Cyclic Codes over a Non-Local Non-Unital Ring

Abstract

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as $H=⟨a,b\mid 2a=2b=0,a^2=0,b^2=b,ab=ba=0⟩.$ This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length $n.$ We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes $H.$ We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length $2n$ over ${\mathbb{F}}_2$ is studied.

1. Introduction

Cyclic codes over finite fields [1,2] and finite rings [3,4,5,6,7,8,9] constitute the most popular class of algebraic codes for many reasons, both practical and theoretical. Their clean algebraic structure enables the practitioner to use efficient encoding and decoding techniques [10,11]. It also permits the theoretician to derive strong lower bounds on the minimum distance (BCH, Hartmann–Tzeng, van Lint/Wilson, Roos) [1,12,13,14]. At the heart of this popularity lies the quotient ring $A\left[x\right]/(x^n-1),$ where x is a formal variable and A the code alphabet (a finite ring). Ideals in this ring are in one-to-one correspondence with cyclic codes of length n over $A.$ This ring is principal and every ideal admits a monic generator, say, $g\left(x\right)$. The roots of $g\left(x\right)$ allow for the computation of various bounds on the minimum distance. The dual of such a code is generated by the quotient of $x^n-1$ by $g\left(x\right).$ In recent years, a theory of codes over non-unitary rings emerged [15,16,17]. The absence of a unit element for the multiplication forbids the researcher investigating cyclic codes over a non-unitary ring A to use the auxiliary ring $A\left[x\right]/(x^n-1),$ simply because $x^n-1$ is not available in $A\left[x\right].$ In such a case, alternative strategies have to be derived. In this paper we study the fundamental algebraic properties (generation, duality) of cyclic codes over a special non-unitary ring without any polynomial formalism.

In particular, we investigate cyclic codes over the non-unitary ring H in the classification of Rhagavendran [18] and the notation of Fine [19]. This ring is semi-local with two maximal ideals. Essentially, every cyclic code over H can be regarded as an ordered pair of binary cyclic codes of the same length. This structure theorem is used to characterize self-dual, quasi-self-dual, and linear complementary dual (LCD) cyclic codes over H. It is also used to classify isomorphism classes of cyclic codes of given length up to length $7.$ A Gray map between codes of length n over H and codes of length $2n$ over ${\mathbb{F}}_2$ is introduced. The Gray image of a cyclic code is proved to be quasi-cyclic of index $2.$

The material is organized as follows. The next section collects basic notations and notions needed for the other sections. Section 3 considers the structure of cyclic codes over H. Section 4 studies the Gray map of cyclic code over H, Section 5 investigates the automorphism group of a cyclic code, and Section 6 applies this information to classifying cyclic codes of a given length up to permutation. Section 7 derives conditions for LCD-ness. Section 8 contains numerical examples. Section 9 concludes this article.

2. Preliminaries

2.1. Binary Codes

The Hamming weight of $x\in {\mathbb{F}}_2^n$ is denoted by $wt\left(x\right)$. The dual of a binary linear code C is signified by $C^{\perp }$ and defined as :

$$C^{\perp }=\left\{y\in {\mathbb{F}}_2^n|\forall x\in C,(x,y)=0\right\},$$

where $(x,y)={\displaystyle \sum _{i=1}^n}{x}_i{y}_i$, denotes the standard inner product. A code C is self-orthogonal if it is included in its dual: $C\subseteq C^{\perp }$. A code is even if all its code words have even weight. All binary self-orthogonal codes are even, but not all even binary codes are self-orthogonal. Two binary codes are equivalent if there is a permutation of coordinates that maps one to the other.

A linear code C of length n over ${\mathbb{F}}_2$ is cyclic provided that, for each vector $c=(c_0\dots c_{n-2}{c}_{n-1})$ in C, the vector $c_{n-1}{c}_0\dots c_{n-2}$, obtained from c by the cyclic shift of coordinates $i\mapsto i+1$$\left(modn\right)$, is also in C.

2.2. Rings and Modules

Following [19], we define a ring on two generators $a,b$ by their relations

$$H=⟨a,b\mid 2a=2b=0,a^2=0,b^2=b,ab=ba=0⟩.$$

Thus, H has characteristic 2 and consists of four elements $H=\left\{0,a,b,c\right\},$ with $c=a+b.$ The addition table is immediate from these definitions

+	0	a	b	c
0	0	a	b	c
a	a	0	c	b
b	b	c	0	a
c	c	b	a	0

The multiplication table is as follows.

×	0	a	b	c
0	0	0	0	0
a	0	0	0	0
b	0	0	b	b
c	0	0	b	b

From this table, we deduce that this ring is commutative and that it does not have an unity element for the multiplication. Two absorbing elements are 0 and a. It is semi-local with two maximal ideals, $J_a=\left\{0,a\right\}$ and $J_b=\left\{0,b\right\}.$ Denote by ${\alpha }_a:H\mapsto {\mathbb{F}}_2$ the reduction map with respect to $J_a$, and by ${\alpha }_b:H\mapsto {\mathbb{F}}_2$ is the map of reduction map with respect to $J_b$. Thus, ${\alpha }_a\left(a\right)={\alpha }_b\left(b\right)=0,$ and ${\alpha }_a\left(b\right)={\alpha }_b\left(a\right)=1$. Note that these two applications are morphisms for addition but for not multiplication. They can be extended in the obvious way into maps from $H^n$ to ${\mathbb{F}}_2^n.$ This ring decomposition induces a code decomposition in the following way. The code C over H can be written as a direct sum (in the sense of modules)

$$C=a{C}_a\oplus b{C}_b,$$

where $C_a={\alpha }_b\left(C\right)$ and $C_b={\alpha }_a\left(C\right).$ Sometimes we will use the inner product notation $(x,r)$ for $x\in {\mathbb{F}}_2^n,r\in H^n$ to mean

$$(x,r)=\sum _{i=1}^n{x}_i{r}_i=\sum _{x_i=1}{r}_i.$$

A linear H-code C of length n is an H-submodule of $H^n$. It can be regarded as the H-span of the rows of a generator matrix. With that code, we attach two binary codes of length n: ${\alpha }_a\left(C\right)$ and ${\alpha }_b\left(C\right).$ An additive code of length n over ${\mathbb{F}}_4$ is an additive subgroup of ${\mathbb{F}}_4^n.$ It is an ${\mathbb{F}}_2$ vector space of size $4^k$ elements for some $k\le n$. Using a generator matrix $G,$ such a code can be cast as the ${\mathbb{F}}_2$-span of the rows of $G.$ With every linear H-code C we associate an additive ${\mathbb{F}}_4$-code ${\varphi }_H\left(C\right)$ by the alphabet substitution

$$0\to 0,a\to \omega ,b\to 1,c\to {\omega }^2,$$

where ${\mathbb{F}}_4$ = ${\mathbb{F}}_2\left[\omega \right],$ extended in the obvious way to ${\mathbb{F}}_4^n.$

We use the notation of the Magma package [20]

$$[<0,1>,\dots ,<i,A_i>,\dots ,<n,A_n>]$$

for the weight distribution of a code, where $A_i$ is the number of code words of weight $i.$ Two H-codes are permutation-equivalent if there is a permutation of coordinates that maps one to the other.

Equip $H^n$ with an inner product by the rule $(x,y)={\displaystyle \sum _{i=1}^n}{x}_i{y}_i.$ The dual $C^{\perp }$ of C is the module defined by

$$C^{\perp }=\left\{y\in H^n|\forall x\in C,(x,y)=0\right\}.$$

Thus, the dual of a module is a module. A code is self-dual if it is equal to its dual.

A code C is self-orthogonal if $\forall x,y\in C,(x,y)=0.$

3. The Structure of Cyclic Codes over $\mathit{H}$

Definition 1. 

A cyclic code C of length n over H is a linear code with property that if $c=(c_0,c_1,\dots ,c_{n-1})$∈C then $\sigma \left(c\right)=(c_{n-1},c_0,\dots ,c_{n-2})$∈C.

Example 1. 

The repetition code of length 2, defined by $R_2=\left\{00,aa,bb,cc\right\}$ is a cyclic code over H.

The dual of $R_2$ is defined by ${R_2}^{\perp }=\left\{00,aa,bb,cc,cb,bc,a0,0a\right\}.$ It is also a cyclic code over $H.$

Example 2. 

Let C = $a{C}_a\oplus b{C}_b$, where $C_a$ and $C_b$ are binary codes generated by the matrix

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

Then C is a cyclic code of length 3 over H such that $C_a=\left\{000,101,110,011\right\}=C_b$$.\Rightarrow$$\left|C_a\right|=4$ and $\left|C_b\right|=4$. Hence, $\left|C\right|=$$\left|C_a\right|$$\left|C_b\right|$ = $4^2$ = 16.

Then $C=\left\{000,a0a,aa0,0aa,b0b,bb0,0bb,c0c,cc0,0cc,cba,abc,cab,acb,bac,bca\right\}$.

The following result is of crucial importance to our study.

Theorem 1. 

If $C=a{C}_a\oplus b{C}_b$ is an arbitrary linear code over $H,$ then C is a cyclic code if and only if $C_a$ and $C_b$ are both cyclic.

Proof. 

(⇒) Let T be the permutation matrix corresponding to the shift.

It can be seen that T commutes with both ${\alpha }_a$ and ${\alpha }_b$. We know $TC\subseteq C$.

Hence, ${\alpha }_a\left(TC\right)=T{\alpha }_a\left(C\right)\subseteq {\alpha }_a\left(C\right)$. This implies that $C_b={\alpha }_a\left(C\right)$ is cyclic.

Same argument with $C_a={\alpha }_b\left(C\right)$.

(⇐) Suppose that $C_a$ and $C_b$ are both cyclic. We want to prove that C is cyclic.

Let $C=a{c}_1+b{c}_2$ be an arbitrary code word in C where $c_1=(x_0,x_1,\dots ,x_{n-1})\in C_a$

and $c_2=(y_0,y_1,\dots ,y_{n-1})\in C_b$.

We need to prove that $c^{\prime }=a{c}_1{ }^{\prime }+b{c}_2{ }^{\prime }\in C$. Then

$$\begin{array}{cc}\hfill c& =a(x_0,x_1,\dots ,x_{n-1})+b(y_0,y_1,\dots ,y_{n-1})\hfill \\ \hfill & =(a{x}_0+b{y}_0,a{x}_1+b{y}_1,\dots ,a{x}_{n-1}+b{y}_{n-1})\hfill \end{array}$$

(1)

As $C_a$ and $C_b$ are both cyclic, then $c_1{ }^{\prime }=(x_{n-1},x_0,\dots ,x_{n-2})\in C_a$ and $c_2{ }^{\prime }=(y_{n-1},y_0,\dots ,$$y_{n-2})\in C_b$.

Hence, (1) will become

$$\begin{array}{cc}\hfill c^{\prime }& =(a{x}_{n-1}+b{y}_{n-1},a{x}_0+b{y}_0,\dots ,a{x}_{n-2}+b{y}_{n-2})\hfill \\ \hfill & =a(x_{n-1},x_0,\dots ,x_{n-2})+b(y_{n-1},y_0,\dots ,y_{n-2})\in C\hfill \end{array}$$

Hence, we proved C is a cyclic code over H. □

To derive more corollaries of Theorem 1, we require some definitions pertaining to the duality of H-codes.

Definition 2. 

• A code of length n isquasi-self-dual if it is self-orthogonal and of size $2^n$.
• A quasi-self-dual code over H with all weights even is called aType IV code.
• AQSD H-code is called quasi-Type IVQT4 if $C_a$ is an even code.
• A linear code C is called an LCD code (Linear Complementary Dual Code) if $C\cap C^{\perp }=0$.

For LCD codes over finite fields see [21].

Corollary 1. 

A code C of length n over H is cyclic QSD if and only if:

1. 

$C_b$ is a cyclic self-orthogonal [n,k] binary code.

2. 

$C_a$ is a cyclic [n,n-k] binary code.

Proof. 

By combining Theorem 1 and [15] (Lemma 1) we obtain the proof. □

Corollary 2. 

A code C over H is cyclic Type IV if $C_b$ is a cyclic self-orthogonal binary code and $C_a=C_b^{\perp }$.

Proof. 

By combining Theorem 1 and [15] (Theorem 5) we obtain the result. □

Corollary 3. 

Every cyclic QT4 code of length n over H is of the form $a{C}_a\oplus b{C}_b$ where:

1. 

$C_b$ is a cyclic self-orthogonal [n,k] binary code.

2. 

$C_a$ is an even cyclic binary [n,n-k] code.

Proof. 

By combining Theorem 1 and [15] (Lemma 2) we obtain the proof. □

Corollary 4. 

Let C be a cyclic code over H, then C is an LCD cyclic code if and only if $C_b$ is LCD cyclic and $C_a=\left\{0\right\}$.

Proof. 

By combining Theorem 1 and [22] (Theorem 5.11) we obtain the result. □

To prepare for the study of self-dual cyclic codes over H, we need the following theorem. The proof is omitted.

Theorem 2 

([22] (Theorem 22)). Let $C^{\perp }$ be the dual code of C over H. Then $C^{\perp }=a{\mathbb{F}}_2^n\oplus b{C}_b^{\perp }$.

The next result follows.

Corollary 5. 

If C is a cyclic code over H, then the dual code $C^{\perp }$ of C is also cyclic.

Proof. 

By Theorem 2, we have $C^{\perp }=a{\mathbb{F}}_2^n\oplus b{C}_b^{\perp }$. Since the dual code of binary cyclic code is also cyclic, by Theorem 1 and the fact that ${\mathbb{F}}_2^n$ is a cyclic code, we obtain the result. □

In the next results, we study the additive code of cyclic codes over H.

Corollary 6. 

If C is cyclic code over $H,$ then the additive code ${\varphi }_H\left(C\right)$ is also cyclic.

Proof. 

The proof is direct from Theorem 1. □

Corollary 7. 

If the binary code ${\varphi }_H\left(C\right)$ is cyclic, then ${\varphi }_H\left(C^{\perp }\right)$ is also cyclic.

Proof. 

By combining Corollary 5 and Corollary 6, we obtain the proof. □

If C is a cyclic code of length n over H, then the relation ${\varphi }_H\left(C^{\perp }\right)={\varphi }_H{\left(C\right)}^{\perp }$ is not true in general as the following example shows.

Example 3. 

Let $C=\left\{00,bb\right\}$ be a cyclic code of length 2 over H then the dual of C is $C^{\perp }=\left\{00,aa,bb,cc,cb,bc,a0,0a\right\}$ and ${\varphi }_H\left(C\right)=\left\{00,11\right\}$.

$${\varphi }_H\left(C^{\perp }\right)=\left\{00,\omega \omega ,11,{\omega }^2{\omega }^2,{\omega }^{2}1,1{\omega }^2,\omega 0,0\omega \right\}and$$

$${\varphi }_H{\left(C\right)}^{\perp }=\left\{00,10,{\omega }^2\omega ,\omega \omega ,\omega {\omega }^2,{\omega }^2{\omega }^2,11,01\right\}.$$

4. Gray Map

Any code word of H can be expressed as $c=ax+by$, where $a,b$ are generators for the ring H and $x,y$ are arbitrary elements in ${\mathbb{F}}_2.$ The Lee weights of $0,a,b,c\in H$ are 0, 1, 1, 2, respectively. The Gray map from H to ${\mathbb{F}}_2\times {\mathbb{F}}_2$ is given by $\mathsf{\Psi }\left(c\right)=(x,y).$ The Gray map is a bijection. This map can be extended to $H^n$ in a natural way. For any $s=(s_0,s_1,\dots ,s_{n-1})\in H^n$, where $s_i=a{u}_i+b{v}_i,0\le i\le n-1,$ we define $\mathsf{\Psi }\left(s\right)=\left(u\right(s),v(s\left)\right),$ where $u\left(s\right)=(u_0,u_1,\dots ,u_{n-1}),v\left(s\right)=(v_0,v_1,\dots ,v_{n-1}).$

Then $\mathsf{\Psi }$ is a weight-preserving map from ($H^n$,Lee weight) to (${\mathbb{F}}_2^{2n}$,Hamming weight), that is, $w_L\left(s\right)=w_H\left(\mathsf{\Psi }\left(s\right)\right)$.

Example 4. 

Let $C=a{C}_a\oplus b{C}_b$, where $C_a$ and $C_b$ are binary codes generated by $G_a=\left(\begin{array}{cc}1& 1\end{array}\right)$ and $G_b=\left(\begin{array}{cc}1& 1\end{array}\right)$, then $C_a=\left\{00,11\right\}=C_b$. Therefore, $C=\left\{00,aa,bb,cc\right\}$ and we can write the code C as $C=\left\{a\left(00\right)+b\left(00\right),a\left(11\right)+b\left(00\right),a\left(00\right)+b\left(11\right),a\left(11\right)+b\left(11\right)\right\}.$ Hence, $\mathsf{\Psi }\left(C\right)=\left\{0000,1100,0011,1111\right\}.$

The following result characterizes the Gray image of an H-code as a function of its two binary components.

Theorem 3. 

Let C be a linear code of length n over H. Then $\mathsf{\Psi }\left(C\right)=C_a\oplus C_b$.

Proof. 

For any $(u_0,u_1,\dots ,u_{n-1},v_0,v_1,\dots ,v_{n-1})\in \mathsf{\Psi }\left(C\right)$, let $c_i=a{u}_i+b{v}_i$, $i=0,1,\dots ,n-1$. Since $\mathsf{\Psi }$ is bijection, $c=(c_0,c_1,\dots ,c_{n-1})\in C$. By the definition of C, we obtain that $(u_0,u_1,\dots ,u_{n-1})\in C_a$, $(v_0,v_1,\dots ,v_{n-1})\in C_b$; therefore, $(u_0,u_1,\dots ,u_{n-1},v_0,v_1,\dots ,v_{n-1})$$\in C_a\oplus C_b$. This implies that $\mathsf{\Psi }\left(C\right)\subseteq C_a\oplus C_b$.

On the other hand, for any $(u_0,u_1,\dots ,u_{n-1},v_0,v_1,\dots ,v_{n-1})\in C_a\oplus C_b$ where $(u_0,u_1,$$\dots ,u_{n-1})\in C_a$, $(v_0,v_1,\dots ,v_{n-1})\in C_b$. Since C is linear, we have $c=a{u}_i+b{v}_i\in C$, and it follows that $\mathsf{\Psi }\left(c\right)=(u_0,u_1,\dots ,u_{n-1},v_0,v_1,\dots ,v_{n-1})$, which gives $C_a\oplus C_b\subseteq \mathsf{\Psi }\left(C\right)$. Therefore, $\mathsf{\Psi }\left(C\right)=C_a\oplus C_b$. □

As a consequence, we characterize the Gray image of the dual code.

Corollary 8. 

Let C be a linear code of length n over H and $\mathsf{\Psi }\left(C\right)=C_a\oplus C_b$ then $\mathsf{\Psi }\left(C^{\perp }\right)=F_2^n\oplus C_b^{\perp }.$

Proof. 

It follows directly from Theorem 2. □

Proposition 1. 

Let $d_H$ and $d_L$ be the minimum Hamming and Lee weights of a linear code C over H, respectively. Then, $d_H=d_L=min\left\{d\left(C_a\right),d\left(C_b\right)\right\}$, where $d\left(C_a\right)$ and $d\left(C_b\right)$ denote the minimum weight of a binary code $C_a$ and $C_b$, respectively.

Proof. 

Since $\mathsf{\Psi }$ is a weight-preserving map, then $d_L\left(C\right)=d_H\left(\mathsf{\Psi }\left(C\right)\right)=d_H(C_a\oplus C_b)=min\left\{d\left(C_a\right),d\left(C_b\right)\right\}$, and $d_H=d_L$ is obvious. □

To prepare for the study of the relation between Gray map and duality, we need the following Corollary and the following Proposition.

Corollary 9 

([22] (Corollary 24)). Let $C=a{C}_a\oplus b{C}_b$ be a linear code of length n over H. Then C is nice if and only if $C_a=\left\{0\right\}.$

Proposition 2. 

Let $C^{\perp }$ be the dual code of C over H. If C is a nice code over H, then $\mathsf{\Psi }\left(C^{\perp }\right)=\mathsf{\Psi }{\left(C\right)}^{\perp }$. Moreover, if C is a self-orthogonal code, so $\mathsf{\Psi }\left(C\right).$

Proof. 

$\left(\subseteq \right)$ For any $c_1=a{u}_1+b{v}_1\in C$, $c_2=a{u}_2+b{v}_2\in C^{\perp }$ where $u_1,u_2,v_1,v_2\in F_2^n$. Since $c_1·c_2=0$ implies $b(v_1·v_2)=0,$ thus we can obtain that $\mathsf{\Psi }\left(c_1\right)·\mathsf{\Psi }\left(c_2\right)=(u_1,v_1)·(u_2,v_2)=u_1{u}_2+v_1{v}_2=0$, which means $\mathsf{\Psi }\left(C^{\perp }\right)\subseteq \mathsf{\Psi }{\left(C\right)}^{\perp }$.

$\left(\supseteq \right)$ Let $\mathsf{\Psi }\left(c_1\right)=\mathsf{\Psi }(a{u}_1+b{v}_1)=(u_1,v_1)\in \mathsf{\Psi }\left(C\right)$ and $\mathsf{\Psi }\left(c_2\right)=\mathsf{\Psi }(a{u}_2+b{v}_2)=(u_2,v_2)\in \mathsf{\Psi }{\left(C\right)}^{\perp }$ where $u_1,u_2,v_1,v_2\in F_2^n$. Since $\mathsf{\Psi }\left(c_1\right)·\mathsf{\Psi }\left(c_2\right)=0$ implies $u_1{u}_2+v_1{v}_2=0$, thus we can obtain that $c_1·c_2=a^2{u}_1{u}_2+ab{u}_1{v}_2+ba{u}_2{v}_1+b^2{v}_1{v}_2=0+0+0+b(v_1·v_2)=0$, which means $\mathsf{\Psi }{\left(C\right)}^{\perp }\subseteq \mathsf{\Psi }\left(C^{\perp }\right)$. Hence, $\mathsf{\Psi }\left(C^{\perp }\right)=\mathsf{\Psi }{\left(C\right)}^{\perp }$. □

Proposition 3. 

If C is an LCD code over H then $\mathsf{\Psi }\left(C\right)$ is also an LCD code over $F_2$.

Proof. 

Since C is an LCD code over H then we have $C\cap C^{\perp }=\left\{0\right\}$ and $C_a$ is nice. We want to prove that $\mathsf{\Psi }\left(C\right)\cap \mathsf{\Psi }{\left(C\right)}^{\perp }=\left\{0\right\}.$

Consider,

$$\begin{array}{ccc}\hfill \left\{0\right\}& =& \mathsf{\Psi }\left(0\right)\hfill \\ & =& \mathsf{\Psi }(C\cap C^{\perp })\hfill \\ & =& \mathsf{\Psi }\left(C\right)\cap \mathsf{\Psi }\left(C^{\perp }\right)\mathrm{because}\mathsf{\Psi }\mathrm{is}\mathrm{a}\mathrm{bijection}\hfill \\ & =& \mathsf{\Psi }\left(C\right)\cap \mathsf{\Psi }{\left(C\right)}^{\perp }\mathrm{by}\mathrm{Proposition}2\hfill \end{array}$$

Hence, $\mathsf{\Psi }\left(C\right)$ is an LCD code over H. □

To prepare for the study of the symmetry of Gray images, we need the following Definition.

Definition 3. 

Let ${\phi }_s$ be the quasi-cyclic shift on ${\left(F_2^n\right)}^s$ given by:

${\phi }_s(x^{\left(1\right)}|x^{\left(2\right)}|\dots |x^{\left(s\right)})=(\sigma \left(x^{\left(1\right)}\right)|\sigma \left(x^{\left(2\right)}\right)|\dots |\sigma \left(x^{\left(s\right)}\right))$. A quasi-cyclic code C of index s and length $ns$ over $F_2$ is a subset of ${\left(F_2^n\right)}^s$ such that ${\phi }_s\left(C\right)=C$, where σ is the cyclic shift in length n.

Corollary 10. 

If C is a cyclic code of length n over H, then $\mathsf{\Psi }\left(C\right)$ is a binary quasi-cyclic code of index 2 and length $2n$.

Proof. 

It follows directly from the definition of quasi-cyclic codes. □

5. Automorphism Group

Let C be a linear code of length n over H and $\sigma$ a permutation of the symmetric group $S_n$ acting on $\left\{0,1,\dots ,n-1\right\}$. We associate with this code a linear code $\sigma \left(C\right)$ defined by:

$\sigma \left(C\right)=\left\{(x_{{\sigma }^{-1}\left(0\right)},\dots ,x_{{\sigma }^{-1}(n-1)})\mid (x_0,\dots ,x_{n-1})\in C)\right\}$

We say that the code C and $C^{\prime }$ are equivalent if there exists a permutation $\sigma \in S_n$ such that $C^{\prime }=\sigma \left(C\right)$.

The automorphism group of C is the subgroup of $S_n$ given by:

$$Aut\left(C\right)=\left\{\sigma \in S_n\mid \sigma \left(C\right)=C\right\}.$$

Example 5. 

Let C = $a{C}_a\oplus b{C}_b$, where $C_a$ and $C_b$ are binary codes generated by the matrix

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

Then C is a linear cyclic code of length 3 over H such that $C_a=\left\{000,101,110,011\right\}=C_b$. Consider $\sigma =(1,3)$ then a generator matrix will become $\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\end{array}\right)$ and we obtain $C_a=\left\{000,101,110,011\right\}=C_b$. Therefore, $\sigma \left(C\right)=C$. So $\sigma \in Aut\left(C\right)$.

Example 6. 

Let $C=a{C}_a\oplus b{C}_b$ where $C_a$ and $C_b$ are binary codes generated by the matrix

$$G=\left(\begin{array}{ccccccc}1& 0& 1& 1& 1& 0& 0\\ 0& 1& 0& 1& 1& 1& 0\\ 0& 0& 1& 0& 1& 1& 1\end{array}\right).$$

Then C is a cyclic code of length 7 over H, where

$C_a=\left\{0000000,1011100,0101110,0010111,1110010,0111001,1001011,1100101\right\}=C_b$.

And let $C^{\prime }=a{C}_a^{\prime }\oplus b{C}_b^{\prime }$, where $C_a^{\prime }$ and $C_b^{\prime }$ are binary codes generated by the matrix

$$G^{\prime }=\left(\begin{array}{ccccccc}1& 1& 1& 0& 1& 0& 0\\ 0& 1& 1& 1& 0& 1& 0\\ 0& 0& 1& 1& 1& 0& 1\end{array}\right).$$

Then $C^{\prime }$ is a cyclic code of length 7 over $H,$ where

$C_a^{\prime }=\left\{0000000,1110100,0111010,0011101,10011100,0100111,1101001,1010011\right\}=C_b^{\prime }$.

Hence, $\sigma \left(C\right)=C^{\prime }$ where $\sigma =\left(264\right)\left(35\right)$. Therefore, C and $C^{\prime }$ are equivalent codes.

The following result shows the relationship between the equivalence of two linear codes over H and that of their constituents.

Lemma 1. 

Let $C=a{C}_a\oplus b{C}_b$ and $C^{\prime }=a{C}_a^{\prime }\oplus b{C}_b^{\prime }$ be two codes over H. Then C and $C^{\prime }$ are equivalent if and only if there exists a permutation $\sigma \in S_n$ such that $\sigma \left(C_a\right)=C_a^{\prime }$ and $\sigma \left(C_b\right)=C_b^{\prime }$.

Proof. 

The first direction: If C and $C^{\prime }$ are equivalent, then $C^{\prime }=\sigma \left(C\right)$

$$\begin{array}{cc}\hfill a{C}_a^{\prime }\oplus b{C}_b^{\prime }& =\sigma (a{C}_a\oplus b{C}_b),\hfill \\ \hfill & =\sigma \left(a{C}_a\right)\oplus \sigma \left(b{C}_b\right),\hfill \\ \hfill & =a\sigma \left(C_a\right)\oplus b\sigma \left(C_b\right).\hfill \end{array}$$

By taking ${\alpha }_a$ and ${\alpha }_b$ for both sides, we obtain: $\sigma \left(C_a\right)=C_a^{\prime }$ and $\sigma \left(C_b\right)=C_b^{\prime }$.

Conversely, suppose that there exists a permutation $\sigma \in S_n$ such that $\sigma \left(C_a\right)=C_a^{\prime }$ and $\sigma \left(C_b\right)=C_b^{\prime }$. Then, we can write:

$$\begin{array}{cc}\hfill \sigma \left(C\right)& =\sigma (a{C}_a\oplus b{C}_b),\hfill \\ \hfill & =\sigma \left(a{C}_a\right)\oplus \sigma \left(b{C}_b\right),\hfill \\ \hfill & =a\sigma \left(C_a\right)\oplus b\sigma \left(C_b\right),\hfill \\ \hfill & =a{C}_a^{\prime }\oplus b{C}_b^{\prime },\hfill \\ \hfill & =C^{\prime }.\hfill \end{array}$$

Hence, C and $C^{\prime }$ are equivalent. □

As consequences from the previous lemma, we derive the following two results.

Proposition 4. 

Let $C=a{C}_a\oplus b{C}_b$ and $C^{\prime }=a{C}_a^{\prime }\oplus b{C}_b^{\prime }$ be two self-dual codes over H. Then C and $C^{\prime }$ are equivalent if and only if there exists a permutation $\sigma \in S_n$ such that $\sigma \left(C_b\right)=C_b^{\prime }$.

Proof. 

By combining Theorem 2 and Lemma 1, we obtain the proof. □

Proposition 5. 

Let $C=a{C}_a\oplus b{C}_b$ and $C^{\prime }=a{C}_a^{\prime }\oplus b{C}_b^{\prime }$ be two LCD codes over H. Then C and $C^{\prime }$ are equivalent if and only if there exists a permutation $\sigma \in S_n$ such that $\sigma \left(C_b\right)=C_b^{\prime }$.

Proof. 

By combining Corollary 4 and Lemma 1, we obtain the proof. □

6. Classification

The following characterization result is easy but essential to understand the classification technique.

Lemma 2. 

A code C of length n over H is cyclic if and only if it is of the form $a{C}_a\oplus b{C}_b$ where:

• $C_a$ is a cyclic $[n,k_1]$ binary code.
• $C_b$ is a cyclic $[n,k_2]$ binary code.

Proof. 

The proof is obtained directly from Theorem 1. □

To classify cyclic codes, we thus have to find all codes that are permutation-equivalent to $a{C}_a\oplus b{C}_b$ for a given pair $(C_a,C_b)$. This is a similar situation to the classification of self-dual codes over $Z_{pq}$ and we follow the method there. Here, SDR stands for System of Distinct Representatives of cyclic code, that is to say, elements that are representative of subsets (here the double cosets) in a set (here the group $S_n$) partition. The following result is an immediate generalization of [23] (Theorem 3.5) from $Z_{pq}$ to H. Its proof is omitted.

Theorem 4 

([15] (Theorem 6)). Let $(C_a,C_b)$ be a pair of codes as defined in Lemma 1, with respective permutation groups A and B. Then, the set

$S_{C_a,C_b}:=\left\{a{C}_a\oplus b\sigma \left(C_b\right)|\sigma \mathrm{runs}\mathrm{over}\mathrm{a}\mathrm{SDR}\mathrm{of}A\setminus S_n/B\right\}$.

forms a set of non-equivalent codes. In particular, $|S_{C_a,C_b}|=|A\setminus S_n/B|$.

The immediate application is the next result.

Corollary 11. 

Let $L_a$ be the set of all non-equivalent $[n,k_1]$ cyclic binary codes. Let $L_b$ be the set of all non-equivalent $[n,k_2]$ cyclic binary codes. Then, the set of all cyclic codes over H is, up to permutation, the disjoint union ${\coprod }_{\begin{array}{c}{C}_a\in L_a\\ C_b\in L_b\end{array}}$$S_{C_a,C_b}$.

Proof. 

By application of the maps ${\alpha }_a$ and ${\alpha }_b$, the following observation is immediate. If the pairs $C_a$,$C_b$ and $C_a^{\prime }$,$C_b^{\prime }$ are distinct, then none of the codes in $S_{C_a,C_b}$ are equivalent to any of the codes in $S_{C_a^{\prime },C_b^{\prime }}$. □

7. LCD Cyclic Codes over $\mathit{H}$

We prepare for the main result of this section by recalling some results on binary cyclic codes.

Definition 4. 

A linear code C of length n over H is said to be reversible if for any code word $c=(c_0,c_1,\dots ,c_{n-1})\in C$ implies $c^r=(c_{n-1},c_{n-2},\dots ,c_1,c_0)\in C.$

Lemma 3 

([24]). If $C_b$ is an $[n,k]$ cyclic code over ${\mathbb{F}}_2$ with generator polynomial $g\left(x\right)$ then it is an LCD code if and only if $g\left(x\right)=\tilde{g}\left(x\right)$ and all the monic irreducible factors of $g\left(x\right)$ have the same multiplicity in $x^n-1$ and $g\left(x\right)$. In particular, for $gcd(n,2)=1$, $C_b$ is an LCD code if and only if it is a reversible code.

Lemma 4 

([25] (Theorem 1)). Let $C_b$ be a cyclic code over a finite field ${\mathbb{F}}_2$ having generator polynomial $g\left(x\right)$. Then $C_b$ is a reversible code if and only if $g\left(x\right)=\tilde{g}\left(x\right).$

Proposition 6 

([22]). Let $C=a{C}_a\oplus b{C}_b$ be any linear code of length n over H. Then C is LCD if and only if $C_b$ is LCD and $C_a=\left\{0\right\}.$

Corollary 12. 

Let $C=a{C}_a\oplus b{C}_b$ be a cyclic code of arbitrary length n over H. Then C is LCD if and only if $C_a=\left\{0\right\}$ and the generator polynomial $g\left(x\right)$ of $C_b$ is self-reciprocal and the multiplicity of each monic irreducible factor of $g\left(x\right)$ is the same in $g\left(x\right)$ and $x^n-1$.

Proof. 

It follows from Lemma 3 and Proposition 6. □

Theorem 5. 

Let n be an odd positive integer and $C=a{C}_a\oplus b{C}_b$ be a cyclic code over H of length n. Then C is LCD if and only if $C_b$ is reversible and $C_a=\left\{0\right\}$.

Proof. 

Let C be an LCD over H. Then $C_b$ is LCD over ${\mathbb{F}}_2$ i.e., $C_b$ is a reversible code over ${\mathbb{F}}_2$ by Lemma 3. Hence, $C=a{C}_a\oplus b{C}_b$ is also a reversible code over H.

Conversely, let C be a reversible code over H. Then $c=a\left\{0\right\}+by\in C$ implies that $c^r\in C$ where $y=(y_0,y_1,\dots ,y_{n-1})\in C_b$. We can write

$$\begin{array}{ccc}\hfill c^r& =& {(a·0+b{y}_0,a·0+b{y}_1,\dots ,a·0+b{y}_{n-1})}^r\hfill \\ & =& (a·0+b{y}_{n-1},a·0+b{y}_{n-2},\dots ,a·0+b{y}_1,a·0+b{y}_0)\hfill \\ & =& a\left\{0\right\}+b{y}^r.\hfill \end{array}$$

This implies that $C_b$ is reversible over ${\mathbb{F}}_2$ and LCD over ${\mathbb{F}}_2$ also. Therefore, C is LCD over H. □

In the following example we will show an LCD cyclic code over $H.$

Example 7. 

The code $C=\left\{000,bbb\right\}$ of length 3 over H is an LCD cyclic where $C_a=\left\{0\right\}$ and $C_b$ is generated by $G_b=\left(\begin{array}{ccc}1& 1& 1\end{array}\right)$.

Then the dual code of C generated by $G=\left(\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\\ 0& b& b\\ b& 0& b\end{array}\right)$. We obtain $C\cap C^{\perp }=\left\{0\right\}$

8. Numerical Results

In the following, we classify, up to equivalence, cyclic codes up to length 7 (Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17 and Table 18). All the computations needed for this section were performed in Magma [20].

The generator matrices of the classified cyclic codes up to n = 7 can be found at: https://www.kau.edu.sa/GetFile.aspx?id=316077&fn=Gen (accessed on 10 September 2023).

• Length 2 (8 Codes)

Table 1. Weight distribution of cyclic codes of length 2.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<2,1>]$
$G_3$ and $G_4$	$[<0,1>,<1,2>,<2,1>]$
$G_5$	$[<0,1>,<2,3>]$
$G_6$ and $G_7$	$[<0,1>,<1,2>,<2,5>]$
$G_8$	$[<0,1>,<1,6>,<2,9>]$

Table 1. Weight distribution of cyclic codes of length 2.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<2,1>]$
$G_3$ and $G_4$	$[<0,1>,<1,2>,<2,1>]$
$G_5$	$[<0,1>,<2,3>]$
$G_6$ and $G_7$	$[<0,1>,<1,2>,<2,5>]$
$G_8$	$[<0,1>,<1,6>,<2,9>]$

Table 2. List of all binary cyclic codes of length 2.

Index	Generator Polynomial
0	$x^2-1$
1	$x-1$
2	1

Table 2. List of all binary cyclic codes of length 2.

Index	Generator Polynomial
0	$x^2-1$
1	$x-1$
2	1

Table 3. Cyclic codes of length 2.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	QSD	Even
$G_1$	2	0	1	Yes	No	Yes
$G_2$	2	1	0	Yes	No	Yes
$G_3$	1	0	2	No	No	No
$G_4$	1	2	0	Yes	Yes	No
$G_5$	2	1	1	Yes	Yes	Yes
$G_6$	1	1	2	No	No	No
$G_7$	1	2	1	Yes	No	No
$G_8$	1	2	2	No	No	No

Table 3. Cyclic codes of length 2.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	QSD	Even
$G_1$	2	0	1	Yes	No	Yes
$G_2$	2	1	0	Yes	No	Yes
$G_3$	1	0	2	No	No	No
$G_4$	1	2	0	Yes	Yes	No
$G_5$	2	1	1	Yes	Yes	Yes
$G_6$	1	1	2	No	No	No
$G_7$	1	2	1	Yes	No	No
$G_8$	1	2	2	No	No	No

• Codes are not self-dual except $G_7$.
• Codes are not LCD except $G_3$.
• Codes neither Type IV nor QT4 except $G_5$.

• Length 3 (15 Codes)

Table 4. Weight distribution of cyclic codes of length 3.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<3,1>]$
$G_3$ and $G_4$	$[<0,1>,<2,3>]$
$G_5$	$[<0,1>,<3,3>]$
$G_6$ and $G_7$	$[<0,1>,<1,3>,<2,3>,<3,1>]$
$G_8$ and $G_9$	$[<0,1>,<2,3>,<3,4>]$
$G_{10}$	$[<0,1>,<2,9>,<3,6>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<1,3>,<2,3>,<3,9>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,3>,<2,15>,<3,13>]$
$G_{15}$	$[<0,1>,<1,9>,<2,27>,<3,27>]$

Table 4. Weight distribution of cyclic codes of length 3.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<3,1>]$
$G_3$ and $G_4$	$[<0,1>,<2,3>]$
$G_5$	$[<0,1>,<3,3>]$
$G_6$ and $G_7$	$[<0,1>,<1,3>,<2,3>,<3,1>]$
$G_8$ and $G_9$	$[<0,1>,<2,3>,<3,4>]$
$G_{10}$	$[<0,1>,<2,9>,<3,6>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<1,3>,<2,3>,<3,9>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,3>,<2,15>,<3,13>]$
$G_{15}$	$[<0,1>,<1,9>,<2,27>,<3,27>]$

Table 5. List of all binary cyclic codes of length 3.

Index	Generator Polynomial
0	$x^3-1$
1	$x-1$
2	$x^2+x+1$
3	1

Table 5. List of all binary cyclic codes of length 3.

Index	Generator Polynomial
0	$x^3-1$
1	$x-1$
2	$x^2+x+1$
3	1

Table 6. Cyclic codes of length 3.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even	LCD
$G_1$	3	0	2	No	No	Yes
$G_2$	3	2	0	Yes	No	No
$G_3$	2	0	1	No	Yes	Yes
$G_4$	2	1	0	Yes	Yes	No
$G_5$	3	2	2	No	No	No
$G_6$	2	0	3	No	Yes	Yes
$G_7$	2	3	0	Yes	Yes	No
$G_8$	2	1	2	No	No	No
$G_9$	2	2	1	No	No	No
$G_{10}$	2	1	1	No	No	No
$G_{11}$	1	2	3	No	No	No
$G_{12}$	1	3	2	No	No	No
$G_{13}$	1	1	3	No	No	No
$G_{14}$	1	3	1	No	No	No
$G_{15}$	1	3	3	No	No	No

Table 6. Cyclic codes of length 3.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even	LCD
$G_1$	3	0	2	No	No	Yes
$G_2$	3	2	0	Yes	No	No
$G_3$	2	0	1	No	Yes	Yes
$G_4$	2	1	0	Yes	Yes	No
$G_5$	3	2	2	No	No	No
$G_6$	2	0	3	No	Yes	Yes
$G_7$	2	3	0	Yes	Yes	No
$G_8$	2	1	2	No	No	No
$G_9$	2	2	1	No	No	No
$G_{10}$	2	1	1	No	No	No
$G_{11}$	1	2	3	No	No	No
$G_{12}$	1	3	2	No	No	No
$G_{13}$	1	1	3	No	No	No
$G_{14}$	1	3	1	No	No	No
$G_{15}$	1	3	3	No	No	No

• Codes are neither self-dual nor Type IV or QT4.
• Codes are not QSD.

• Length 4 (24 Codes)

Table 7. Weight distribution of cyclic codes of length 4.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<4,1>]$
$G_3$ and $G_4$	$[<0,1>,<2,2>,<4,1>]$
$G_5$	$[<0,1>,<4,3>]$
$G_6$ and $G_7$	$[<0,1>,<2,6>,<4,1>]$
$G_8$ and $G_9$	$[<0,1>,<2,2>,<4,5>]$
$G_{10}$	$[<0,1>,<2,6>,<4,9>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<2,6>,<4,9>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,4>,<2,6>,<3,4>,<4,1>]$
$G_{15}$ and $G_{16}$	$[<0,1>,<1,4>,<2,6>,<3,4>,<4,17>]$
$G_{17}$ and $G_{18}$	$[<0,1>,<2,10>,<3,8>,<4,13>]$
$G_{19}$ and $G_{20}$	$[<0,1>,<1,4>,<2,14>,<3,20>,<4,25>]$
$G_{21}$	$[<0,1>,<2,18>,<3,24>,<4,21>]$
$G_{22}$ and $G_{23}$	$[<0,1>,<1,4>,<2,30>,<3,52>,<4,41>]$
$G_{24}$	$[<0,1>,<1,12>,<2,54>,<3,108>,<4,81>]$

Table 7. Weight distribution of cyclic codes of length 4.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<4,1>]$
$G_3$ and $G_4$	$[<0,1>,<2,2>,<4,1>]$
$G_5$	$[<0,1>,<4,3>]$
$G_6$ and $G_7$	$[<0,1>,<2,6>,<4,1>]$
$G_8$ and $G_9$	$[<0,1>,<2,2>,<4,5>]$
$G_{10}$	$[<0,1>,<2,6>,<4,9>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<2,6>,<4,9>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,4>,<2,6>,<3,4>,<4,1>]$
$G_{15}$ and $G_{16}$	$[<0,1>,<1,4>,<2,6>,<3,4>,<4,17>]$
$G_{17}$ and $G_{18}$	$[<0,1>,<2,10>,<3,8>,<4,13>]$
$G_{19}$ and $G_{20}$	$[<0,1>,<1,4>,<2,14>,<3,20>,<4,25>]$
$G_{21}$	$[<0,1>,<2,18>,<3,24>,<4,21>]$
$G_{22}$ and $G_{23}$	$[<0,1>,<1,4>,<2,30>,<3,52>,<4,41>]$
$G_{24}$	$[<0,1>,<1,12>,<2,54>,<3,108>,<4,81>]$

Table 8. List of all binary cyclic codes of length 4.

Index	Generator Polynomial
0	$x^4-1$
1	$x+1$
2	$x^2+1$
3	$x^3+x^2+x+1$
4	1

Table 8. List of all binary cyclic codes of length 4.

Index	Generator Polynomial
0	$x^4-1$
1	$x+1$
2	$x^2+1$
3	$x^3+x^2+x+1$
4	1

Table 9. Cyclic codes of length 4.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even
$G_1$	4	0	3	Yes	Yes
$G_2$	4	3	0	Yes	Yes
$G_3$	2	0	2	Yes	Yes
$G_4$	2	2	0	Yes	Yes
$G_5$	4	3	3	Yes	Yes
$G_6$	2	0	1	No	Yes
$G_7$	2	1	0	Yes	Yes
$G_8$	2	3	2	Yes	Yes
$G_9$	2	2	3	Yes	Yes
$G_{10}$	2	2	2	Yes	Yes
$G_{11}$	2	3	1	No	Yes
$G_{12}$	2	1	3	Yes	Yes
$G_{13}$	1	0	4	No	No
$G_{14}$	1	4	0	Yes	No
$G_{15}$	1	3	4	No	No
$G_{16}$	1	4	3	Yes	No
$G_{17}$	2	2	1	No	No
$G_{18}$	2	1	2	Yes	No
$G_{19}$	1	2	4	No	No
$G_{20}$	1	4	2	Yes	No
$G_{21}$	2	1	1	No	No
$G_{22}$	1	1	4	No	No
$G_{23}$	1	4	1	No	No
$G_{24}$	1	4	4	No	No

Table 9. Cyclic codes of length 4.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even
$G_1$	4	0	3	Yes	Yes
$G_2$	4	3	0	Yes	Yes
$G_3$	2	0	2	Yes	Yes
$G_4$	2	2	0	Yes	Yes
$G_5$	4	3	3	Yes	Yes
$G_6$	2	0	1	No	Yes
$G_7$	2	1	0	Yes	Yes
$G_8$	2	3	2	Yes	Yes
$G_9$	2	2	3	Yes	Yes
$G_{10}$	2	2	2	Yes	Yes
$G_{11}$	2	3	1	No	Yes
$G_{12}$	2	1	3	Yes	Yes
$G_{13}$	1	0	4	No	No
$G_{14}$	1	4	0	Yes	No
$G_{15}$	1	3	4	No	No
$G_{16}$	1	4	3	Yes	No
$G_{17}$	2	2	1	No	No
$G_{18}$	2	1	2	Yes	No
$G_{19}$	1	2	4	No	No
$G_{20}$	1	4	2	Yes	No
$G_{21}$	2	1	1	No	No
$G_{22}$	1	1	4	No	No
$G_{23}$	1	4	1	No	No
$G_{24}$	1	4	4	No	No

• Codes are not self-dual except $G_{20}$.
• Codes are not QSD except $G_{10}$,$G_{12}$ and $G_{14}$.
• Codes are not Type IV except $G_{10}$ and $G_{12}$.
• Codes are not QT4 except $G_{10}$ and $G_{12}$.
• Codes are not LCD except $G_{13}$.

• Length 5 (15 Codes)

Table 10. Weight distribution of cyclic codes of length 5.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<5,1>]$
$G_3$ and $G_4$	$[<0,1>,<2,10>,<4,5>]$
$G_5$	$[<0,1>,<5,3>]$
$G_6$ and $G_7$	$[<0,1>,<1,5>,<2,10>,<3,10>,<4,5>,<5,1>]$
$G_8$ and $G_9$	$[<0,1>,<2,10>,<4,5>,<5,16>]$
$G_{10}$	$[<0,1>,<2,30>,<3,60>,<4,105>,<5,60>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<1,5>,<2,10>,<3,10>,<4,5>,<5,33>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,5>,<2,50>,<3,130>,<4,205>,<5,121>]$
$G_{15}$	$[<0,1>,<1,15>,<2,90>,<3,270>,<4,405>,<5,243>]$

Table 10. Weight distribution of cyclic codes of length 5.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<5,1>]$
$G_3$ and $G_4$	$[<0,1>,<2,10>,<4,5>]$
$G_5$	$[<0,1>,<5,3>]$
$G_6$ and $G_7$	$[<0,1>,<1,5>,<2,10>,<3,10>,<4,5>,<5,1>]$
$G_8$ and $G_9$	$[<0,1>,<2,10>,<4,5>,<5,16>]$
$G_{10}$	$[<0,1>,<2,30>,<3,60>,<4,105>,<5,60>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<1,5>,<2,10>,<3,10>,<4,5>,<5,33>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,5>,<2,50>,<3,130>,<4,205>,<5,121>]$
$G_{15}$	$[<0,1>,<1,15>,<2,90>,<3,270>,<4,405>,<5,243>]$

Table 11. List of all binary cyclic codes of length 5.

Index	Generator Polynomial
0	$x^5-1$
1	$x+1$
2	$x^4+x^3+x^2+x+1$
3	1

Table 11. List of all binary cyclic codes of length 5.

Index	Generator Polynomial
0	$x^5-1$
1	$x+1$
2	$x^4+x^3+x^2+x+1$
3	1

Table 12. Cyclic codes of length 5.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even	LCD
$G_1$	5	0	2	No	No	Yes
$G_2$	5	2	0	Yes	No	No
$G_3$	2	0	1	No	Yes	Yes
$G_4$	2	1	0	Yes	Yes	No
$G_5$	5	2	2	No	No	No
$G_6$	1	0	3	No	No	Yes
$G_7$	1	3	0	Yes	No	No
$G_8$	2	1	2	No	No	No
$G_9$	2	2	1	No	No	No
$G_{10}$	2	1	1	No	No	No
$G_{11}$	1	2	3	No	No	No
$G_{12}$	1	3	2	No	No	No
$G_{13}$	1	1	3	No	No	No
$G_{14}$	1	3	1	No	No	No
$G_{15}$	1	3	3	No	No	No

Table 12. Cyclic codes of length 5.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even	LCD
$G_1$	5	0	2	No	No	Yes
$G_2$	5	2	0	Yes	No	No
$G_3$	2	0	1	No	Yes	Yes
$G_4$	2	1	0	Yes	Yes	No
$G_5$	5	2	2	No	No	No
$G_6$	1	0	3	No	No	Yes
$G_7$	1	3	0	Yes	No	No
$G_8$	2	1	2	No	No	No
$G_9$	2	2	1	No	No	No
$G_{10}$	2	1	1	No	No	No
$G_{11}$	1	2	3	No	No	No
$G_{12}$	1	3	2	No	No	No
$G_{13}$	1	1	3	No	No	No
$G_{14}$	1	3	1	No	No	No
$G_{15}$	1	3	3	No	No	No

• Codes are neither self-dual nor Type IV or QT4.
• Codes are not QSD except $G_7$.

• Length 6 (80 Codes)

Table 13. Weight distribution of cyclic codes of length 6.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<6,1>]$
$G_3$ and $G_4$	$[<0,1>,<3,2>,<6,1>]$
$G_5$ and $G_6$	$[<0,1>,<4,3>]$
$G_7$ and $G_8$	$[<0,1>,<2,3>,<4,3>,<6,1>]$
$G_9$ and $G_{10}$	$[<0,1>,<2,3>,<3,8><4,3>,<6,1>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<2,6>,<4,9>],$
$G_{13}$ and $G_{14}$	$[<0,1>,<2,15>,<4,15>,<6,1>]$
$G_{15}$ and $G_{16}$	$[<0,1>,<1,6>,<2,15>,<3,20>,<4,15>,<5,6><6,1>]$
$G_{17}$	$[<0,1>,<6,3>]$
$G_{18}$ and $G_{19}$	$[<0,1>,<3,2>,<6,5]$
$G_{20}$ and $G_{21}$	$[<0,1>,<4,3>,<6,4>]$
$G_{22}$ and $G_{23}$	$[<0,1>,<2,3>,<4,3>,<6,9>]$
$G_{24}$ and $G_{25}$	$[<0,1>,<2,3>,<3,8>,<4,3>,<6,17>]$
$G_{26}$ and $G_{27}$	$[<0,1>,<2,6>,<4,9>,<6,16>]$
$G_{28}$ and $G_{29}$	$[<0,1>,<1,3>,<2,6>,<3,10>,<4,9>,<5,3>,<6,32>]$
$G_{30}$ and $G_{31}$	$[<0,1>,<1,6>,<2,15>,<3,20>,<4,15>,<5,6>,<6,65>]$
$G_{32}$	$[<0,1>,<3,6>,<6,9>]$
$G_{33}$ and $G_{34}$	$[<0,1>,<3,2>,<4,3>,<5,6><6,4>]$
$G_{35}$ and $G_{36}$	$[<0,1>,<2,3>,<3,2>,<4,9>,<5,6>,<6,11>]$
$G_{37}$ and $G_{38}$	$[<0,1>,<2,3>,<3,12>,<4,15>,<5,12>,<6,21>]$
$G_{39}$ and $G_{40}$	$[<0,1>,<2,6>,<3,8>,<4,9>,<5,24>,<6,16>]$
$G_{41}$ and $G_{42}$	$[<0,1>,<2,15>,<3,8>,<4,39>,<5,24>,<6,41>]$
$G_{43}$ and $G_{44}$	$[<0,1>,<1,6>,<2,15>,<3,36>,<4,63>,<5,54>,<6,81>]$
$G_{45}$	$[<0,1>,<4,9>,<6,6>]$
$G_{46}$ and $G_{47}$	$[<0,1>,<2,3>,<4,15>,<6,13>]$
$G_{48}$ and $G_{49}$	$[<0,1>,<2,3>,<3,8>,<4,15>,<5,24>,<6,13>]$
$G_{50}$ and $G_{51}$	$[<0,1>,<2,6>,<4,21>,<5,24>,<6,12>]$
$G_{52}$ and $G_{53}$	$[<0,1>,<2,15>,<4,39>,<5,48>,<6,25>]$
$G_{54}$ and $G_{55}$	$[<0,1>,<1,6>,<2,15>,<3,20>,<4,63>,<5,102>,<6,49>]$
$G_{56}$	$[<0,1>,<2,9>,<4,27>,<6,27>]$
$G_{57}$ and $G_{58}$	$[<0,1>,<2,9>,<3,8>,<4,51>,<5,24>,<6,35>]$
$G_{59}$ and $G_{60}$	$[<0,1>,<2,9>,<3,12>,<4,39>,<5,36>,<6,31>]$
$G_{61}$ and $G_{62}$	$[<0,1>,<2,21>,<3,24>,<4,75>,<5,72>,<6,63>]$
$G_{63}$ and $G_{64}$	$[<0,1>,<1,6>,<2,27>,<3,68>,<4,135>,<5,150>,<6,125>]$
$G_{65}$	$[<0,1>,<2,9>,<3,24>,<4,99>,<5,72>,<6,51>]$
$G_{66}$ and $G_{67}$	$[<0,1>,<2,9>,<3,32>,<4,75>,<5,96>,<6,43>]$
$G_{68}$ and $G_{69}$	$[<0,1>,<2,21>,<3,56>,<4,171>,<5,168>,<6,95>]$
$G_{70}$ and $G_{71}$	$[<0,1>,<1,6>,<2,27>,<3,132>,<4,327>,<5,342>,<6,189>]$
$G_{72}$	$[<0,1>,<2,18>,<3,12>,<4,81>,<5,108>,<6,36>]$
$G_{73}$ and $G_{74}$	$[<0,1>,<2,27>,<3,48>,<4,159>,<5,192>,<6,85>]$
$G_{75}$ and $G_{76}$	$[<0,1>,<1,6>,<2,39>,<3,116>,<4,303>,<5,390>,<6,169>]$
$G_{77}$	$[<0,1>,<2,45>,<3,120>,<4,315>,<5,360>,<6,183>]$
$G_{78}$ and $G_{79}$	$[<0,1>,<1,6>,<2,75>,<3,260>,<4,615>,<5,726>,<6,365>]$
$G_{80}$	$[<0,1>,<1,18>,<2,135>,<3,540>,<4,1215>,<5,1458>,<6,729>]$

Table 13. Weight distribution of cyclic codes of length 6.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<6,1>]$
$G_3$ and $G_4$	$[<0,1>,<3,2>,<6,1>]$
$G_5$ and $G_6$	$[<0,1>,<4,3>]$
$G_7$ and $G_8$	$[<0,1>,<2,3>,<4,3>,<6,1>]$
$G_9$ and $G_{10}$	$[<0,1>,<2,3>,<3,8><4,3>,<6,1>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<2,6>,<4,9>],$
$G_{13}$ and $G_{14}$	$[<0,1>,<2,15>,<4,15>,<6,1>]$
$G_{15}$ and $G_{16}$	$[<0,1>,<1,6>,<2,15>,<3,20>,<4,15>,<5,6><6,1>]$
$G_{17}$	$[<0,1>,<6,3>]$
$G_{18}$ and $G_{19}$	$[<0,1>,<3,2>,<6,5]$
$G_{20}$ and $G_{21}$	$[<0,1>,<4,3>,<6,4>]$
$G_{22}$ and $G_{23}$	$[<0,1>,<2,3>,<4,3>,<6,9>]$
$G_{24}$ and $G_{25}$	$[<0,1>,<2,3>,<3,8>,<4,3>,<6,17>]$
$G_{26}$ and $G_{27}$	$[<0,1>,<2,6>,<4,9>,<6,16>]$
$G_{28}$ and $G_{29}$	$[<0,1>,<1,3>,<2,6>,<3,10>,<4,9>,<5,3>,<6,32>]$
$G_{30}$ and $G_{31}$	$[<0,1>,<1,6>,<2,15>,<3,20>,<4,15>,<5,6>,<6,65>]$
$G_{32}$	$[<0,1>,<3,6>,<6,9>]$
$G_{33}$ and $G_{34}$	$[<0,1>,<3,2>,<4,3>,<5,6><6,4>]$
$G_{35}$ and $G_{36}$	$[<0,1>,<2,3>,<3,2>,<4,9>,<5,6>,<6,11>]$
$G_{37}$ and $G_{38}$	$[<0,1>,<2,3>,<3,12>,<4,15>,<5,12>,<6,21>]$
$G_{39}$ and $G_{40}$	$[<0,1>,<2,6>,<3,8>,<4,9>,<5,24>,<6,16>]$
$G_{41}$ and $G_{42}$	$[<0,1>,<2,15>,<3,8>,<4,39>,<5,24>,<6,41>]$
$G_{43}$ and $G_{44}$	$[<0,1>,<1,6>,<2,15>,<3,36>,<4,63>,<5,54>,<6,81>]$
$G_{45}$	$[<0,1>,<4,9>,<6,6>]$
$G_{46}$ and $G_{47}$	$[<0,1>,<2,3>,<4,15>,<6,13>]$
$G_{48}$ and $G_{49}$	$[<0,1>,<2,3>,<3,8>,<4,15>,<5,24>,<6,13>]$
$G_{50}$ and $G_{51}$	$[<0,1>,<2,6>,<4,21>,<5,24>,<6,12>]$
$G_{52}$ and $G_{53}$	$[<0,1>,<2,15>,<4,39>,<5,48>,<6,25>]$
$G_{54}$ and $G_{55}$	$[<0,1>,<1,6>,<2,15>,<3,20>,<4,63>,<5,102>,<6,49>]$
$G_{56}$	$[<0,1>,<2,9>,<4,27>,<6,27>]$
$G_{57}$ and $G_{58}$	$[<0,1>,<2,9>,<3,8>,<4,51>,<5,24>,<6,35>]$
$G_{59}$ and $G_{60}$	$[<0,1>,<2,9>,<3,12>,<4,39>,<5,36>,<6,31>]$
$G_{61}$ and $G_{62}$	$[<0,1>,<2,21>,<3,24>,<4,75>,<5,72>,<6,63>]$
$G_{63}$ and $G_{64}$	$[<0,1>,<1,6>,<2,27>,<3,68>,<4,135>,<5,150>,<6,125>]$
$G_{65}$	$[<0,1>,<2,9>,<3,24>,<4,99>,<5,72>,<6,51>]$
$G_{66}$ and $G_{67}$	$[<0,1>,<2,9>,<3,32>,<4,75>,<5,96>,<6,43>]$
$G_{68}$ and $G_{69}$	$[<0,1>,<2,21>,<3,56>,<4,171>,<5,168>,<6,95>]$
$G_{70}$ and $G_{71}$	$[<0,1>,<1,6>,<2,27>,<3,132>,<4,327>,<5,342>,<6,189>]$
$G_{72}$	$[<0,1>,<2,18>,<3,12>,<4,81>,<5,108>,<6,36>]$
$G_{73}$ and $G_{74}$	$[<0,1>,<2,27>,<3,48>,<4,159>,<5,192>,<6,85>]$
$G_{75}$ and $G_{76}$	$[<0,1>,<1,6>,<2,39>,<3,116>,<4,303>,<5,390>,<6,169>]$
$G_{77}$	$[<0,1>,<2,45>,<3,120>,<4,315>,<5,360>,<6,183>]$
$G_{78}$ and $G_{79}$	$[<0,1>,<1,6>,<2,75>,<3,260>,<4,615>,<5,726>,<6,365>]$
$G_{80}$	$[<0,1>,<1,18>,<2,135>,<3,540>,<4,1215>,<5,1458>,<6,729>]$

Table 14. List of all binary cyclic codes of length 6.

Index	Generator Polynomial
0	$x^6-1$
1	$x+1$
2	$x^2+1$
3	$x^2+x+1$
4	$x^3+1$
5	$x^4+x^2+x+1$
6	$x^4+x^3+x+1$
7	$x^5+x^4+x^3+x^2+x+1$
8	1

Table 14. List of all binary cyclic codes of length 6.

Index	Generator Polynomial
0	$x^6-1$
1	$x+1$
2	$x^2+1$
3	$x^2+x+1$
4	$x^3+1$
5	$x^4+x^2+x+1$
6	$x^4+x^3+x+1$
7	$x^5+x^4+x^3+x^2+x+1$
8	1

Table 15. Cyclic codes of length 6.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even
$G_1$	6	0	7	Yes	Yes
$G_2$	6	7	0	Yes	Yes
$G_3$	3	0	5	No	No
$G_4$	3	5	0	Yes	No
$G_5$	4	0	6	Yes	Yes
$G_6$	4	6	0	Yes	Yes
$G_7$	2	0	4	Yes	Yes
$G_8$	2	4	0	Yes	Yes
$G_9$	2	0	3	No	No
$G_{10}$	2	3	0	Yes	No
$G_{11}$	2	0	2	No	Yes
$G_{12}$	2	2	0	Yes	Yes
$G_{13}$	2	0	1	No	Yes
$G_{14}$	2	1	0	Yes	Yes
$G_{15}$	1	0	8	No	No
$G_{16}$	1	8	0	Yes	No
$G_{17}$	6	7	7	Yes	Yes
$G_{18}$	3	7	5	No	No
$G_{19}$	3	5	7	Yes	No
$G_{20}$	4	7	6	Yes	Yes
$G_{21}$	4	6	7	Yes	Yes
$G_{22}$	2	7	4	Yes	Yes
$G_{23}$	2	4	7	Yes	Yes
$G_{24}$	2	7	3	No	No
$G_{25}$	2	3	7	Yes	No
$G_{26}$	2	7	2	No	Yes
$G_{27}$	2	2	7	Yes	Yes
$G_{28}$	1	7	1	No	No
$G_{29}$	1	1	7	Yes	No
$G_{30}$	1	7	8	No	No
$G_{31}$	1	8	7	Yes	No
$G_{32}$	3	5	5	No	No
$G_{33}$	3	5	6	Yes	No
$G_{34}$	3	6	5	No	No
$G_{35}$	2	5	4	Yes	No
$G_{36}$	2	4	5	No	No
$G_{37}$	2	5	3	No	No
$G_{38}$	2	3	5	No	No
$G_{39}$	2	5	2	No	No
$G_{40}$	2	2	5	No	No
$G_{41}$	2	5	1	No	No
$G_{42}$	2	1	5	No	No
$G_{43}$	1	5	8	No	No
$G_{44}$	1	8	5	No	No
$G_{45}$	4	6	6	Yes	Yes
$G_{46}$	2	6	4	Yes	Yes
$G_{47}$	2	4	6	Yes	Yes
$G_{48}$	2	6	3	No	No
$G_{49}$	2	3	6	Yes	No
$G_{50}$	2	6	2	No	No
$G_{51}$	2	2	6	Yes	No
$G_{52}$	2	6	1	No	No
$G_{53}$	2	1	6	Yes	No
$G_{54}$	1	6	8	No	No
$G_{55}$	1	8	6	Yes	No
$G_{56}$	2	4	4	Yes	Yes
$G_{57}$	2	4	3	No	No
$G_{58}$	2	3	4	Yes	No
$G_{59}$	2	4	2	No	No
$G_{60}$	2	2	4	Yes	No
$G_{61}$	2	4	1	No	No
$G_{62}$	2	1	4	Yes	No
$G_{63}$	1	4	8	No	No
$G_{64}$	1	8	4	Yes	No
$G_{65}$	2	3	3	No	No
$G_{66}$	2	3	2	No	No
$G_{67}$	2	2	3	No	No
$G_{68}$	2	3	1	No	No
$G_{69}$	2	1	3	No	No
$G_{70}$	1	3	8	No	No
$G_{71}$	1	8	3	No	No
$G_{72}$	2	2	2	No	No
$G_{73}$	2	2	1	No	No
$G_{74}$	2	1	2	No	No
$G_{75}$	1	2	8	No	No
$G_{76}$	1	8	2	No	No
$G_{77}$	2	1	1	No	No
$G_{78}$	1	1	8	No	No
$G_{79}$	1	8	1	No	No
$G_{80}$	1	8	8	No	No

Table 15. Cyclic codes of length 6.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even
$G_1$	6	0	7	Yes	Yes
$G_2$	6	7	0	Yes	Yes
$G_3$	3	0	5	No	No
$G_4$	3	5	0	Yes	No
$G_5$	4	0	6	Yes	Yes
$G_6$	4	6	0	Yes	Yes
$G_7$	2	0	4	Yes	Yes
$G_8$	2	4	0	Yes	Yes
$G_9$	2	0	3	No	No
$G_{10}$	2	3	0	Yes	No
$G_{11}$	2	0	2	No	Yes
$G_{12}$	2	2	0	Yes	Yes
$G_{13}$	2	0	1	No	Yes
$G_{14}$	2	1	0	Yes	Yes
$G_{15}$	1	0	8	No	No
$G_{16}$	1	8	0	Yes	No
$G_{17}$	6	7	7	Yes	Yes
$G_{18}$	3	7	5	No	No
$G_{19}$	3	5	7	Yes	No
$G_{20}$	4	7	6	Yes	Yes
$G_{21}$	4	6	7	Yes	Yes
$G_{22}$	2	7	4	Yes	Yes
$G_{23}$	2	4	7	Yes	Yes
$G_{24}$	2	7	3	No	No
$G_{25}$	2	3	7	Yes	No
$G_{26}$	2	7	2	No	Yes
$G_{27}$	2	2	7	Yes	Yes
$G_{28}$	1	7	1	No	No
$G_{29}$	1	1	7	Yes	No
$G_{30}$	1	7	8	No	No
$G_{31}$	1	8	7	Yes	No
$G_{32}$	3	5	5	No	No
$G_{33}$	3	5	6	Yes	No
$G_{34}$	3	6	5	No	No
$G_{35}$	2	5	4	Yes	No
$G_{36}$	2	4	5	No	No
$G_{37}$	2	5	3	No	No
$G_{38}$	2	3	5	No	No
$G_{39}$	2	5	2	No	No
$G_{40}$	2	2	5	No	No
$G_{41}$	2	5	1	No	No
$G_{42}$	2	1	5	No	No
$G_{43}$	1	5	8	No	No
$G_{44}$	1	8	5	No	No
$G_{45}$	4	6	6	Yes	Yes
$G_{46}$	2	6	4	Yes	Yes
$G_{47}$	2	4	6	Yes	Yes
$G_{48}$	2	6	3	No	No
$G_{49}$	2	3	6	Yes	No
$G_{50}$	2	6	2	No	No
$G_{51}$	2	2	6	Yes	No
$G_{52}$	2	6	1	No	No
$G_{53}$	2	1	6	Yes	No
$G_{54}$	1	6	8	No	No
$G_{55}$	1	8	6	Yes	No
$G_{56}$	2	4	4	Yes	Yes
$G_{57}$	2	4	3	No	No
$G_{58}$	2	3	4	Yes	No
$G_{59}$	2	4	2	No	No
$G_{60}$	2	2	4	Yes	No
$G_{61}$	2	4	1	No	No
$G_{62}$	2	1	4	Yes	No
$G_{63}$	1	4	8	No	No
$G_{64}$	1	8	4	Yes	No
$G_{65}$	2	3	3	No	No
$G_{66}$	2	3	2	No	No
$G_{67}$	2	2	3	No	No
$G_{68}$	2	3	1	No	No
$G_{69}$	2	1	3	No	No
$G_{70}$	1	3	8	No	No
$G_{71}$	1	8	3	No	No
$G_{72}$	2	2	2	No	No
$G_{73}$	2	2	1	No	No
$G_{74}$	2	1	2	No	No
$G_{75}$	1	2	8	No	No
$G_{76}$	1	8	2	No	No
$G_{77}$	2	1	1	No	No
$G_{78}$	1	1	8	No	No
$G_{79}$	1	8	1	No	No
$G_{80}$	1	8	8	No	No

• Codes are not self-dual except $G_{64}$.
• Codes are not QSD except $G_{16}$,$G_{29}$, $G_{49}$, $G_{51}$ and $G_{56}$.
• Codes are not Type IV except $G_{56}$.
• Codes are not QT4 except $G_{51}$ and $G_{56}$.
• Codes are not LCD except $G_3$, $G_{11}$ and $G_{15}$.

• Length 7 (63 Codes)

Table 16. Weight distribution of cyclic codes of length 7.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<7,1>]$
$G_3$ and $G_4$	$[<0,1>,<4,7>]$
$G_5$ and $G_6$	$[<0,1>,<4,7>]$
$G_7$ and $G_8$	$[<0,1>,<3,7>,<4,7>,<7,1>]$
$G_9$ and $G_{10}$	$[<0,1>,<3,7>,<4,7>,<7,1>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<2,21>,<4,35>,<6,7>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,35>,$
	$<5,21>,<6,7><7,1>]$
$G_{15}$	$[<0,1>,<7,3>]$
$G_{16}$ and $G_{17}$	$[<0,1>,<4,7>,<7,8>]$
$G_{18}$ and $G_{19}$	$[<0,1>,<4,7>,<7,8>]$
$G_{20}$ and $G_{21}$	$[<0,1>,<3,7>,<4,7>,<7,17>]$
$G_{22}$ and $G_{23}$	$[<0,1>,<3,7>,<4,7>,<7,17>]$
$G_{24}$ and $G_{25}$	$[<0,1>,<2,21>,<4,35>,<6,7>,<7,64>]$
$G_{26}$ and $G_{27}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,35>,<5,21>,<6,7>,<7,129>]$
$G_{28}$	$[<0,1>,<4,21>,<6,42>]$
$G_{29}$ and $G_{30}$	$[<0,1>,<4,14>,<5,21>,<6,21>,<7,7>$
$G_{31}$ and $G_{32}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{33}$ and $G_{34}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{35}$ and $G_{36}$	$[<0,1>,<2,21>,<4,91>,<5,168>,<6,175>,<7,56>]$
$G_{37}$ and $G_{38}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,147>,<5,357>,<6,343>,<7,113>]$
$G_{39}$	$[<0,1>,<4,21>,<6,42>]$
$G_{40}$ and $G_{41}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{42}$ and $G_{43}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{44}$ and $G_{45}$	$[<0,1>,<2,21>,<4,91>,<5,168>,<6,175>,<7,56>]$
$G_{46}$ and $G_{47}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,147>,<5,357>,<6,343>,<7,113>]$
$G_{48}$	$[<0,1>,<3,21>,<4,21>,<5,126>,<6,42>,<7,45>]$
$G_{49}$ and $G_{50}$	$[<0,1>,<3,14>,<4,49>,<5,84>,<6,70>,<7,38>]$
$G_{51}$ and $G_{52}$	$[<0,1>,<2,21>,<3,28>,<4,203>,<5,336>,<6,287>,<7,148>]$
$G_{53}$ and $G_{54}$	$[<0,1>,<1,7>,<2,21>,<3,91>,<4,371>,<5,693>,<6,567>,<7,297>]$
$G_{55}$	$[<0,1>,<3,21>,<4,21>,<5,126>,<6,42>,<7,45>]$
$G_{56}$ and $G_{57}$	$[<0,1>,<2,21>,<3,28>,<4,203>,<5,336>,<6,287>,<7,148>]$
$G_{58}$ and $G_{59}$	$[<0,1>,<1,7>,<2,21>,<3,91>,<4,371>,<5,693>,<6,567>,<7,297>]$
$G_{60}$	$[<0,1>,<2,63>,<3,210>,<4,735>,<5,1260>,<6,1281>,<7,546>]$
$G_{61}$ and $G_{62}$	$[<0,1>,<1,7>,<2,105>,<3,455>,<4,1435>,<5,2541>,<6,2555>,<7,1093>]$
$G_{63}$	$[<0,1>,<1,21>,<2,189>,<3,945>,<4,2835>,<5,5103>,<6,5103>,<7,2187>]$

Table 16. Weight distribution of cyclic codes of length 7.

Generator Matrix	Weight Distribution
$G_1$ and $G_2$	$[<0,1>,<7,1>]$
$G_3$ and $G_4$	$[<0,1>,<4,7>]$
$G_5$ and $G_6$	$[<0,1>,<4,7>]$
$G_7$ and $G_8$	$[<0,1>,<3,7>,<4,7>,<7,1>]$
$G_9$ and $G_{10}$	$[<0,1>,<3,7>,<4,7>,<7,1>]$
$G_{11}$ and $G_{12}$	$[<0,1>,<2,21>,<4,35>,<6,7>]$
$G_{13}$ and $G_{14}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,35>,$
	$<5,21>,<6,7><7,1>]$
$G_{15}$	$[<0,1>,<7,3>]$
$G_{16}$ and $G_{17}$	$[<0,1>,<4,7>,<7,8>]$
$G_{18}$ and $G_{19}$	$[<0,1>,<4,7>,<7,8>]$
$G_{20}$ and $G_{21}$	$[<0,1>,<3,7>,<4,7>,<7,17>]$
$G_{22}$ and $G_{23}$	$[<0,1>,<3,7>,<4,7>,<7,17>]$
$G_{24}$ and $G_{25}$	$[<0,1>,<2,21>,<4,35>,<6,7>,<7,64>]$
$G_{26}$ and $G_{27}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,35>,<5,21>,<6,7>,<7,129>]$
$G_{28}$	$[<0,1>,<4,21>,<6,42>]$
$G_{29}$ and $G_{30}$	$[<0,1>,<4,14>,<5,21>,<6,21>,<7,7>$
$G_{31}$ and $G_{32}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{33}$ and $G_{34}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{35}$ and $G_{36}$	$[<0,1>,<2,21>,<4,91>,<5,168>,<6,175>,<7,56>]$
$G_{37}$ and $G_{38}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,147>,<5,357>,<6,343>,<7,113>]$
$G_{39}$	$[<0,1>,<4,21>,<6,42>]$
$G_{40}$ and $G_{41}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{42}$ and $G_{43}$	$[<0,1>,<3,7>,<4,21>,<5,42>,<6,42>,<7,15>]$
$G_{44}$ and $G_{45}$	$[<0,1>,<2,21>,<4,91>,<5,168>,<6,175>,<7,56>]$
$G_{46}$ and $G_{47}$	$[<0,1>,<1,7>,<2,21>,<3,35>,<4,147>,<5,357>,<6,343>,<7,113>]$
$G_{48}$	$[<0,1>,<3,21>,<4,21>,<5,126>,<6,42>,<7,45>]$
$G_{49}$ and $G_{50}$	$[<0,1>,<3,14>,<4,49>,<5,84>,<6,70>,<7,38>]$
$G_{51}$ and $G_{52}$	$[<0,1>,<2,21>,<3,28>,<4,203>,<5,336>,<6,287>,<7,148>]$
$G_{53}$ and $G_{54}$	$[<0,1>,<1,7>,<2,21>,<3,91>,<4,371>,<5,693>,<6,567>,<7,297>]$
$G_{55}$	$[<0,1>,<3,21>,<4,21>,<5,126>,<6,42>,<7,45>]$
$G_{56}$ and $G_{57}$	$[<0,1>,<2,21>,<3,28>,<4,203>,<5,336>,<6,287>,<7,148>]$
$G_{58}$ and $G_{59}$	$[<0,1>,<1,7>,<2,21>,<3,91>,<4,371>,<5,693>,<6,567>,<7,297>]$
$G_{60}$	$[<0,1>,<2,63>,<3,210>,<4,735>,<5,1260>,<6,1281>,<7,546>]$
$G_{61}$ and $G_{62}$	$[<0,1>,<1,7>,<2,105>,<3,455>,<4,1435>,<5,2541>,<6,2555>,<7,1093>]$
$G_{63}$	$[<0,1>,<1,21>,<2,189>,<3,945>,<4,2835>,<5,5103>,<6,5103>,<7,2187>]$

Table 17. List of all binary cyclic codes of length 7.

Index	Generator Polynomial
0	$x^7-1$
1	$x+1$
2	$x^3+x+1$
3	$x^3+x^2+1$
4	$x^4+x^3+x^2+1$
5	$x^4+x^2+x+1$
6	$x^6+x^5+x^4+x^3+x^2+x+1$
7	1

Table 17. List of all binary cyclic codes of length 7.

Index	Generator Polynomial
0	$x^7-1$
1	$x+1$
2	$x^3+x+1$
3	$x^3+x^2+1$
4	$x^4+x^3+x^2+1$
5	$x^4+x^2+x+1$
6	$x^6+x^5+x^4+x^3+x^2+x+1$
7	1

Table 18. Cyclic codes of length 7.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even
$G_1$	7	0	6	No	No
$G_2$	7	6	0	Yes	No
$G_3$	4	0	4	Yes	Yes
$G_4$	4	4	0	Yes	Yes
$G_5$	4	0	5	Yes	Yes
$G_6$	4	5	0	Yes	Yes
$G_7$	3	0	2	No	No
$G_8$	3	2	0	Yes	No
$G_9$	3	0	3	No	No
$G_{10}$	3	3	0	Yes	No
$G_{11}$	2	0	1	No	Yes
$G_{12}$	2	1	0	Yes	Yes
$G_{13}$	1	0	7	No	No
$G_{14}$	1	7	0	Yes	No
$G_{15}$	7	6	6	No	No
$G_{16}$	4	6	4	Yes	No
$G_{17}$	4	4	6	No	No
$G_{18}$	4	6	5	Yes	No
$G_{19}$	4	5	6	No	No
$G_{20}$	3	6	2	No	No
$G_{21}$	3	2	6	No	No
$G_{22}$	3	6	3	No	No
$G_{23}$	3	3	6	No	No
$G_{24}$	2	6	1	No	No
$G_{25}$	2	1	6	No	No
$G_{26}$	1	6	7	No	No
$G_{27}$	1	7	6	No	No
$G_{28}$	4	4	4	Yes	Yes
$G_{29}$	4	4	5	Yes	No
$G_{31}$	3	4	2	No	No
$G_{32}$	3	2	4	Yes	No
$G_{33}$	3	4	3	No	No
$G_{34}$	3	3	4	Yes	No
$G_{35}$	2	4	1	No	No
$G_{36}$	2	1	4	Yes	No
$G_{37}$	1	4	7	No	No
$G_{38}$	1	7	4	Yes	No
$G_{39}$	4	5	5	Yes	Yes
$G_{42}$	3	5	3	No	No
$G_{43}$	3	3	5	Yes	No
$G_{44}$	2	5	1	No	No
$G_{45}$	2	1	5	Yes	No
$G_{46}$	1	5	7	No	No
$G_{47}$	1	7	5	Yes	No
$G_{48}$	3	2	2	No	No
$G_{49}$	3	2	3	No	No
$G_{51}$	2	2	1	No	No
$G_{52}$	2	1	2	No	No
$G_{53}$	1	2	7	No	No
$G_{54}$	1	7	2	No	No
$G_{55}$	3	3	3	No	No
$G_{56}$	2	3	1	No	No
$G_{57}$	2	1	3	No	No
$G_{58}$	1	3	7	No	No
$G_{59}$	1	7	3	No	No
$G_{60}$	2	1	1	No	No
$G_{61}$	1	1	7	No	No
$G_{62}$	1	7	1	No	No
$G_{63}$	1	7	7	No	No

Table 18. Cyclic codes of length 7.

Generator Matrix	${\mathit{d}}_{\mathit{H}}$	Index ${\mathit{C}}_{\mathit{a}}$	Index ${\mathit{C}}_{\mathit{b}}$	Self-Orthogonal	Even
$G_1$	7	0	6	No	No
$G_2$	7	6	0	Yes	No
$G_3$	4	0	4	Yes	Yes
$G_4$	4	4	0	Yes	Yes
$G_5$	4	0	5	Yes	Yes
$G_6$	4	5	0	Yes	Yes
$G_7$	3	0	2	No	No
$G_8$	3	2	0	Yes	No
$G_9$	3	0	3	No	No
$G_{10}$	3	3	0	Yes	No
$G_{11}$	2	0	1	No	Yes
$G_{12}$	2	1	0	Yes	Yes
$G_{13}$	1	0	7	No	No
$G_{14}$	1	7	0	Yes	No
$G_{15}$	7	6	6	No	No
$G_{16}$	4	6	4	Yes	No
$G_{17}$	4	4	6	No	No
$G_{18}$	4	6	5	Yes	No
$G_{19}$	4	5	6	No	No
$G_{20}$	3	6	2	No	No
$G_{21}$	3	2	6	No	No
$G_{22}$	3	6	3	No	No
$G_{23}$	3	3	6	No	No
$G_{24}$	2	6	1	No	No
$G_{25}$	2	1	6	No	No
$G_{26}$	1	6	7	No	No
$G_{27}$	1	7	6	No	No
$G_{28}$	4	4	4	Yes	Yes
$G_{29}$	4	4	5	Yes	No
$G_{31}$	3	4	2	No	No
$G_{32}$	3	2	4	Yes	No
$G_{33}$	3	4	3	No	No
$G_{34}$	3	3	4	Yes	No
$G_{35}$	2	4	1	No	No
$G_{36}$	2	1	4	Yes	No
$G_{37}$	1	4	7	No	No
$G_{38}$	1	7	4	Yes	No
$G_{39}$	4	5	5	Yes	Yes
$G_{42}$	3	5	3	No	No
$G_{43}$	3	3	5	Yes	No
$G_{44}$	2	5	1	No	No
$G_{45}$	2	1	5	Yes	No
$G_{46}$	1	5	7	No	No
$G_{47}$	1	7	5	Yes	No
$G_{48}$	3	2	2	No	No
$G_{49}$	3	2	3	No	No
$G_{51}$	2	2	1	No	No
$G_{52}$	2	1	2	No	No
$G_{53}$	1	2	7	No	No
$G_{54}$	1	7	2	No	No
$G_{55}$	3	3	3	No	No
$G_{56}$	2	3	1	No	No
$G_{57}$	2	1	3	No	No
$G_{58}$	1	3	7	No	No
$G_{59}$	1	7	3	No	No
$G_{60}$	2	1	1	No	No
$G_{61}$	1	1	7	No	No
$G_{62}$	1	7	1	No	No
$G_{63}$	1	7	7	No	No

• Codes are neither self-dual nor Type IV or QT4.
• Codes are not LCD except $G_1$, $G_{11}$ and $G_{13}$.
• Codes are not QSD except $G_{14}$, $G_{32}$,$G_{34}$,$G_{41}$ and $G_{43}$.

Remark 1. 

In the case of length equal to 7, there are four equivalent codes, and they are $G_{34}$ equivalent to $G_{41}$, $G_{29}$ equivalent to $G_{30}$, $G_{33}$ equivalent to $G_{40}$, and $G_{49}$ equivalent to $G_{50}$.

In Table 19, we summarized the number of codes according to the Hamming weight.

Table 19. Number of cyclic codes over H.

$\mathit{d}\setminus \mathit{n}$	2	3	4	5	6	7
1	5	7	9	7	19	15
2	3	5	12	5	46	13
3		3			7	20
4			3		5	12
5				3
6					3
7						3
Total	8	15	24	15	80	63

9. Conclusions and Open Problems

In this article, we have studied cyclic codes over the non-unitary ring $H.$ We have given criteria for a cyclic code over H to be self-dual or quasi-self-dual. We have derived an algorithm to classify cyclic codes of given length, based on the classification of cyclic binary codes of that length and the knowledge of their automorphism group. A Gray map allows us to construct quasi-cyclic codes of index 2 from cyclic codes over $H.$

In the future, we plan to study the same questions over other non-unitary rings, possibly of characteristic larger than $2.$