with}p{x}_1\ne 0,p{x}_2=p{x}_3=0.$$

Proof. 

Since R is Frobenius, $Soc\left(R\right)=J^2\ne 0$ and ${dim}_{\mathbb{F}}\left(J^2\right)=1$. Moreover, as $p\in J$, we always have $pJ\subseteq J^2$. If $pJ=0$, then $p\in An{n}_R\left(J\right)=Soc\left(R\right)=J^2$, contradicting the assumption $p\in J\setminus J^2$. Therefore, $pJ\ne 0$, and since $J^2$ is 1-dimensional, it follows that

$$pJ=J^2.$$

Because $\left|R\right|=p^{6r}$ and $|R/J|=p^r$, we have $\left|J\right|=p^{5r}$; hence,

$$|J/J^2|=p^{4r},$$

so ${dim}_{\mathbb{F}}(J/J^2)=4.$ Choose $p,x_1,x_2,x_3\in J$ whose images form an $\mathbb{F}$-basis of $J/J^2$.

Let

$$e={dim}_{\mathbb{F}}\left(\mathrm{Im}(p:J/J^2\to J^2)\right).$$

Since $pJ=J^2\ne 0$, we have $e\ge 1$, while ${dim}_{\mathbb{F}}\left(J^2\right)=1$ forces $e\le 1$. Thus, $e=1$.

If two independent generators $x_1,x_2\in J/J^2$ satisfied $p{x}_1\ne 0$ and $p{x}_2\ne 0$, then $p{x}_1$ and $p{x}_2$ would span the same 1-dimensional space $J^2$, so there exists $\beta \in {\Gamma }^{\times }\left(r\right)$ with $p(x_1-\beta x_2)=0$, while $x_1-\beta x_2\ne 0$ in $J/J^2$, contradicting $e=1$.

Hence, after replacing generators if necessary, exactly one generator has a nonzero p-multiple, and we may assume

$$p{x}_1\ne 0,p{x}_2=p{x}_3=0.$$

  □

Construction

Let $\{x_1,x_2,x_3\}$ be a generating set of the $R_0$-module U. Define

$$R=R_0\oplus U,$$

(24)

with multiplication

$$(a_0+\sum _{i=1}^3{a}_i{x}_i)·(b_0+\sum _{j=1}^3{b}_j{x}_j)=a_0{b}_0+\sum _i(a_0{b}_i+b_0{a}_i)x_i+\sum _{i,j}{a}_i{b}_j{\alpha }_{ij}p{x}_1.$$

(25)

Define the $R_0$–bilinear map $\beta :U\times U\to R$ by

$$\beta \left(\sum _i{a}_i{x}_i,\sum _j{b}_j{x}_j\right)=\left(\sum _{i,j}{a}_i{b}_j{\alpha }_{ij}p\right)x_1\in \langle p{x}_1\rangle .$$

Then, (25) reads $(a,u)·(b,v)=(ab,av+bu)+\beta (u,v).$

One can check that R with this multiplication is a Frobenius local ring with invariants $(p,n=2,r,6,k=3)$. Moreover, any Frobenius local ring with such invariants can be constructed in this way.

Proposition 4.

$R\cong T$ if and only if there exist $D\in {\mathrm{GL}}_3\left(\mathbb{F}\right)$ and $\sigma \in \mathrm{Aut}\left(\mathbb{F}\right)$ such that

$$M^{\prime }=D^t\sigma \left(M\right)D.$$

Proof. 

As before, we may assume the isomorphism restricts to $\sigma$ on $R_0$ and induces D on $J/J^2$. Because the target of the structural form is $\langle p{x}_1\rangle$ and $x_1$ can be normalized (by replacing the basis with one carrying $x_1$ to a scalar multiple of $y_1$), the scalar factor $\beta \in {\Gamma }^{\times }\left(r\right)$ can be absorbed by rescaling $x_1$. Hence the congruence has no extra scalar factor in this case. □

Theorem 5.

The structural matrix M of a ring R constructed as above is congruent to exactly one of the following canonical forms:

(i) 

If $\mathrm{char}\left(\mathbb{F}\right)\ne 2,$ 

$$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& \alpha \end{array}\right).$$

(26)

(ii) 

If $\mathrm{char}\left(\mathbb{F}\right)=2,$ 

$$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right).$$

(27)

Proof. 

Since $n=2$ and $p\notin J^2$, we compute $x_i·x_j$ for $1\le i,j\le 3$. First, note that

$$x_1^2=\beta p{x}_1,$$

for some $\beta \in \Gamma \left(r\right)$. If $\beta \ne 0$ and $\mathrm{char}\left(\mathbb{F}\right)\ne 2$, we can adjust $x_1$ so that $x_1^2=0$ by completing the square.

Similarly, let

$$x_2{x}_3=\gamma p{x}_1,$$

where $\gamma \in {\Gamma }^{\times }\left(r\right)$. By scaling, we may assume $x_2{x}_3=p{x}_1$. Also,

$$x_1{x}_2=0,x_1{x}_3=0.$$

Therefore, when $\mathrm{char}\left(\mathbb{F}\right)\ne 2$,

$$x_1{x}_i=0,x_2{x}_3=p{x}_1,x_2^2={\alpha }_{22}p{x}_1,x_3^2={\alpha }_{33}p{x}_1.$$

(28)

This gives the structural matrix

$$M=\left(\begin{array}{ccc}0& 0& 0\\ 0& {\alpha }_{22}& \delta \\ 0& \delta & {\alpha }_{33}\end{array}\right).$$

(29)

For $\mathrm{char}\left(\mathbb{F}\right)=2$, the structural matrix takes the form

$$M=\left(\begin{array}{ccc}\delta & 0& 0\\ 0& {\delta }_2& {\delta }_1\\ 0& {\delta }_1& {\delta }_3\end{array}\right),$$

(30)

with $\delta ,{\delta }_1,{\delta }_2,{\delta }_3\in \{0,1\}$.

In the case $\mathrm{char}\left(\mathbb{F}\right)\ne 2$, the matrix M can be reduced to its lower $2\times 2$ block

$$\left(\begin{array}{cc}{\alpha }_{22}& \delta \\ \delta & {\alpha }_{33}\end{array}\right),$$

so we may apply the same reduction procedure as in Lemma 1 and Theorem 2, but now with respect to $S_2$ instead of $S_4$. This yields three nonzero congruence classes of matrices:

$$\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\left(\begin{array}{cc}1& 0\\ 0& \alpha \end{array}\right).$$

Including the zero matrix gives the total number of classes listed in the statement.

For $\mathrm{char}\left(\mathbb{F}\right)=2$, the classification proceeds similarly using $S_3$ instead of $S_4$, and we omit the repetitive calculations. □

Example 1.

In this case, the zero matrix appears as the structural matrix of the ring

$${\displaystyle \frac{R_0[x_1,x_2,x_3]}{\langle x_1^2,x_2^2,x_3^2,x_1{x}_2,x_1{x}_3,x_2{x}_3,p{x}_2,p{x}_3\rangle }},$$

which can be written more compactly as

$${\displaystyle \frac{R_0[x_1,x_2,x_3]}{\langle x_i^2,x_i{x}_j,p{x}_2,p{x}_3\rangle }}.$$

Remark 6.

The dichotomy between $p\in J^2$ and $p\in J\setminus J^2$ exhausts all possibilities in characteristic $p^2$. The first case contributes an additional orbit in odd characteristic (rank 1 nonsquare), while the second case replaces the target $p{R}_0$ by the mixed target $\langle p{x}_1\rangle$, altering the congruence group and the resulting canonical forms.

5. Frobenius Local Rings of Characteristic p³

This section treats the remaining case $n=3$. Unlike the previous cases $n=1,2$, where multiple structural configurations may occur, in this case there is only one possible scenario: $p\in J$. If $p\notin J$, then necessarily $n<3$, which contradicts the current assumption. Therefore, in this setting, p is part of a generating system of the Jacobson radical.

Since $\left|R\right|=p^{6r}$, the radical J is generated by

$$\{p,x_1,x_2,x_3\},$$

and one directly verifies that $J^2$ is generated by $p^2$. These generators fully determine the multiplicative structure of R once the products are specified.

5.1. Construction

Let U be an $R_0$–module generated by $x_1,x_2,x_3$. Define

$$R=R_0\oplus U.$$

(31)

Define multiplication on R by

$$(a_0+\sum _{i=1}^3{a}_i{x}_i)·(b_0+\sum _{j=1}^3{b}_j{x}_j)=a_0{b}_0+\sum _i(a_0{b}_i+b_0{a}_i)x_i+\sum _{i,j}{a}_i{b}_j{\alpha }_{ij}{p}^2,$$

(32)

where ${\alpha }_{ij}\in \mathbb{F}$.

This multiplication endows R with a ring structure. The ring R satisfies all three conditions of property (P). Indeed,

$$J=p{R}_0+U$$

is an ideal of R. If $u\in R\setminus J$, then u is a unit of R, so J is maximal and R is a local ring. From (32) we see that

$$J^2=\langle p^2\rangle ,J^3=0.$$

Therefore, R satisfies property (P). Conversely, every ring with these invariants can be constructed in this way.

5.2. Classification

We now classify the isomorphism classes of Frobenius local rings with property (P) when $n=3$.

Theorem 6.

There are 7 non-isomorphic classes of Frobenius local rings with property (P) and $n=3$ when $\mathrm{char}\left(\mathbb{F}\right)\ne 2,$ and 4 such classes when $\mathbb{F}={\mathbb{F}}_{2^r}$.

Proof. 

First, observe that $p{x}_i=0$ for all i, since the additive order of p is $p^3$ and $\left|R\right|=p^{6r}$. Also,

$$x_i{x}_j={\alpha }_{ij}{p}^2.$$

(33)

This induces a structural matrix M of size $3\times 3$:

$$M=\left(\begin{array}{ccc}{\alpha }_{11}& {\alpha }_{12}& {\alpha }_{13}\\ {\alpha }_{21}& {\alpha }_{22}& {\alpha }_{23}\\ {\alpha }_{31}& {\alpha }_{32}& {\alpha }_{33}\end{array}\right).$$

(34)

Let T be another ring with the same invariants as R, and let $M^{\prime }$ be its structural matrix. Then, as in previous sections,

$$R\cong T⟺M^{\prime }=\beta D^t\sigma \left(M\right)D,$$

for some $\sigma \in \mathrm{Aut}\left(\mathbb{F}\right)$, $D\in \mathrm{GL}(3,\mathbb{F})$, and $\beta \in {\Gamma }^{\times }\left(r\right)$.

Consider the canonical blocks

$$\begin{array}{cc}\hfill A& =\left(\begin{array}{ccc}{\alpha }_{11}& 0& 0\\ 0& {\alpha }_{22}& 0\\ 0& 0& {\alpha }_{33}\end{array}\right),B^{\prime }=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right).\hfill \end{array}$$

(35)

From Lemma 1, but working on $S_3$ instead of $S_4$, M is congruent to

$$M\cong A+\nu B^{\prime },$$

where $\nu \in \mathbb{F}$.

If $\mathrm{char}\left(\mathbb{F}\right)\ne 2$, then $\nu =0$, so $M\cong A$. Repeating the argument used in the proof of Theorem 2, we deduce that

$$A\cong \alpha I_{r_0}+I_{3-r_0},0\le r_0\le 3,$$

and $\alpha I_{r_0}$ is congruent either to $I_{r_0}$ or to $\alpha I_1+I_{r_0-1}$. Hence, $I_{r_0}$ and $\alpha I_1+I_{r_0-1}$ give two distinct non-congruent forms of $A_0$.

Classification of congruence classes. We summarize the resulting classes as follows:

(i)

$\mathrm{char}\left(\mathbb{F}\right)\ne 2$:

$$\begin{array}{cc}\hfill & \left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccc}\alpha & 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).\hfill \end{array}$$

(36)

(ii)

$\mathrm{char}\left(\mathbb{F}\right)=2$: depending on whether the rank of M is odd or even, M is congruent to $A_0$ or to $B^{\prime }$ (cf. Theorem 2). We obtain the classes:

$$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right).$$

(37)

□

The classification scheme presented not only yields an explicit enumeration of isomorphism classes but also exhibits a consistent structural correspondence with the theory of symmetric bilinear forms. In particular, the Frobenius condition translates into a symmetry constraint on the matrix M, which significantly simplifies the analysis. This reduction converts what initially appears to be an algebraic classification problem into a linear-algebraic one governed by congruence under the general linear group.

Additionally, the independence of $N(p,n,m)$ from the residue field extension degree r underscores that the determining factors of the classification lie in the nilpotent structure and the parity of the matrix rank rather than in the size of the ground field.

Remark 7.

The results demonstrate that for characteristic $p^3$, the Frobenius condition enforces a tight structural correlation between $J^2$ and the underlying bilinear form on $J/J^2$. Consequently, the enumeration of isomorphism types reduces to counting distinct congruence classes of symmetric matrices in $S_3\left(\mathbb{F}\right)$ up to $\mathrm{GL}(3,\mathbb{F})$-equivalence. The dependence of the total count on the parity of m for $p=2$ further reflects the degeneracy between symmetric and alternating classifications in characteristic two.

Example 2.

For instance, in the classification table, the ring

$$R={\displaystyle \frac{R_0[x_1,x_2,x_3,x_4]}{\langle x_i^2,p{x}_i\rangle }}$$

means that $x_i^2=0$ and $p{x}_i=0$ for all $i=1,2,3,4$.

Corollary 4.

If $n=3,$ then

$$N_F(p,n=3,m)=\left\{\begin{array}{cc}2m-5,\hfill & \mathit{if}p\ne 2,\hfill \\ \left\{\begin{array}{cc}{\displaystyle \frac{3m-6}{2}}+1,\hfill & \mathit{if}m\mathit{is}\mathit{odd},\hfill \\ m-1,\hfill & \mathit{if}m\mathit{is}\mathit{even}.\hfill \end{array}\right.\hfill & \mathit{if}p=2.\hfill \end{array}\right.$$

(38)

Remark 8.

The tables list all possible congruence classes of the symmetric bilinear form determining the multiplication on $J/J^2$. Two rings are isomorphic if and only if their associated matrices lie in the same congruence class under the action of $G{L}_{m-\iota \left(n\right)}\left({\mathbb{F}}_p\right)$. Consequently, any two rings that correspond to different rows of the tables are automatically non-isomorphic. Each row represents a distinct canonical form and therefore a distinct isomorphism class of Frobenius local rings satisfying property (P).

6. Conclusions

The classification of finite local rings of order $p^{mr}$ is a central and long-standing problem in ring theory. While complete results are available for small nilpotency indices ($m\le 5$), the general classification for $m\ge 6$ remains widely open. In this work, we address the first nontrivial case beyond this boundary by giving a complete enumeration of a well-defined and structurally rich subclass of Frobenius local rings of order $p^{6r}$ satisfying property (P), namely: $J^3=0,J^2\ne 0, \mathrm{and} J^2$ is principal.

A key point distinguishing the case $m=6$ from the previously studied case $m=5$ is that ${dim}_{\mathbb{F}}(J/J^2)$ increases from 3 to 4, which leads to a fundamentally different classification problem. In particular, the ring structure is governed by symmetric bilinear forms on a 4-dimensional vector space over the residue field, rather than on a 3-dimensional one. As a result, the enumeration requires a complete analysis of congruence classes of $4\times 4$ symmetric matrices, yielding new canonical forms and counting formulas that cannot be obtained by a simple parameter shift from earlier results.

Within this framework, we established explicit and uniform formulas for the number $N(p,n,m)$ of isomorphism classes in characteristics p, $p^2$, and $p^3$. An important outcome is that $N(p,n,m)$ depends linearly on the nilpotency index m and is independent of the residue field extension degree r. This phenomenon reflects the fact that the classification is controlled entirely by the dimension of $J/J^2$ and the rank-type invariants of the associated symmetric bilinear form rather than by the size of the residue field.

The results are summarized by explicit canonical matrices and corresponding ring structures, presented systematically in

Table 1. Frobenius local rings of order $2^{6r}$ and characteristic 2. Here, x denotes a fixed generator of $J^2$, and all unspecified products $x_i{x}_j$ are zero unless explicitly stated.

Matrix Representative M	Ring Structure
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{2^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2,x_3^2,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{2^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2-x,x_3^2,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{2^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2-x,x_3^2-x,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{2^r}[x_1,x_2,x_3,x_4,x]}{\langle x_i^2-x,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{2^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2,x_2^2,x_3^2,x_4^2,x_1{x}_2-x,x_3{x}_4-x,x_{i}x\rangle }}$
$\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{2^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2,x_2^2,x_3^2,x_4^2,x_1{x}_2-x,x_{i}x,x_i{x}_j\rangle }}$

,

Table 2. Frobenius local rings of order $p^{6r}$ and characteristic p for $p\ne 2$. Here, x denotes a fixed generator of $J^2$, all unspecified products $x_i{x}_j$ are zero unless stated, and $x_{i}x=0$ since $J^3=0$.

Matrix Representative M	Ring Structure
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{p^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2,x_3^2,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{p^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2-x,x_3^2,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{p^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2-x,x_3^2-x,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{p^r}[x_1,x_2,x_3,x_4,x]}{\langle x_i^2-x,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{p^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2-\alpha x,x_3^2,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{p^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2-\alpha x,x_3^2-x,x_4^2,x_{i}x,x_i{x}_j\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$	${\displaystyle \frac{{\mathbb{F}}_{p^r}[x_1,x_2,x_3,x_4,x]}{\langle x_1^2-x,x_2^2-\alpha x,x_3^2-x,x_4^2-x,x_{i}x,x_i{x}_j\rangle }}$

,

Table 3. Frobenius local rings of order $p^{6r}$ and characteristic $p^2$ with $p\in J\setminus J^2$.

$\mathit{p}=2$		$\mathit{p}\ne 2$
$\mathit{M}$	Ring Structure	$\mathit{M}$	Ring Structure
$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3]}{\langle x_i^2,2x_i\rangle }}$	$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3]}{\langle x_i^2,p{x}_i\rangle }}$
$\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2-2x_1,x_3^2,2x_2,2x_3\rangle }}$	$\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2-p{x}_1,x_3^2,p{x}_2,p{x}_3\rangle }}$
$\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2-2x_1,x_3^2-2x_1,2x_2,2x_3\rangle }}$	$\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2-p{x}_1,x_3^2-p{x}_1,p{x}_2,p{x}_3\rangle }}$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3]}{\langle x_1^2-2x_1,x_2^2-2x_1,x_3^2-2x_1,2x_2,2x_3\rangle }}$	$\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& \alpha \end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2-p{x}_1,x_3^2-\alpha p{x}_1,p{x}_2,p{x}_3\rangle }}$
$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2,x_3^2,x_2{x}_3-2x_1,2x_2,2x_3\rangle }}$

,

Table 4. Frobenius local rings of order $p^{6r}$ and characteristic $p^2$ with $p\in J^2$. In this case, $J^2=\langle p\rangle$ and $J·J^2=0$ implies $p{x}_i=0$ for all i.

$\mathit{p}=2$		$\mathit{p}\ne 2$
$\mathit{M}$	Ring Structure	$\mathit{M}$	Ring Structure
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-2,x_2^2,x_3^2,x_4^2,x_i{x}_j,2x_i\rangle }}$	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-p,x_2^2,x_3^2,x_4^2,x_i{x}_j,p{x}_i\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-2,x_2^2-2,x_3^2,x_4^2,x_i{x}_j,2x_i\rangle }}$	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-p,x_2^2-p,x_3^2,x_4^2,x_i{x}_j,p{x}_i\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-2,x_2^2,x_3^2-2,x_4^2,x_i{x}_j,2x_i\rangle }}$	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-p,x_2^2,x_3^2-p,x_4^2,x_i{x}_j,p{x}_i\rangle }}$
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3,x_4]}{\langle x_i^2-2,x_i{x}_j,2x_i\rangle }}$	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_i^2-p,x_i{x}_j,p{x}_i\rangle }}$
$\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3,x_4]}{\langle x_i^2,x_1{x}_2-2,x_3{x}_4-2,2x_i\rangle }}$	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-p,x_2^2-\alpha p,x_3^2-p,x_4^2,x_i{x}_j,p{x}_i\rangle }}$
$\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^2,r)[x_1,x_2,x_3,x_4]}{\langle x_i^2,x_1{x}_2-2,x_i{x}_j,2x_i\rangle }}$	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-p,x_2^2-\alpha p,x_3^2-p,x_4^2-p,x_i{x}_j,p{x}_i\rangle }}$
		$\left(\begin{array}{cccc}\alpha & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-\alpha p,x_2^2,x_3^2,x_4^2,x_i{x}_j,p{x}_i\rangle }}$
		$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^2,r)[x_1,x_2,x_3,x_4]}{\langle x_1^2-p,x_2^2-\alpha p,x_3^2,x_4^2,x_i{x}_j,p{x}_i\rangle }}$

and

Table 5. Frobenius local rings of order $p^{6r}$ and characteristic $p^3$.

$\mathit{p}=2$		$\mathit{p}\ne 2$
$\mathit{M}$	Ring Structure	$\mathit{M}$	Ring Structure
$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^3,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2,x_3^2,2x_i\rangle }}$	$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^3,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2,x_3^2,p{x}_i\rangle }}$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^3,r)[x_1,x_2,x_3]}{\langle x_1^2-4,x_2^2,x_3^2,2x_i\rangle }}$	$\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^3,r)[x_1,x_2,x_3]}{\langle x_1^2-p^2,x_2^2,x_3^2,p{x}_i\rangle }}$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(2^3,r)[x_1,x_2,x_3]}{\langle x_1^2-4,x_2^2-4,x_3^2,2x_i\rangle }}$	$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^3,r)[x_1,x_2,x_3]}{\langle x_1^2-p^2,x_2^2-p^2,x_3^2,p{x}_i\rangle }}$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(2^3,r)[x_1,x_2,x_3]}{\langle x_1^2-4,x_2^2-4,x_3^2-4,2x_i\rangle }}$	$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(p^3,r)[x_1,x_2,x_3]}{\langle x_1^2-p^2,x_2^2-p^2,x_3^2-p^2,p{x}_i\rangle }}$
$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)$	${\displaystyle \frac{GR(2^3,r)[x_1,x_2,x_3]}{\langle x_1^2,x_2^2,x_3^2,x_2{x}_3-4,2x_i\rangle }}$	$\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^3,r)[x_1,x_2,x_3]}{\langle x_1^2-\alpha p^2,x_2^2,x_3^2,p{x}_i\rangle }}$
		$\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)$	${\displaystyle \frac{GR(p^3,r)[x_1,x_2,x_3]}{\langle x_1^2-\alpha p^2,x_2^2-p^2,x_3^2,p{x}_i\rangle }}$
		$\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$	${\displaystyle \frac{GR(p^3,r)[x_1,x_2,x_3]}{\langle x_1^2-\alpha p^2,x_2^2-p^2,x_3^2-p^2,p{x}_i\rangle }}$

. These tables provide a complete correspondence between bilinear-form congruence classes and isomorphism classes of Frobenius local rings with property (P) and $J^3=0$.

Finally, this work clarifies the role of fixed-order classification as a meaningful and effective approach to the broader classification problem. While a uniform solution for arbitrary orders $p^{qr}$ remains a challenging open problem, the methods developed here provide a concrete foundation for extending the analysis to higher nilpotency indices (such as $J^4=0$ and beyond), as well as for applications in algebraic coding theory, where Frobenius rings play a central role in duality-preserving constructions.