Some restrictions on weight enumerators of singly even self-dual codes II

TL;DR

This paper establishes restrictions on the number of vectors of a specific weight in the shadow of singly even self-dual codes, narrowing down possible weight enumerators for certain code parameters.

Contribution

It introduces new restrictions on vector counts in the shadow of singly even self-dual codes, eliminating some potential weight enumerators for specific code lengths and minimum distances.

Findings

Eliminates some possible weight enumerators for codes with parameters (62,12), (72,14), (82,16), (90,16), (100,18)

Provides restrictions on the number of vectors of weight d/2+1 in the shadow

Narrows the classification of singly even self-dual codes based on weight enumerators

Abstract

In this note, we give some restrictions on the number of vectors of weight $d/2+1$ in the shadow of a singly even self-dual $[n,n/2,d]$ code. This eliminates some of the possible weight enumerators of singly even self-dual $[n,n/2,d]$ codes for $(n,d)=(62,12)$, $(72,14)$, $(82,16)$, $(90,16)$ and $(100,18)$.

Tables

Table 1. Table 1: Parameters satisfying ( 2 )

$n$	$d\left(n\right)$	$d\left(C\right)$	$d\left(S\right)$	$B_{d\left(S\right)}$
42	8	8	5	$\le 42$
62	12	12	7	$\le 48$
70	14	12	7	$\le 52$
82	16	16	9	$\le 74$
90	16	16	9	$\le 76$
98	18	16	9	$\le 78$

Table 2. Table 2: Parameters satisfying ( 5 )

$n$	$d\left(n\right)$	$d$	$s$	Proposition 10	Proposition 7
72	14	14	8	$B_s\le 14$ (iv)	$B_s\le 16$, $\ne 15$
100	18	18	10	$B_s\le 18$ (iv)	$B_s\le 20$, $\ne 19$
108	-	18	10	$B_s\le 18$ (iv)	$B_s\le 20$, $\ne 19$
116	-	18	10	$B_s\le 18$ (iv)	$B_s\le 20$, $\ne 19$
128	-	22	12	$B_s\le 16$ (v)	$B_s\le 21$

Equations

$$R_i=\{(x,y)\in X\times X\mid |x\cap y|=d-i\}.$$

$$a_i=\frac{1}{|Y|}|(Y\times Y)\cap R_i|(0\le i\le d).$$

$$(1,0,\dots ,0,0,|Y|-1,0)Q$$

$$q_j^{(v)}(0)+(|Y|-1)q_j^{(v)}(d-1)\ge 0(1\le j\le d).$$

$$|Y|\le M_{v,d},$$

$$M_{v,d}=\min \{1-\frac{q_j^{(v)}(0)}{q_j^{(v)}(d-1)}\mid 1\le j\le d\text{ and }{q}_j^{(v)}(d-1)<0\}.$$

$$M_{v,d}=\{\begin{array}{ll}2& \text{if v=2d−1,}\\ 1& \text{if d≤v≤2d−2,}\\ 0& \text{if 0≤v≤d−1,}\end{array}$$

$$n\equiv 2(\mathrm{mod}4)\text{ and }d(S)=\frac{d(C)}{2}+1.$$

$$|x\cap y|=\{\begin{array}{ll}1& \text{ if }x,y\in Y_i,x\ne y,\\ 0& \text{ if }x\in Y_1,y\in Y_3.\end{array}$$

$$|Y_i|\le M_{|S_i|,d(S)}.$$

$$B_{d(S)}=|Y_1|+|Y_3|\le \max \{M_{v,d(S)}+M_{n-v,d(S)}\mid 0\le v\le n/2\}.$$

$$\begin{array}[]{lll}W_{42}&=&1+(84+8\beta)y^{8}+(1449-24\beta)y^{10}+\cdots,\\ S_{42}&=&\beta y^{5}+(896-8\beta)y^{9}+(48384+28\beta)y^{13}+\cdots,\\ \end{array}$$

$$\begin{array}[]{lll}W_{62}&=&1+(1860+32\beta)y^{12}+(28055-160\beta)y^{14}+\cdots,\\ S_{62}&=&\beta y^{7}+(1116-12\beta)y^{11}+(171368+66\beta)y^{15}+\cdots,\end{array}$$

$$\begin{array}[]{lll}W_{82}&=&1+(39524+128\alpha)y^{16}+(556985-896\alpha)y^{18}+\cdots,\\ S_{82}&=&\alpha y^{9}+(1640-\alpha)y^{13}+(281424+120\alpha)y^{17}+\cdots,\\ \end{array}$$

$$\begin{array}[]{lll}W_{90}&=&1+(9180+8\beta)y^{16}+(-512\alpha-24\beta+224360)y^{18}+\cdots,\\ S_{90}&=&\alpha y^{9}+(\beta-18\alpha)y^{13}+(112320+153\alpha-16\beta)y^{17}+\cdots,\end{array}$$

$$n\equiv 0(\mathrm{mod}4)\text{ and }s=\frac{d}{2}+1.$$

$$B_s\le \frac{2n}{d+2}.$$

$$B_s\le d+2,B_s\ne d+1.$$

$$B_s\le 2\frac{2n-d-2}{d}.$$

$$l_1$$

$$l_2$$

$$|{\mathcal{B}}_i|\le l_1(i=1,3).$$

$$|{\mathcal{B}}_i|\le s.$$

$$l_1^{\prime }$$

$$l_2^{\prime }$$

$$\lceil \frac{n-d+2}{d+2}\rceil =\lceil \frac{n/4-(s/2-1)}{s/2}\rceil \le \frac{n}{2s},$$

$$l_1^{\prime }\le l_1,$$

$$4\lceil \frac{n-d+2}{d+2}\rceil -2\le \frac{2n}{s}-2<2\frac{2n-d-2}{d}.$$

$$2\frac{2n-d-2}{d}-\left(d+2-\sqrt{(d+2{)}^2-4n}\right)$$

$$=\frac{\sqrt{(d+2{)}^2-4n}}{d}(d-\sqrt{(d+2{)}^2-4n})$$

$$\ge 0.$$

$$k=\lceil \frac{n-d+2}{2s}\rceil .$$

Full text

Keywords: self-dual code, weight enumerator, shadow

Mathematics Subject Classification: 94B05

1 Introduction

Let $C$ be a singly even self-dual code and let $C_{0}$ denote the subcode of codewords having weight $\equiv 0\pmod{4}$. Then $C_{0}$ is a subcode of codimension $1$. The shadow $S$ of $C$ is defined to be $C_{0}^{\perp}\setminus C$. Shadows for self-dual codes were introduced by Conway and Sloane [6] in order to derive new upper bounds for the minimum weight of singly even self-dual codes, and to provide restrictions on the weight enumerators of singly even self-dual codes. The largest possible minimum weights of singly even self-dual codes of lengths $n\leq 72$ were given in [6, Table I]. The work was extended to lengths $74\leq n\leq 100$ in [9, Table VI]. We denote by $d(n)$ the largest possible minimum weight given in [6, Table I] and [9, Table VI] throughout this note. The possible weight enumerators of singly even self-dual codes having minimum weight $d(n)$ were also given in [6] for lengths $n\leq 64$ and $n=72$ (see also [9] for length $72$), and the work was extended to lengths up to $100$ in [9]. It is a fundamental problem to find which weight enumerators actually occur among the possible weight enumerators (see [6] and [11]).

Some restrictions on the number of vectors of weight $d/2$ in the shadow of a singly even self-dual $[n,n/2,d]$ code were given in [10]. Also, some restrictions on the number of vectors of weight $d/2+1$ in the shadow of a singly even self-dual $[n,n/2,d]$ code were given in [2] for $n\equiv 0\pmod{4}$. In this note, we improve the result in [2] about the restriction on the number of vectors of weight $d/2+1$ in the shadow of a singly even self-dual $[n,n/2,d]$ code for $n\equiv 0\pmod{4}$. We also give a restriction on the number of vectors of weight $d/2+1$ in the shadow of a singly even self-dual $[n,n/2,d]$ code for $n\equiv 2\pmod{4}$. These restrictions eliminate some of the possible weight enumerators determined in [6] and [9] for the parameters $(n,d)=(62,12)$, $(72,14)$, $(82,16)$, $(90,16)$ and $(100,18)$

2 Preliminaries

A (binary) $[n,k]$ code $C$ is a $k$-dimensional vector subspace of $\mathbb{F}_{2}^{n}$, where $\mathbb{F}_{2}$ denotes the finite field of order $2$. All codes in this note are binary. The parameter $n$ is called the length of $C$. The weight $\operatorname{wt}(x)$ of a vector $x\in\mathbb{F}_{2}^{n}$ is the number of non-zero components of $x$. A vector of $C$ is a codeword of $C$. The minimum non-zero weight of all codewords in $C$ is called the minimum weight $d(C)$ of $C$ and an $[n,k]$ code with minimum weight $d$ is called an $[n,k,d]$ code. The dual code $C^{\perp}$ of a code $C$ of length $n$ is defined as $C^{\perp}=\{x\in\mathbb{F}_{2}^{n}\mid x\cdot y=0\text{ for all }y\in C\},$ where $x\cdot y$ is the standard inner product. A code $C$ is called self-dual if $C=C^{\perp}$. A self-dual code $C$ is doubly even if all codewords of $C$ have weight divisible by four, and singly even if there exists at least one codeword of weight $\equiv 2\pmod{4}$. Rains [12] showed that the minimum weight $d$ of a self-dual code $C$ of length $n$ is bounded by $d\leq 4\lfloor{\frac{n}{24}}\rfloor+6$ if $n\equiv 22\pmod{24}$, $d\leq 4\lfloor{\frac{n}{24}}\rfloor+4$ otherwise. In addition, if $n\equiv 0\pmod{24}$ and $C$ is singly even, then $d\leq 4\lfloor{\frac{n}{24}}\rfloor+2$. A self-dual code meeting the bound is called extremal. Let $A_{i}$ and $B_{i}$ be the numbers of vectors of weight $i$ in $C$ and $S$, respectively. The weight enumerators of $C$ and $S$ are given by $\sum_{i=0}^{n}A_{i}y^{i}$ and $\sum_{i=d(S)}^{n-d(S)}B_{i}y^{i}$, respectively, where $d(S)$ denotes the minimum weight of $S$.

Let $C$ be a singly even self-dual code of length $n$ and let $S$ be the shadow of $C$. Let $C_{0}$ denote the subcode of codewords having weight $\equiv 0\pmod{4}$. There are cosets $C_{1},C_{2},C_{3}$ of $C_{0}$ such that $C_{0}^{\perp}=C_{0}\cup C_{1}\cup C_{2}\cup C_{3}$, where $C=C_{0}\cup C_{2}$ and $S=C_{1}\cup C_{3}$.

Lemma 1** (Conway and Sloane [6]).**

Let $x_{1},y_{1}$ be vectors of $C_{1}$ and let $x_{3}$ be a vector of $C_{3}$. Then $x_{1}+y_{1}\in C_{0}$, $x_{1}+x_{3}\in C_{2}$ and $\operatorname{wt}(x_{1})\equiv\operatorname{wt}(x_{3})\equiv\frac{n}{2}\pmod{4}$.

Lemma 2** (Brualdi and Pless [5]).**

Let $x_{1},y_{1}$ be vectors of $C_{1}$ and let $x_{3}$ be a vector of $C_{3}$.

Suppose that $n\equiv 0\pmod{4}$. Then $x_{1}\cdot y_{1}=0$ and $x_{1}\cdot x_{3}=1$. 2. 2)

Suppose that $n\equiv 2\pmod{4}$. Then $x_{1}\cdot y_{1}=1$ and $x_{1}\cdot x_{3}=0$.

3 $n\equiv 2\pmod{4}$ and $d(S)=\frac{d(C)}{2}+1$

Recall that the Johnson graph $J(v,d)$ has the collection $X$ of all $d$-subsets of $\{1,2,\dots,v\}$ as vertices, and two distinct vertices are adjacent whenever they share $d-1$ elements in common. Assume $v\geq 2d$ and set

[TABLE]

Then $\{R_{i}\}_{i=0}^{d}$ is a partition of $X\times X$. The following lemma is known as Delsarte’s inequalities since it is the basis of Delsarte’s linear programming bound. We refer the reader to [7] for an explicit formula for the second eigenmatrix $Q$ appearing in the lemma.

Lemma 3** ([4, Proposition 2.5.2]).**

Let $Y$ be a subset of vertices of $J(v,d)$, and set

[TABLE]

If we denote by $Q=(q^{(v)}_{j}(i))$ the second eigenmatrix of $J(v,d)$, then every entry of the vector $(a_{0},\dots,a_{d})Q$ is nonnegative.

Suppose that $Y$ is a subset of vertices of $J(v,d)$ such that two distinct members intersect at exactly one element. Then by Lemma 3, every entry of the vector

[TABLE]

is nonnegative, i.e.,

[TABLE]

Thus, we obtain

[TABLE]

where

[TABLE]

If we define

[TABLE]

then (1) also holds for all $v,d$.

Now, let $C$ be a singly even self-dual code of length $n$ and let $S$ be the shadow of $C$. For the remainder of this section, we assume that

[TABLE]

By Lemma 1, $d(C)\equiv n-2\pmod{8}$, and hence $d(S)$ is odd.

For each of $i=1,3$, let $Y_{i}$ be the set of supports of vectors of weight $d(S)$ in $C_{i}$, and let $S_{i}$ be the union of the members of $Y_{i}$. From Lemma 2 and (2), we have the following:

[TABLE]

Then by (1), we have

[TABLE]

It follows from (3) that $S_{1}\cap S_{3}=\emptyset$. Thus, we have

[TABLE]

For $42\leq n\leq 98$ and $d(C)=d(n)$, the parameters $(n,d(C),d(S))$ satisfying Condition (2) are listed in Table 1, where the values $d(n)$ are also listed in the table. For some lengths $n$, the existence of a singly even self-dual code of length $n$ and minimum weight $d(n)$ is currently not known. In this case, we consider the case $d(C)=d(n)-2$. We calculated the upper bound (4), where the results are listed in Table 1. This calculation was done by the program written in Magma [1], where the program is listed in Appendix A.

We discuss the possible weight enumerators for the case $d(n)=d(C)$ in Table 1. The possible weight enumerators $W_{42}$ and $S_{42}$ of an extremal singly even self-dual $[42,21,8]$ code with $d(S)\geq 5$ and its shadow are as follows [6]:

[TABLE]

respectively, where $\beta$ is an integer. It was shown in [3] that $0\leq\beta\leq 42$. Table 1 gives an alternative proof.

The possible weight enumerators $W_{62}$ and $S_{62}$ of an extremal singly even self-dual $[62,31,12]$ code with $d(S)\geq 7$ and its shadow are as follows [6] (see also [8]):

[TABLE]

respectively, where $\beta$ is an integer with $0\leq\beta\leq 93$. Table 1 gives the following:

Proposition 4**.**

If there exists an extremal singly even self-dual $[62,31,12]$ code with weight enumerator $W_{62}$, then $0\leq\beta\leq 48$.

It is known that there exists an extremal singly even self-dual $[62,31,12]$ code with weight enumerator $W_{62}$ for $\beta=0,2,9,10,15,16$ (see [13]).

The possible weight enumerators $W_{82}$ and $S_{82}$ of an extremal singly even self-dual $[82,41,16]$ code with $d(S)\geq 9$ and its shadow are as follows [9]:

[TABLE]

respectively, where $\alpha$ is an integer with $0\leq\alpha\leq\lfloor\frac{556985}{896}\rfloor=621$. Table 1 gives the following:

Proposition 5**.**

If there exists an extremal singly even self-dual $[82,41,16]$ code with weight enumerator $W_{82}$, then $0\leq\alpha\leq 74$.

It is unknown whether there exists an extremal singly even self-dual code for any of these cases.

The possible weight enumerators $W_{90}$ and $S_{90}$ of an extremal singly even self-dual $[90,45,16]$ code with $d(S)\geq 9$ and its shadow are as follows [9]:

[TABLE]

respectively, where $\alpha$ and $\beta$ are integers with $0\leq\alpha\leq\frac{1}{18}\beta\leq\lfloor\frac{224360}{24}\rfloor=9348$. Table 1 gives the following:

Proposition 6**.**

If there exists an extremal singly even self-dual $[90,45,16]$ code with weight enumerator $W_{90}$, then $0\leq\alpha\leq 76$.

It is unknown whether there exists an extremal singly even self-dual code for any of these cases.

4 $n\equiv 0\pmod{4}$ and $d(S)=\frac{d(C)}{2}+1$

Let $C$ be a singly even self-dual code of length $n$ and let $S$ be the shadow of $C$. In this section, we write $d(C)=d$ and $d(S)=s$ for short, and assume that

[TABLE]

By Lemma 1, $d\equiv n-2\pmod{8}$, and hence $s$ is even.

Proposition 7** ([2]).**

Suppose that $n\equiv 0\pmod{4}$ and $s=\frac{d}{2}+1$. Let $B_{s}$ denote the number of vectors of weight $s$ in $S$.

1. (i)

If $2n>(d+2)^{2}$, then

[TABLE] 2. (ii)

If $(d+2)^{2}\leq 4n\leq 2(d+2)^{2}$, then

[TABLE] 3. (iii)

If $4n<(d+2)^{2}$, then

[TABLE]

The above proposition was essentially established by showing $B_{s}\leq\max\{l_{1},l_{2}\}$, where

[TABLE]

We recall part of the proof of Proposition 7 for later use. Denote the set of all vectors in $C_{i}$ of weight $s$ by $\mathcal{B}_{i}$ ($i=1,3$). Denote by $v*w$ the entrywise product of two vectors $v,w$. If $v,w\in\mathcal{B}_{i}$, then $\operatorname{wt}(v*w)=0$ and hence these vectors have disjoint supports. This implies

[TABLE]

If $v\in\mathcal{B}_{1}$ and $w\in\mathcal{B}_{3}$, then $\operatorname{wt}(v*w)=1$. Thus, if $\mathcal{B}_{1}$ and $\mathcal{B}_{3}$ are both nonempty, then

[TABLE]

Using the following lemmas, we give an improvement of the upper bound by showing $B_{s}\leq\max\{l^{\prime}_{1},l^{\prime}_{2}\}$, where

[TABLE]

Since

[TABLE]

we have

[TABLE]

and

[TABLE]

The latter implies $l^{\prime}_{2}\leq l_{2}$ provided $4n\geq(d+2)^{2}$. If $4n<(d+2)^{2}$, then

[TABLE]

Thus $l^{\prime}_{2}\leq l_{2}$ holds in this case as well. Therefore, the bound $B_{s}\leq\max\{l^{\prime}_{1},l^{\prime}_{2}\}$ which will be shown in Proposition 10 below is an improvement of the bound given in Proposition 7.

Lemma 8**.**

Let

[TABLE]

If $n$ is not divisible by $2s$, then $|\mathcal{B}_{i}|\leq 2k-1$ for $i=1,3$.

Proof.

Suppose, to the contrary, $|\mathcal{B}_{i}|\geq 2k$. Then the sum of the all-one vector and the $2k$ vectors of weight $s$ belongs to $C_{0}$ and has weight $n-2ks\leq d-2$. This forces $n-2ks=0$, contradicting the assumption. ∎

Lemma 9**.**

Let $n$ and $s$ be positive integers with $n<s^{2}$. Then

[TABLE]

Proof.

Since $n<s^{2}$, we have

[TABLE]

The function $f(x)=(n-x^{2})/(s-x)$ defined on the interval $[0,s)$ has maximum $f(\alpha)=2\alpha$, where $\alpha=s-\sqrt{s^{2}-n}$. Thus, we have

[TABLE]

Define $a,b\in\mathbb{Z}$ by $a=\left\lfloor\alpha\right\rfloor$ and

[TABLE]

Then $a+b=\left\lfloor 2\alpha\right\rfloor=2s-\left\lceil 2\sqrt{s^{2}-n}\,\right\rceil$. Since $\alpha<s$, we have $b\leq s$. It remains to show $s(a+b)-ab\leq n$, or equivalently,

[TABLE]

Observe

[TABLE]

If $\alpha-\left\lfloor\alpha\right\rfloor<\frac{1}{2}$, then

[TABLE]

Thus, (10) holds.

If $\alpha-\left\lfloor\alpha\right\rfloor\geq\frac{1}{2}$, then

[TABLE]

Thus

[TABLE]

Since $ab-s(a+b)+n$ is an integer, (10) holds. ∎

Proposition 10**.**

Suppose that $n\equiv 0\pmod{4}$ and $s=\frac{d}{2}+1$. Let $B_{s}$ denote the number of vectors of weight $s$ in $S$. Then

[TABLE]

More precisely,

1. (i)

If $2n>d^{2}+6d$, then

[TABLE] 2. (ii)

If $(d+2)^{2}<2n\leq d^{2}+6d$, then

[TABLE] 3. (iii)

If $d^{2}+8d-4<4n\leq 2(d+2)^{2}$, then

[TABLE] 4. (iv)

If $(d+2)^{2}\leq 4n\leq d^{2}+8d-4$, then

[TABLE] 5. (v)

If $4n<(d+2)^{2}$, then

[TABLE]

Proof.

If one of $\mathcal{B}_{1}$ and $\mathcal{B}_{3}$ is empty, then (6) and Lemma 8 imply $B_{s}\leq l^{\prime}_{1}$. If $\mathcal{B}_{1}$ and $\mathcal{B}_{3}$ are both nonempty, then by (7), we have $B_{s}\leq 2s=d+2$. Moreover, suppose $n<s^{2}$. Observe

[TABLE]

and this is at most $n$. By (7), we can apply Lemma 9 to conclude

[TABLE]

Thus $B_{s}\leq l^{\prime}_{2}$. Therefore, (11) holds.

Next, we determine $\max\{l^{\prime}_{1},l^{\prime}_{2}\}$. If $2n>d^{2}+6d$, then

[TABLE]

so

[TABLE]

Thus $\max\{l^{\prime}_{1},l^{\prime}_{2}\}=l^{\prime}_{1}$, and (i) holds.

Next suppose $(d+2)^{2}<2n\leq d^{2}+6d$. Since

[TABLE]

we have $l^{\prime}_{2}=d+2$. Since

[TABLE]

we have

[TABLE]

These imply

[TABLE]

and (ii) holds.

Next suppose $(d+2)^{2}\leq 4n\leq 2(d+2)^{2}$. We claim

[TABLE]

Indeed, since $(d+4)/4=(s+1)/2\notin\mathbb{Z}$, we have

[TABLE]

Since $4n\geq(d+2)^{2}$ and $d\neq 4$, we have $n\geq 3d-2$. Thus

[TABLE]

This, together with $2n\leq(d+2)^{2}$ implies $l_{1}\leq l^{\prime}_{2}$. Therefore, $\max\{l^{\prime}_{1},l^{\prime}_{2}\}=l^{\prime}_{2}$. Now (iii) and (iv) hold by Proposition 7 (ii).

Finally, suppose $4n<(d+2)^{2}$. Then it is easy to verify

[TABLE]

hence $\max\{l^{\prime}_{1},l^{\prime}_{2}\}=l^{\prime}_{2}$ by (9). Thus (v) holds. ∎

Remark 11*.*

In Proposition 10 (v), it is sometimes possible to draw a stronger conclusion

[TABLE]

This is when a pair $\{a,b\}$ achieving the maximum in Lemma 9 is unique. For the parameters $(n,d,s)=(128,22,12)$, we necessarily have $|\mathcal{B}_{i}|=8$ for $i=1,3$. In general, a pair $\{a,b\}$ achieving the maximum in Lemma 9 is not unique. For example, when $(n,d,s)=(120,22,12)$, both $\{6,8\}$ and $\{7,7\}$ achieve the maximum.

For only the parameters $(n,d,s)=(72,14,8)$ and $(100,18,10)$, Proposition 10 gives an improvement over Proposition 7, for $44\leq n\leq 100$ and $d=d(n)$. The bounds on $B_{s}$ obtained by Proposition 10 are listed in Table 2 for these parameters, together with the part of Proposition 10 used, where the bounds by Proposition 7 are listed in the last column. The values $d(n)$ are also listed in the table.

We discuss the possible weight enumerators for the case $d(n)=d$ in Table 2. The possible weight enumerators of an extremal singly even self-dual $[72,36,14]$ code with $s\geq 8$ and the shadow are as follows:

[TABLE]

respectively, where $\alpha$ is an integer with $0\leq\alpha\leq\lfloor\frac{546}{14}\rfloor=39$ [9]. We remark that Conway and Sloane [6] give only two weight enumerators as the possible weight enumerators of an extremal singly even self-dual $[72,36,14]$ code with $s\geq 8$ without reason, namely $\alpha=0,1$ in $W_{72}$. Table 2 shows the following:

Proposition 12**.**

If there exists an extremal singly even self-dual $[72,36,14]$ code with weight enumerator $W_{72}$, then $0\leq\alpha\leq 14$.

It is unknown whether there exists an extremal singly even self-dual code for any of these cases.

The possible weight enumerators of a singly even self-dual $[100,50,18]$ code with $s\geq 10$ and the shadow are as follows:

[TABLE]

respectively, where $\alpha,\beta$ are integers with $0\leq\alpha\leq\frac{-1}{20}\beta\leq\frac{5225}{32}$ [9]. Table 2 shows the following:

Proposition 13**.**

If there exists a singly even self-dual $[100,50,18]$ code with weight enumerator $W_{100}$, then $0\leq\alpha\leq 18$.

It is unknown whether there exists a singly even self-dual $[100,50,18]$ code for any of these cases.

We give more sets of parameters for which the bound on $B_{s}$ obtained by Proposition 10 improves the bound obtained by Proposition 7:

[TABLE]

These bounds are also listed in Table 2.

Acknowledgment. This work was supported by JSPS KAKENHI Grant Number 15H03633.

Appendix Appendix A

HahnPolynomial:=function(v,k,l,x) return (Binomial(v,l)-Binomial(v,l-1))* &+[ (-1)^iBinomial(l,i)Binomial(v+1-l,i) Binomial(k,i)^(-1)Binomial(v-k,i)^(-1) Binomial(x,i) : i in [0..l] ]; end function; Qmatrix:=function(v,k) return Matrix(Rationals(),k+1,k+1, [[HahnPolynomial(v,k,l,x) : l in [0..k] ]: x in [0..k]]); end function; boundM:=function(v,ds) if v le ds-1 then return 0; elif v le ds2-2 then return 1; elif v eq ds*2-1 then return 2; else Q:=Qmatrix(v,ds); return Min( { 1-Q[1][i+1]/Q[ds][i+1] : i in [0..ds] | Q[ds][i+1] lt 0 } ); end if; end function; res:=function(n,ds) bounds:=[ Floor(boundM(v,ds)+boundM(n-v,ds)): v in {0..(n div 2)} ]; max:=Max(bounds); return max; end function;

X:=[[42,5],[62,7],[70,7],[82,9],[90,9],[98,9]]; [res(x[1],x[2]): x in X] eq [42,48,52,74,76,78];