Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4)

Abstract

Binary self-dual codes and additive self-dual codes over $GF\left(4\right)$ contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over $GF\left(2\right)$ and $GF\left(4\right)$ with minimal shadow. In particular, we prove that there is no near-extremal Type I $[24m,12m,2m+2]$ binary self-dual code with minimal shadow if $m\ge 323$, and we prove that there is no near-extremal Type I $(6m+1,2^{6m+1},2m+1)$ additive self-dual code over $GF\left(4\right)$ with minimal shadow if $m\ge 22$.

1. Introduction

There are many interesting classes of codes in coding theory, such as cyclic codes, quadratic residue codes, algebraic geometry codes and self-dual codes. This paper focuses on self-dual codes, which, while of interest themselves, are closely related to other mathematical structures such as block designs, lattices, modular forms and sphere packings (for example, see [1]).

There are several types of self-dual codes. Among them, binary self-dual codes and additive self-dual codes over $GF\left(4\right)$ have common points. Firstly, there are Type I and Type II codes in both classes. Secondly, there are shadow codes in both classes. Using shadow theory, E. M.Rains provided an upper bound to the minimum distances of Type I codes in both classes [2]. If a code meets this bound, then it is called an extremal code.

For extremal Type II codes, there is a systematic nonexistence proof [3]. However, for extremal Type I codes, no such nonexistence proof exists. Research has also been conducted on extremal Type I codes with minimal shadow. S. Bouyuklieva and W. Willems studied the nonexistence of extremal Type I binary codes with minimal shadow [4]. Impressed by the results, S. Han studied the nonexistence of extremal Type I additive codes over $GF\left(4\right)$ with minimal shadow [5]. Recently, S. Bouyuklieva, M. Harada and A. Munemasa studied the nonexistence of near-extremal Type I binary self-dual codes with minimal shadow [6].

In this paper, we cover the missing case of the nonexistence of near-extremal Type I binary self-dual codes with minimal shadow, which was not covered in [6], and we apply the technique to near-extremal Type I additive codes over $GF\left(4\right)$ with minimal shadow. The main contribution of this paper is three-fold. Firstly, it provides a comprehensive presentation of the nonexistence of extremal and near-extremal Type I codes over $GF\left(2\right)$ and $GF\left(4\right)$. Secondly, we prove that there is no near-extremal Type I $[24m,12m,2m+2]$ binary self-dual code with minimal shadow if $m\ge 323$. Thirdly, we prove that there is no near-extremal Type I $(6m+1,2^{6m+1},2m+1)$ additive self-dual code over $GF\left(4\right)$ with minimal shadow if $m\ge 22$.

The rest of the paper is organized as follows. In Section 2, we deal with binary self-dual codes with minimal shadow. We consider the nonexistence of extremal Type I binary self-dual codes with minimal shadow. In Section 3, we consider the nonexistence of near-extremal Type I binary self-dual codes with minimal shadow. In Section 4, we deal with additive self-dual codes over $GF\left(4\right)$ with minimal shadow. We consider the nonexistence of extremal Type I additive self-dual codes over $GF\left(4\right)$ with minimal shadow. In Section 5, we consider the nonexistence of near-extremal Type I additive self-dual codes over $GF\left(4\right)$ with minimal shadow. All computer calculations in this study were performed using the mathematical software Maple.

2. Extremal Type I Binary Self-Dual Codes with Minimal Shadow

In this section, we deal with binary self-dual codes with minimal shadow. First, we discuss basic facts about binary self-dual codes. Secondly, we consider the nonexistence of extremal Type I binary self-dual codes with minimal shadow.

A binary linear code C is a subspace of a vector space $GF{\left(2\right)}^n$, and the vectors in C are called codewords. The weight of a codeword $u=(u_1,u_2,\cdots ,u_n)$ in $GF{\left(2\right)}^n$ is the number of nonzero $u_j$. The minimum distance of C is the smallest nonzero weight of any codeword in C. If the dimension of C is k and the minimum distance in C is d, we say C is an $[n,k,d]$ code.

The scalar product in $GF{\left(2\right)}^n$ is defined by:

$$(u,v)=\sum _{j=1}^n{u}_j{v}_j,$$

(1)

where the sum is evaluated in $GF\left(2\right)$. The dual code of a binary linear code C is defined by:

$$C^{\perp }=\{v\in GF{\left(2\right)}^n:(v,c)=0\mathrm{for}\mathrm{all}c\in C\}.$$

(2)

If $C\subseteq C^{\perp }$, we say C is self-orthogonal, and if $C=C^{\perp }$, we say C is self-dual.

A binary code is even if all its codewords have even weights. Clearly, self-dual binary codes are even. In addition, some of these codes have all codewords of weights divisible by four. A self-dual code with all codewords of weights divisible by four is called doubly-even or Type II; a self-dual code where some codewords have weights not divisible by four is called singly-even or Type I. Bounds on the minimum distance of binary self-dual codes were provided in [2].

Theorem 1.

([2]) Let C be an $[n,n/2,d]$ binary self-dual code. Then, $d\le 4[n/24]+4$ if $n\neg \equiv 22\left(mod24\right)$. If $n\equiv 22\left(mod24\right)$, then $d\le 4[n/24]+6$, and if the equality holds, C can be obtained by shortening a Type II code of length $n+2$. If $24|n$ and $d=4[n/24]+4$, then C is Type II.

A code meeting the bounds of Theorem 1, i.e., for which equality holds within the bounds, is called extremal. From Theorem 1, note that there is no extremal Type I code of length $n=24m$ ($m\ge 1$). There is a systematic proof for the nonexistence of extremal Type II codes if the code length is sufficiently large [3].

Theorem 2.

([3]) Let C be an extremal binary Type II code of length $n=24m+8\ell$. Then, the code C does not exist if $m\ge 154$ (for $\ell =0$), $m\ge 159$ (for $\ell =1$) and $m\ge 164$ (for $\ell =2$).

The proof of Theorem 1 for Type I codes is formulated using a shadow code. In [7], the concept of a shadow code was introduced. The shadow code of a self-dual code C is defined as follows: let $C^{\left(0\right)}$ be the subset of C consisting of all codewords whose weights are multiples of four, and let $C^{\left(2\right)}=C\setminus C^{\left(0\right)}.$ The shadow code of C is defined by:

$$S=S\left(C\right)=\{u\in GF{\left(2\right)}^n:(u,v)=0\mathrm{for}\mathrm{all}v\in C^{\left(0\right)},(u,v)=1\mathrm{for}\mathrm{all}v\in C^{\left(2\right)}\}.$$

(3)

The weight enumerator of a code is given by:

$$W_C(x,y)=\sum _{i=0}^n{A}_i{x}^{n-i}{y}^i,$$

(4)

where there are $A_i$ codewords of weight i in C. The following lemma is needed in this paper:

Lemma 1.

[7] Let C be a Type I binary self-dual code of length n and minimum weight d. Let $S\left(y\right)={\sum }_{i=0}^n{b}_i{y}^i$ be the weight enumerator of $S\left(C\right)$. Then:

1. 

$b_0=0$

2. 

$b_i\le 1$ for $i<d/2$

Let C be a Type I binary self-dual code of length $n=24m+8\ell +2r$ where $\ell =0,1,2$ and $r=0,1,2,3$. By Gleason’s theorem [8,9,10], we can calculate the weight enumerator of C as follows for suitable constants $c_i$:

$$W_C(x,y)=\sum _{i=0}^{[n/8]}{c}_i{(x^2+y^2)}^{n/2-4i}{\left\{x^2{y}^2{(x^2-y^2)}^2\right\}}^i.$$

(5)

Using the shadow code theory [7], we can calculate the weight enumerator of shadow code $S\left(C\right)$:

$$W_S(x,y)=\sum _{i=0}^{[n/8]}{(-1)}^i{2}^{n/2-6i}{c}_i{\left(xy\right)}^{n/2-4i}{(x^4-y^4)}^{2i}.$$

(6)

We rewrite Equations (5) and (6) to the following:

$$W_C(1,y)=\sum _{j=0}^{12m+4\ell +r}{a}_j{y}^{2j}=\sum _{i=0}^{3m+\ell }{c}_i{(1+y^2)}^{12m+4\ell +r-4i}{\left\{y^2{(1-y^2)}^2\right\}}^i,$$

(7)

$$W_S(1,y)=\sum _{j=0}^{6m+2\ell }{b}_j{y}^{4j+r}=\sum _{i=0}^{3m+\ell }{(-1)}^i{c}_i{2}^{12m+4\ell +r-6i}{y}^{12m+4\ell +r-4i}{(1-y^4)}^{2i}.$$

(8)

Note that all $a_j$ and $b_j$ must be nonnegative integers. One can write $c_i$ as a linear combination of the $a_j$ for $0\le j\le i$, and one can write $c_i$ as a linear combination of $b_j$ for $0\le j\le 3m+\ell -i$, as follows for suitable constants ${\alpha }_{ij}$ and ${\beta }_{ij}$:

$$c_i=\sum _{j=0}^i{\alpha }_{ij}{a}_j=\sum _{j=0}^{3m+\ell -i}{\beta }_{ij}{b}_j.$$

(9)

In our computation, we need to calculate ${\alpha }_{i0}$ and ${\beta }_{ij}$. The following formula can be found in [2] for $i>0$:

$${\alpha }_{i0}=-\frac{n}{2i}\left[\mathrm{coeff}.\mathrm{of}{y}^{i-1}\mathrm{in}{(1+y)}^{-(n/2)-1+4i}{(1-y)}^{-2i}\right]$$

(10)

and:

$${\beta }_{ij}={(-1)}^i{2}^{-\frac{n}{2}+6i}\frac{k-j}{i}\left(\genfrac{}{}{0pt}{}{k+i-j-1}{k-i-j}\right),$$

(11)

where $k=3m+\ell$. Note that $a_0=c_0={\alpha }_{00}=1$. Now, we introduce the definition of a code with minimal shadow:

Definition 1.

Let C be a Type I binary self-dual code of length $n=24m+8\ell +2r$ with $\ell =0,1,2$ and $r=0,1,2,3$. Then, C is a code with minimal shadow if:

1. 

$d\left(S\right)=r$ for $r>0$ and

2. 

$d\left(S\right)=4$ for $r=0$

where $d\left(S\right)$ is the minimum weight of S.

Let C be an extremal Type I binary self-dual code with a minimal shadow of length n. Then, the following facts can be found in [4]: For $a_i$, we have $a_0=1,a_1=a_2=\cdots =a_{2m+1}=0$. Moreover, if $n\equiv 22(mod24)$, then $a_{2m+2}=0$. For $b_j$, we have $b_0=1$ if (i) $r=1$ and $m\ge 0$ and (ii) $r=2,3$ and $m\ge 1$. Furthermore, we have $b_0=0,b_1=1$ if $r=0$ and $m\ge 2$. If $r>0$, then $b_1=b_2=\cdots =b_{m-1}=0$. If $r=0$, then $b_2=b_3=\cdots =b_{m-1}=0$. Moreover, if $n=24m+8l+2$, then $b_m=0$. Using these facts, we have the following lemma:

Lemma 2.

Using the above notations, we have the following results:

1. 

If $n=24m+2$ ($m\ge 0$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m\le i\le 3m$.

2. 

If $n=24m+4$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+1\le i\le 3m$.

3. 

If $n=24m+6$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+1\le i\le 3m$.

4. 

If $n=24m+8$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i1}$ for $2m+2\le i\le 3m+1$.

5. 

If $n=24m+10$ ($m\ge 0$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+1\le i\le 3m+1$.

6. 

If $n=24m+12$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+1$.

7. 

If $n=24m+14$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+1$.

8. 

If $n=24m+16$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i1}$ for $2m+3\le i\le 3m+2$.

9. 

If $n=24m+18$ ($m\ge 0$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+2$.

10. 

If $n=24m+20$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+3\le i\le 3m+2$.

11. 

If $n=24m+22$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+2$, $c_i={\beta }_{i0}$ for $2m+3\le i\le 3m+2$.

Proof. 

Let C be an extremal Type I binary self-dual code with minimal shadow of length $n=24m+2$. We can rewrite Equation (9) as follows:

$$c_i=\sum _{j=0}^i{\alpha }_{ij}{a}_j=\sum _{j=0}^{3m-i}{\beta }_{ij}{b}_j.$$

(12)

Then, we have:

$$c_i=\sum _{j=0}^i{\alpha }_{ij}{a}_j={\alpha }_{i0}\mathrm{for}i=0,1,2,\cdots ,2m+1$$

(13)

and:

$$c_i=\sum _{j=0}^{3m-i}{\beta }_{ij}{b}_j={\beta }_{i0}\mathrm{for}i=2m,2m+1,\cdots ,3m.$$

(14)

Therefore, the first statement is proven. The other cases can be proven similarly. □

Using Lemma 2, we have the following theorem:

Theorem 3.

Let C be an extremal Type I binary self-dual code of length n with minimal shadow. Then, the weight enumerator of C is unique if $n\neg \equiv 24m+16,24m+20$.

Proof. 

Suppose that $n\neg \equiv 24m+16,24m+20$. From Lemma 2, we can see that $c_i$ can be calculated by Equations (10) and (11), and they depend only on the length n for all $i,(0\le i\le [n/8\left]\right)$, except the following cases. By [7], we know that:

• $n=24m+4$: If $m=0$, then $n=4$. For this case, there is no extremal code.
• $n=24m+6$: If $m=0$, then $n=6$. For this case, there is no extremal code.
• $n=24m+8$: If $m=0$, then $n=8$. For this case, there is no extremal Type I code. If $m=1$, then $n=32$. For this case, there are three extremal Type I codes. They have the same weight enumerator: $W_C(1,y)=1+364y^8+2048y^{10}+6720y^{12}+14336y^{14}+18598y^{16}+\cdots$, $W_S(1,y)=8y^4+592y^8+13944y^{12}+36448y^{16}+\cdots$. We can see that the codes have minimal shadow.
• $n=24m+12$: If $m=0$, then $n=12$. For this case, there is a unique extremal Type I code. The weight enumerator is the following: $W_C(1,y)=1+15y^4+32y^6+\cdots$, $W_S(1,y)=6y^2+5y^6+\cdots$. We can see that the code has minimal shadow.
• $n=24m+22$: If $m=0$, then $n=22$. For this case, there is a unique extremal Type I code. The weight enumerator is the following: $W_C(1,y)=1+77y^6+330y^8+616y^{10}+\cdots$, $W_S(1,y)=352y^7+1344y^{11}+\cdots$.

This completes the proof. □

The following nonexistence theorems are proven in [4].

Theorem 4.

[4] Extremal self-dual codes of lengths $n=24m+2,24m+4,24m+6,24m+10$ and $24m+22$ with minimal shadow do not exist.

Theorem 5.

[4] There are no extremal Type I binary self-dual codes of length n with minimal shadow if:

1. 

$n=24m+8$ and $m\ge 53$;

2. 

$n=24m+12$ and $m\ge 142$;

3. 

$n=24m+14$ and $m\ge 146$;

4. 

$n=24m+16$ and $m\ge 164$;

5. 

$n=24m+18$ and $m\ge 157$.

Remark 1.

Currently, $n=24m+20$ is the unique untouched code length for the nonexistence or an explicit bound for the length n of an extremal Type I binary self-dual code with minimal shadow.

3. Near-Extremal Type I Binary Self-Dual Codes with Minimal Shadow

In this section, we consider the nonexistence of near-extremal Type I binary self-dual codes with minimal shadow. We start with the following definition:

Definition 2.

Let C be an $[n,n/2,d]$ Type I binary self-dual code. Then, C is a near-extremal code if:

1. 

$d=4[n/24]+2$ for $n\neg \equiv 22(mod24)$; and

2. 

$d=4[n/24]+4$ for $n\equiv 22(mod24)$.

Let C be a near-extremal Type I binary self-dual code with minimal shadow. Then, we have the following: $a_0=1,a_1=a_2=\cdots =a_{2m}=0$. Moreover, if $n\equiv 22(mod24)$, then $a_{2m+1}=0$.

By Lemma 1, $b_0=1$ if (i) $r=1,2$ and $m\ge 1$, (ii) $r=3$, $n\neg \equiv 22(mod24)$ and $m\ge 2$ and (iii) $r=3$, $n\equiv 22(mod24)$ and $m\ge 1$. In addition, $b_0=0,b_1=1$ if $r=0$ and $m\ge 2$.

If $r=1,2$ or $r=3$ and $n\equiv 22(mod24)$, then $b_1=b_2=\cdots =b_{m-1}=0$. Otherwise, S would contain a vector v of weight less than or equal to $4m-4+r$, and if $u\in S$ is a vector of weight r, then $u+v\in C$ with $wt(u+v)\le 4m-4+2r$, a contradiction with a minimum distance of C. If $r=3$ and $n\neg \equiv 22(mod24)$, then $b_1=b_2=\cdots =b_{m-2}=0$. Furthermore, if $r=0$, then $b_2=b_3=\cdots =b_{m-1}=0$. The proofs are similar to the above case. Using this fact, we have the following lemma:

Lemma 3.

Using the above notations, we have the following results:

1. 

If $n=24m$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i1}$ for $2m+1\le i\le 3m$.

2. 

If $n=24m+2$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+1\le i\le 3m$.

3. 

If $n=24m+4$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+1\le i\le 3m$.

4. 

If $n=24m+6$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m$.

5. 

If $n=24m+8$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i1}$ for $2m+2\le i\le 3m+1$.

6. 

If $n=24m+10$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+1$.

7. 

If $n=24m+12$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+1$.

8. 

If $n=24m+14$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+3\le i\le 3m+1$.

9. 

If $n=24m+16$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i1}$ for $2m+3\le i\le 3m+2$.

10. 

If $n=24m+18$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+3\le i\le 3m+2$.

11. 

If $n=24m+20$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+3\le i\le 3m+2$.

12. 

If $n=24m+22$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+3\le i\le 3m+2$.

Proof. 

The proof is similar to the one for Lemma 2. □

Using Lemma 3, we have the following theorem [6]:

Theorem 6.

[6] Let C be a near-extremal Type I binary self-dual code with minimal shadow of length n. Then, we have the following:

1. 

The weight enumerator of C is uniquely determined if $n=24m+2,24m+4,24m+10$.

2. 

The code C does not exist if:

(a) 

$n=24m+2$ and $m\ge 155$

(b) 

$n=24m+4$ and $m\ge 156$

(c) 

$n=24m+10$ and $m\ge 160$

The missing case in Theorem 6 is the code length $n=24m$. We can prove similar results for the missing case using the following theorem:

Theorem 7.

Let C be a $[24m,12m,4m+2]$ near-extremal Type I binary self-dual code with minimal shadow. Then, we have the following:

1. 

The weight enumerator of C is uniquely determined.

2. 

The code C does not exist if $m\ge 323$.

Proof. 

From Lemma 2, we can see that $c_i$ can be calculated by Equations (10) and (11), and they depend only on the length n for all $i,(0\le i\le [n/8\left]\right)$ unless $m=1$. If $m=1$, then $n=24$. For this case, there is a unique near-extremal Type I code [7]. The weight enumerator is the following: $W_C(1,y)=1+64y^6+375y^8+960y^{10}+1296y^{12}+\cdots$. $W_S(1,y)=6y^4+744y^8+2596y^{12}+\cdots$. We can see that the code has minimal shadow. This proves the first statement.

For the second statement, from Equation (9) and the fact that $c_i={\alpha }_{i,0}$ for $0\le i\le 2m$, we have:

$$c_{2m}={\alpha }_{2m,0}={\beta }_{2m,1}+{\beta }_{2m,m}{b}_m.$$

(15)

Therefore, we get:

$$b_m={\beta }_{2m,m}^{-1}({\alpha }_{2m,0}-{\beta }_{2m,1}).$$

(16)

Using Equations (10) and (11), we have:

$${\beta }_{2m,m}=1,{\alpha }_{2m,0}=6\left(\genfrac{}{}{0pt}{}{5m-1}{m-1}\right),{\beta }_{2m,1}=\frac{3m-1}{2m}\left(\genfrac{}{}{0pt}{}{5m-2}{m-1}\right).$$

(17)

From this, we get:

$$b_m=6\left(\genfrac{}{}{0pt}{}{5m-1}{m-1}\right)-\frac{3m-1}{2m}\left(\genfrac{}{}{0pt}{}{5m-2}{m-1}\right).$$

(18)

From Equation (9) and the fact that $c_i={\alpha }_{i,0}$ for $0\le i\le 2m$, we have:

$$c_{2m-1}={\alpha }_{2m-1,0}={\beta }_{2m-1,1}+{\beta }_{2m-1,m}{b}_m+{\beta }_{2m-1,m+1}{b}_{m+1}.$$

(19)

From this, we get:

$$b_{m+1}={\beta }_{2m-1,m+1}^{-1}({\alpha }_{2m-1,0}-{\beta }_{2m-1,1}-{\beta }_{2m-1,m}{b}_m).$$

(20)

Using Equations (10) and (11), we have:

$${\beta }_{2m-1,m+1}=-2^{-6},$$

(21)

$${\alpha }_{2m-1,0}=-\frac{24m}{2(2m-1)}\left[\left(\genfrac{}{}{0pt}{}{5m+3}{m-1}\right)+\left(\genfrac{}{}{0pt}{}{5m+2}{m-2}\right)\left(\genfrac{}{}{0pt}{}{7}{2}\right)+\left(\genfrac{}{}{0pt}{}{5m+1}{m-3}\right)\left(\genfrac{}{}{0pt}{}{7}{4}\right)+\left(\genfrac{}{}{0pt}{}{5m}{m-4}\right)\left(\genfrac{}{}{0pt}{}{7}{6}\right)\right]$$

(22)

and:

$${\beta }_{2m-1,1}=-2^{-6}\times \frac{3m-1}{2m-1}\left(\genfrac{}{}{0pt}{}{5m-3}{m}\right),{\beta }_{2m-1,m}=-\frac{m}{16}.$$

(23)

Therefore, we get:

$$b_{m+1}=\frac{64(6m-1)(5m-1)(5m-3)!}{(4m+4)!(m-1)!}{h}_0\left(m\right),$$

(24)

where:

$$h_0\left(m\right)=-64m^5+20640m^4-9388m^3+582m^2-49m-3.$$

(25)

We can see that $h_0\left(m\right)<0$ if $m\ge 323$. Therefore, if $m\ge 323$, then $b_{m+1}<0$. This is a contradiction. □

Remark 2.

The definition of near-extremal Type II binary self-dual codes and the corresponding nonexistence proof can be found in [11].

4. Extremal Type I Additive Self-Dual Codes over $\mathit{GF}\left(\mathbf{4}\right)$ with Minimal Shadow

In this section, we deal with additive self-dual codes over $GF\left(4\right)$ with minimal shadow. First, we discuss basic facts about additive self-dual codes over $GF\left(4\right)$. Then, we consider the nonexistence of extremal Type I additive self-dual codes over $GF\left(4\right)$ with minimal shadow.

An additive code C over $GF\left(4\right)$ of length n is an additive subgroup of $GF{\left(4\right)}^n$. The weight of a vector $u=(u_1,u_2,\cdots ,u_n)$ in $GF{\left(4\right)}^n$ and the minimum distance of C are defined the same way as for binary linear codes. C is a k-dimensional $GF\left(2\right)$-subspace of $GF{\left(4\right)}^n$ and thus has $2^k$ codewords. It is denoted as an $(n,2^k)$ code, and if its minimum distance is d, the code is an ($n,2^k,d)$ code.

The trace map, $\mathrm{Tr}:GF\left(4\right)\to GF\left(2\right)$, is defined by $\mathrm{Tr}\left(x\right)=x+x^2$. The Hermitian trace inner product of two vectors over $GF\left(4\right)$ of length n, $u=(u_1,u_2,\dots ,u_n)$ and $v=(v_1,v_2,\dots ,v_n)$ is given by:

$$u\ast v=\sum _{i=1}^n\mathrm{Tr}\left(u_i{v_i}^2\right)=\sum _{i=1}^n(u_i{v_i}^2+{u_i}^2{v}_i)\left(\mathrm{mod}2\right).$$

(26)

We define the dual of the code Cwith respect to the Hermitian trace inner product as follows:

$$C^{\perp }=\{u\in GF{\left(4\right)}^n:u\ast c=0\mathrm{for}\mathrm{all}c\in C\}.$$

(27)

If $C\subseteq C^{\perp }$, we say C is self-orthogonal, and if $C=C^{\perp }$, we say C is self-dual. If C is self-dual, then it must be an $(n,2^n)$ code.

We distinguish between two types of additive self-dual codes over $GF\left(4\right)$. A code is Type II if all codewords have even weights, otherwise it is Type I. Bounds on the minimum distance of additive self-dual codes over $GF\left(4\right)$ were provided in [2,2].

Theorem 8.

[2,2] Let C be an $(n,2^n,d)$ additive self-dual code over $GF\left(4\right)$. If C is Type I, then:

$$d\le \left\{\begin{array}{cc}2[n/6]+1,\hfill & \mathrm{if}n\equiv 0(mod6);\hfill \\ 2[n/6]+3,\hfill & \mathrm{if}n\equiv 5(mod6);\hfill \\ 2[n/6]+2,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(28)

If C is Type II, then:

$$d\le 2[n/6]+2.$$

(29)

A code that meets the appropriate bound is called extremal. There is a systematic proof for the nonexistence of extremal Type II codes if the code length is sufficiently large.

Theorem 9.

Let C be an extremal Type II additive self-dual code over $GF\left(4\right)$ of length n. Then, the code C does not exist if $n=6m(m\ge 17)$, $n=6m+2(m\ge 20)$ and $n=6m+4(m\ge 22)$.

Proof. 

The Gleason polynomials of Type II additive self-dual codes over $GF\left(4\right)$ are the same as the ones for Type IV Hermitian self-dual linear codes over $GF\left(4\right)$ (see [1], Section 7.7, for examples). Both have the same upper bounds on the minimum distance and the same definition of extremal codes w.r.t. minimum distance. There is a nonexistence theorem for Type IV Hermitian self-dual linear codes over $GF\left(4\right)$ that is the same as the above statements [3]. The proof is formulated with Gleason polynomials, so that the nonexistence statements are still valid for Type II additive self-dual codes over $GF\left(4\right)$. □

The proof of Theorem 8 for Type I codes is formulated using a shadow code, which is defined as follows: Let C be an additive self-dual code over $GF\left(4\right)$ and $C_0$ be the subset of C consisting of all codewords whose weights are multiples of two. Then, $C_0$ is a subgroup of C. The shadow code of an additive code C over $GF\left(4\right)$ is defined by:

$$S=C_0^{\perp }\setminus C.$$

(30)

Alternately, it can be defined as:

$$S=\{u\in GF{\left(4\right)}^n|u\ast v=0\mathrm{for}\mathrm{all}v\in C_0,u\ast v=1\mathrm{for}\mathrm{all}v\in C\setminus C_0\}.$$

(31)

The following lemmas for shadow codes can be found in [5]:

Lemma 4.

[5] Let C be a Type I additive self-dual code over $GF\left(4\right)$ and S be the shadow code of C. If $u,v\in S$, then $u+v\in C$.

Lemma 5.

[5] Let C be an additive self-dual code over $GF\left(4\right)$ of length n and minimum weight d. Let $S\left(y\right)={\sum }_{r=0}^n{B}_r{y}^r$ be the weight enumerator of S. Then:

1. 

$B_0=0$

2. 

$B_r\le 1$ for $r<d/2$

Let C be a Type I additive self-dual code over $GF\left(4\right)$. By [2], the weight enumerator of C, $W_C(x,y)$, and its shadow code weight enumerator, $W_S(x,y)$, are given by:

$$W_C(x,y)=\sum _{i=0}^{[n/2]}{c}_i{(x+y)}^{n-2i}{\left\{y(x-y)\right\}}^i,$$

(32)

$$W_S(x,y)=\sum _{i=0}^{[n/2]}{(-1)}^i{2}^{n-3i}{c}_i{y}^{n-2i}{(x^2-y^2)}^i,$$

(33)

for suitable constants $c_i$. We rewrite Equations (32) and (33) to the following:

$$W_C(1,y)=\sum _{j=0}^n{a}_j{y}^j=\sum _{i=0}^{[n/2]}{c}_i{(1+y)}^{n-2i}{\left\{y(1-y)\right\}}^i$$

(34)

and:

$$W_S(1,y)=\sum _{j=0}^{[n/2]}{b}_j{y}^{2j+t}=\sum _{i=0}^{[n/2]}{(-1)}^i{2}^{n-3i}{c}_i{y}^{n-2i}{(1-y^2)}^i,$$

(35)

where $t=0$ if n is even and $t=1$ if n is odd. Note that all $a_j$ and $b_j$ must be nonnegative integers. One can write $c_i$ as a linear combination of the $a_j$ for $0\le j\le i$, and one can write $c_i$ as a linear combination of $b_j$ for $0\le j\le [n/2]-i$ in the following form for suitable constants ${\alpha }_{ij}$ and ${\beta }_{ij}$:

$$c_i=\sum _{j=0}^i{\alpha }_{ij}{a}_j=\sum _{j=0}^{[n/2]-i}{\beta }_{ij}{b}_j.$$

(36)

In our computation, we need to calculate ${\alpha }_{i0}$ and ${\beta }_{ij}$. The following formulas can be found in [2] for $i>0$:

$${\alpha }_{i0}=-\frac{n}{i}\left[\mathrm{coeff}.\mathrm{of}{y}^{i-1}\mathrm{in}{(1+y)}^{-n-1+2i}{(1-y)}^{-i}\right]$$

(37)

and:

$${\beta }_{ij}={(-1)}^i{2}^{3i-n}\left(\genfrac{}{}{0pt}{}{k-j}{i}\right),$$

(38)

where $k=[n/2]$. Note that $a_0=c_0={\alpha }_{00}=1$. Now, we will introduce the definition of a code with minimal shadow:

Definition 3.

([5]) Let C be a Type I additive self-dual code over $GF\left(4\right)$ of length $n=6m+r(0\le r\le 5)$. Then, C is a code with minimal shadow if:

1. 

$d\left(S\right)=1$ if $r=1,3,5$; and

2. 

$d\left(S\right)=2$ if $r=0,2,4$,

where $d\left(S\right)$ is the minimum weight of S.

Let C be an extremal Type I additive self-dual code over $GF\left(4\right)$ with minimal shadow of length $n=6m+r$. Then, the following facts can be found in [5]:

Suppose that $r=0$. Then, $a_0=1$, $a_1=a_2=\cdots =a_{2m}=0$, $b_0=0$, $b_1=1$ if $m\ge 2$, and $b_2=b_3=\cdots =b_{m-1}=0$.

Suppose that $r=1,3$. Then, $a_0=1$, $a_1=a_2=\cdots =a_{2m+1}=0$, $b_0=1$ if $m\ge 1$, and $b_1=b_2=\cdots =b_{m-1}=0$.

Suppose that $r=2,4$. Then, $a_0=1$, $a_1=a_2=\cdots =a_{2m+1}=0$, $b_0=0$, $b_1=1$ if $m\ge 2$, and $b_2=b_3=\cdots =b_{m-1}=0$.

Suppose that $r=5$. Then, $a_0=1$, $a_1=a_2=\cdots =a_{2m+2}=0$, $b_0=1$, and $b_1=b_2=\cdots =b_{m-1}=b_m=0$. Using this fact, we have the following lemma:

Lemma 6.

[5] Using the above notations, we have the following results:

1. 

If $n=6m$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i1}$ for $2m+1\le i\le 3m$.

2. 

If $n=6m+1$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+1\le i\le 3m$.

3. 

If $n=6m+2$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i1}$ for $2m+2\le i\le 3m+1$.

4. 

If $n=6m+3$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+1$.

5. 

If $n=6m+4$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i1}$ for $2m+3\le i\le 3m+2$.

6. 

If $n=6m+5$ ($m\ge 0$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+2$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+2$.

Using Lemma 6, we have the following theorems [5]:

Theorem 10.

[5] Extremal Type I additive self-dual codes over $GF\left(4\right)$ with minimal shadows of lengths $n=6m,6m+1,6m+2,6m+3$ and $6m+5$ have uniquely-determined weight enumerators.

Theorem 11.

[5] Extremal Type I additive self-dual codes over $GF\left(4\right)$ with minimal shadows of lengths $n=6m+1$ and $n=6m+5$ do not exist.

Theorem 12.

[5] There are no extremal Type I additive self-dual codes over $GF\left(4\right)$ with minimal shadow if:

1. 

$n=6m$ and $m\ge 40$;

2. 

$n=6m+2$ and $m\ge 6$;

3. 

$n=6m+3$ and $m\ge 22$.

Remark 3.

Currently, $n=6m+4$ is the unique untouched code length for the nonexistence or an explicit bound for the length n of an extremal Type I additive self-dual code over $GF\left(4\right)$ with minimal shadow.

5. Near-Extremal Type I Additive Self-Dual Codes over $\mathit{GF}\left(\mathbf{4}\right)$ with Minimal Shadow

In this section, we consider the nonexistence of near-extremal Type I additive self-dual codes over $GF\left(4\right)$ with minimal shadow. We start with the following definition:

Definition 4.

Let C be an $(n,2^n,d)$ Type I additive self-dual code over $GF\left(4\right)$. Then, C is a near-extremal code if Cis Type I and $d=2[n/6]$ if $n\equiv 0(mod6)$, $d=2[n/6]+2$ if $n\equiv 5(mod6)$ and $d=2[n/6]+1$ otherwise.

Let C be a near-extremal Type I additive self-dual code over $GF\left(4\right)$ with a minimal shadow of length $n=6m+r$. Then, we have the following facts:

Suppose that $r=0$. Then, $a_0=1$, $a_1=a_2=\cdots =a_{2m-1}=0$ and $b_0=0$. By Lemma 5, $b_1=1$ if $m\ge 3$. We have $b_2=b_3=\cdots =b_{m-2}=0$. Otherwise, S would contain a vector v of weight less than or equal to $2m-4$, and if $u\in S$ is a vector of weight two, then $u+v\in C$ with $wt(u+v)\le 2m-4+2=2m-2$, a contradiction with the minimum distance of C.

Suppose that $r=1,3$. Then, $a_0=1$ and $a_1=a_2=\cdots =a_{2m}=0$. By Lemma 5, $b_0=1$ if $m\ge 1$. We have $b_1=b_2=\cdots =b_{m-1}=0$. The proof is similar to the above case.

Suppose that $r=2,4$. Then, $a_0=1$, $a_1=a_2=\cdots =a_{2m}=0$ and $b_0=0$. By Lemma 5, $b_1=1$ if $m\ge 2$. We have $b_2=b_3=\cdots =b_{m-1}=0$. The proof is similar to the above case.

Suppose that $r=5$. Then, $a_0=1$ and $a_1=a_2=\cdots =a_{2m+1}=0$. By Lemma 5, $b_0=1$ if $m\ge 1$. We have $b_1=b_2=\cdots =b_{m-1}=0$. The proof is similar to the above case. Using this fact, we have the following lemma:

Lemma 7.

Using the above notations, we have the following results:

1. 

If $n=6m$ ($m\ge 3$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m-1$, $c_i={\beta }_{i1}$ for $2m+2\le i\le 3m$.

2. 

If $n=6m+1$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+1\le i\le 3m$.

3. 

If $n=6m+2$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i1}$ for $2m+2\le i\le 3m+1$.

4. 

If $n=6m+3$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i0}$ for $2m+2\le i\le 3m+1$.

5. 

If $n=6m+4$ ($m\ge 2$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, $c_i={\beta }_{i1}$ for $2m+3\le i\le 3m+2$.

6. 

If $n=6m+5$ ($m\ge 1$), then $c_i={\alpha }_{i0}$ for $0\le i\le 2m+1$, $c_i={\beta }_{i0}$ for $2m+3\le i\le 3m+2$.

Proof. 

Let C be an near-extremal Type I additive self-dual code over $GF\left(4\right)$ with a minimal shadow of length $n=6m(m\ge 3)$. We rewrite Equation (36) as follows:

$$c_i=\sum _{j=0}^i{\alpha }_{ij}{a}_j=\sum _{j=0}^{3m-i}{\beta }_{ij}{b}_j.$$

(39)

Then, we have:

$$c_i=\sum _{j=0}^i{\alpha }_{ij}{a}_j={\alpha }_{i0}\mathrm{for}i=0,1,2,\dots ,2m-1$$

(40)

and:

$$c_i=\sum _{j=0}^{3m-i}{\beta }_{ij}{b}_j={\beta }_{i1}\mathrm{for}i=2m+2,2m+2,\dots ,3m.$$

(41)

Therefore, the first statement is proven. The other cases can be proven similarly. □

Using Lemma 7, we have the following theorem:

Theorem 13.

Let C be a near-extremal Type I additive self-dual code over $GF\left(4\right)$ with a minimal shadow of length $n=6m+1$. Then, we have the following:

1. 

The weight enumerator of C is uniquely determined.

2. 

The code C does not exist if $m\ge 22$.

Proof. 

From Lemma 7, we can see that $c_i$ can be calculated by Equations (37) and (38), and the values depend only on the length n for all $i,(0\le i\le [n/3\left]\right)$ unless $m=0$. If $m=0$, then there is only one code for that code length [12]. This proves the first statement.

For the second statement, from Equation (36) and the fact that $c_i={\alpha }_{i,0}$ for $0\le i\le 2m$, we have:

$$c_{2m}={\alpha }_{2m,0}={\beta }_{2m,0}+{\beta }_{2m,m}{b}_m.$$

(42)

Therefore, we get:

$$b_m={\beta }_{2m,m}^{-1}({\alpha }_{2m,0}-{\beta }_{2m,0}).$$

(43)

Using Equations (37) and (38), we have:

$${\beta }_{2m,m}=\frac{1}{2},{\alpha }_{2m,0}=\frac{6m+1}{m}\left(\genfrac{}{}{0pt}{}{3m}{m-1}\right),{\beta }_{2m,0}=\frac{1}{2}\left(\genfrac{}{}{0pt}{}{3m}{2m}\right).$$

(44)

Therefore, we get:

$$b_m=\frac{12m+2}{m}\left(\genfrac{}{}{0pt}{}{3m}{m-1}\right)-\left(\genfrac{}{}{0pt}{}{3m}{2m}\right).$$

(45)

From Equation (36) and the fact that $c_i={\alpha }_{i0}$ for $0\le i\le 2m$, we have:

$$c_{2m-1}={\alpha }_{2m-1,0}={\beta }_{2m-1,0}+{\beta }_{2m-1,m}{b}_m+{\beta }_{2m-1,m+1}{b}_{m+1}.$$

(46)

Therefore, we get:

$$b_{m+1}={\beta }_{2m-1,m+1}^{-1}({\alpha }_{2m-1,0}-{\beta }_{2m-1,0}-{\beta }_{2m-1,m}{b}_m).$$

(47)

Using Equations (37) and (38), we have:

$${\beta }_{2m-1,m+1}=-\frac{1}{16},{\alpha }_{2m-1,0}=-\frac{6m+1}{2m-1}\left[\left(\genfrac{}{}{0pt}{}{3m+2}{m-1}\right)+10\left(\genfrac{}{}{0pt}{}{3m+1}{m-2}\right)+5\left(\genfrac{}{}{0pt}{}{3m}{m-3}\right)\right]$$

(48)

and:

$${\beta }_{2m-1,0}=-\frac{1}{16}\left(\genfrac{}{}{0pt}{}{3m}{2m-1}\right),{\beta }_{2m-1,m}=-\frac{m}{8}.$$

(49)

Therefore, we get:

$$\begin{array}{ccc}\hfill b_{m+1}& =& 16·\frac{6m+1}{2m-1}\left[\left(\genfrac{}{}{0pt}{}{3m+2}{m-1}\right)+10\left(\genfrac{}{}{0pt}{}{3m+1}{m-2}\right)+5\left(\genfrac{}{}{0pt}{}{3m}{m-3}\right)\right]\hfill \\ & & -\left(\genfrac{}{}{0pt}{}{3m}{2m-1}\right)-2m\left[\frac{12m+2}{m}\left(\genfrac{}{}{0pt}{}{3m}{m-1}\right)-\left(\genfrac{}{}{0pt}{}{3m}{2m}\right)\right].\hfill \end{array}$$

(50)

From this, we have:

$$b_{m+1}=\frac{\left(3m\right)!}{(2m+3)!(m-1)!}{h}_1\left(m\right),$$

(51)

where:

$$h_1\left(m\right)=-88m^3+1864m^2-34m-62.$$

(52)

We can see that $h_1\left(m\right)<0$ if $m\ge 22$. Therefore, if $m\ge 22$, then $b_{m+1}<0$. This is a contradiction. □

Remark 4.

The definition of near-extremal Type II additive self-dual codes over $GF\left(4\right)$ and the corresponding nonexistence proof can be found in [11].

6. Summary

In this paper, we provided a comprehensive presentation of extremal and near-extremal Type I self-dual codes over $GF\left(2\right)$ and $GF\left(4\right)$ with minimal shadow. We discussed recent research results for these codes. We also proved that there is no near-extremal Type I $[24m,12m,2m+2]$ binary self-dual code with minimal shadow if $m\ge 323$, and we proved that there is no near-extremal Type I $(6m+1,2^{6m+1},2m+1)$ additive self-dual code over $GF\left(4\right)$ with minimal shadow if $m\ge 22$.