Binary extremal self-dual codes of length $60$ and related codes

TL;DR

This paper classifies specific self-dual codes of length 60, constructs new extremal codes with previously unknown weight enumerators, and explores restrictions on weight enumerators of related codes.

Contribution

It provides a classification of four-circulant singly even self-dual codes of length 60 and constructs new extremal codes with novel weight enumerators.

Findings

New extremal singly even self-dual [60,30,12] codes constructed.

Existence of extremal codes with previously unknown weight enumerators.

Restrictions on weight enumerators of certain self-dual codes with shadow of minimum weight 1.

Abstract

We give a classification of four-circulant singly even self-dual $[60,30,d]$ codes for $d=10$ and $12$. These codes are used to construct extremal singly even self-dual $[60,30,12]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual $[60,30,12]$ codes, we also construct extremal singly even self-dual $[58,29,10]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight $1$.

Tables

Table 1. Table 1: Extremal four-circulant self-dual [ 60 , 30 , 12 ] 60 30 12 [60,30,12] codes C 60 , i subscript 𝐶 60 𝑖 C_{60,i}

Code	$r_A$	$r_B$	$\beta$
$C_{60,1}$	$(1,0,1,1,1,0,0,0,0,1,1,1,0,1,1)$	$(0,0,0,0,0,0,1,0,1,0,0,1,0,0,1)$	0
$C_{60,2}$	$(0,1,1,1,1,0,0,1,1,1,1,1,1,1,0)$	$(0,1,0,0,1,0,0,0,0,0,0,1,1,1,1)$	0
$C_{60,3}$	$(1,1,0,0,1,0,0,1,0,1,1,0,1,1,1)$	$(0,0,0,0,1,0,0,0,0,0,0,1,1,0,1)$	0
$C_{60,4}$	$(1,1,1,0,0,1,0,1,0,1,0,0,1,1,1)$	$(1,0,1,1,1,1,0,0,0,0,0,1,1,0,1)$	0
$C_{60,5}$	$(1,1,1,1,0,1,1,0,1,1,1,0,1,1,0)$	$(1,1,0,1,0,1,1,1,0,0,0,1,1,1,1)$	0
$C_{60,6}$	$(1,1,1,1,1,1,1,1,0,1,1,1,1,1,0)$	$(0,1,0,1,1,1,0,0,0,0,0,1,1,1,1)$	0
$C_{60,7}$	$(0,1,1,0,0,1,1,1,0,1,1,0,1,1,1)$	$(0,1,0,0,1,1,0,0,0,0,0,1,1,1,1)$	0
$C_{60,8}$	$(0,0,1,1,0,0,1,0,1,1,0,0,1,1,1)$	$(0,1,0,1,1,1,1,0,0,0,0,1,1,1,1)$	0
$C_{60,9}$	$(0,1,1,0,1,0,0,1,0,0,1,0,1,1,0)$	$(1,0,1,0,0,0,0,0,0,0,0,1,1,1,1)$	10
$C_{60,10}$	$(0,1,1,1,0,1,0,1,1,1,1,0,1,1,0)$	$(0,1,0,1,0,1,1,0,0,0,0,1,1,0,1)$	10
$C_{60,11}$	$(0,0,1,1,1,1,1,1,1,1,0,0,1,1,0)$	$(0,1,0,1,0,1,1,0,0,0,0,1,1,0,1)$	10
$C_{60,12}$	$(1,1,0,0,1,0,0,0,0,1,1,0,1,1,1)$	$(0,0,0,0,0,1,0,0,0,0,0,1,1,1,1)$	10
$C_{60,13}$	$(1,1,1,1,1,1,0,1,1,0,1,0,1,1,0)$	$(0,0,1,0,0,1,0,0,0,0,0,1,1,1,1)$	10

Table 2. Table 2: Extremal self-dual [ 60 , 30 , 12 ] 60 30 12 [60,30,12] neighbors D 60 , i subscript 𝐷 60 𝑖 D_{60,i}

$C$	$D$	$\mathrm{supp}\left(x\right)$	$W$
$D_{60,1}$	$C_{60,1}$	$\{1,31,32,38,42,43,46,47,48,50,51,55\}$	$\beta =0$
$D_{60,2}$	$C_{60,1}$	$\{2,3,8,33,35,39,40,41,46,50,54,59\}$	$\beta =0$
$D_{60,3}$	$C_{60,1}$	$\{4,8,9,32,42,43,48,51,53,54,56,60\}$	$\beta =2$
$D_{60,4}$	$C_{60,2}$	$\{2,32,34,38,40,43,49,52,54,55,57,59\}$	$\beta =0$
$D_{60,5}$	$C_{60,4}$	$\{1,31,35,39,40,41,42,43,50,52,54,55\}$	$\beta =0$
$D_{60,6}$	$C_{60,10}$	$\{2,32,38,41,43,49,51,52,54,55,56,60\}$	$\beta =10$
$D_{60,7}$	$C_{60,12}$	$\{3,7,10,32,35,36,38,46,53,55,58,60\}$	$\beta =10$

Table 3. Table 3: Extremal self-dual [ 60 , 30 , 12 ] 60 30 12 [60,30,12] neighbors E 60 , i subscript 𝐸 60 𝑖 E_{60,i} and F 60 subscript 𝐹 60 F_{60}

$C$	$D$	$\mathrm{supp}\left(x\right)$	$W$
$E_{60,1}$	$D_{60,2}$	$\{2,3,6,31,32,37,39,40,46,47,54,57\}$	$\beta =0$
$E_{60,2}$	$D_{60,6}$	$\{1,2,5,7,8,40,43,46,47,50,51,60\}$	$\beta =10$
$E_{60,3}$	$D_{60,6}$	$\{1,4,5,8,36,38,39,40,48,53,55,60\}$	$\beta =10$
$E_{60,4}$	$D_{60,6}$	$\{3,32,33,34,37,46,48,52,56,57,58,60\}$	$\beta =10$
$F_{60}$	$E_{60,3}$	$\{1,2,5,35,37,40,45,49,50,55,57,59\}$	$\beta =10$

Table 4. Table 4: Four-circulant self-dual [ 60 , 30 , 10 ] 60 30 10 [60,30,10] codes G 60 , i subscript 𝐺 60 𝑖 G_{60,i}

Code	$r_A$	$r_B$
$G_{60,1}$	$(0,1,1,1,0,1,0,0,0,0,0,0,0,0,1)$	$(1,1,0,0,1,0,0,1,1,1,0,1,0,0,1)$
$G_{60,2}$	$(0,0,0,1,0,0,0,1,1,1,1,0,0,0,1)$	$(0,0,0,0,0,1,0,1,1,1,0,1,0,1,1)$
$G_{60,3}$	$(1,1,1,1,1,0,0,1,0,1,0,1,0,0,0)$	$(0,1,1,0,1,1,0,1,1,1,0,1,0,0,1)$
$G_{60,4}$	$(1,1,1,1,1,1,0,0,1,0,1,0,1,0,0)$	$(1,1,1,0,1,1,1,1,1,1,0,1,0,1,1)$
$G_{60,5}$	$(0,0,0,1,0,0,0,1,1,1,1,0,0,0,1)$	$(0,1,1,0,1,1,1,1,1,1,0,1,0,1,1)$
$G_{60,6}$	$(0,0,1,1,0,1,1,0,1,0,0,1,0,0,0)$	$(1,1,1,1,1,0,1,1,1,1,0,1,0,0,1)$
$G_{60,7}$	$(0,0,1,1,0,0,1,1,0,0,1,1,0,0,0)$	$(0,0,1,0,1,0,1,0,1,1,0,1,0,0,1)$
$G_{60,8}$	$(0,0,0,0,0,0,1,1,1,1,0,1,0,0,0)$	$(0,1,1,1,0,1,0,0,0,1,0,1,0,1,1)$
$G_{60,9}$	$(0,1,1,0,1,0,0,0,0,0,0,0,0,0,1)$	$(1,0,1,1,1,1,0,0,0,1,0,1,0,1,1)$
$G_{60,10}$	$(0,0,0,1,1,0,1,0,1,0,1,1,0,0,0)$	$(1,0,1,0,0,0,1,0,1,1,0,1,0,0,1)$
$G_{60,11}$	$(1,0,1,0,1,0,1,1,1,0,0,0,0,0,1)$	$(0,0,1,0,0,0,1,0,1,1,0,1,0,0,1)$
$G_{60,12}$	$(1,1,1,1,1,0,0,1,0,1,0,1,0,0,1)$	$(1,0,1,0,0,1,0,1,1,1,0,1,0,0,1)$
$G_{60,13}$	$(0,0,1,0,0,0,0,0,0,1,0,1,0,0,0)$	$(0,0,0,0,0,0,1,1,0,1,0,1,0,1,1)$

Table 5. Table 5: Extremal self-dual [ 60 , 30 , 12 ] 60 30 12 [60,30,12] neighbors H 60 , i subscript 𝐻 60 𝑖 H_{60,i} , J 60 , i subscript 𝐽 60 𝑖 J_{60,i} , K 60 , i subscript 𝐾 60 𝑖 K_{60,i} and L 60 , i subscript 𝐿 60 𝑖 L_{60,i}

$C$	$D$	$\mathrm{supp}\left(x\right)$	$W$
$H_{60,1}$	$G_{60,1}$	$\{4,30,32,36,37,40,42,43,47,51,52,54,57,58\}$	$\beta =10$
$H_{60,2}$	$G_{60,2}$	$\{1,2,4,32,36,39,40,42,49,50,55,56,57,58\}$	$\beta =0$
$H_{60,3}$	$G_{60,3}$	$\{1,2,32,33,34,35,37,44,45,48,51,54,55,59\}$	$\beta =0$
$H_{60,4}$	$G_{60,4}$	$\{2,30,32,34,36,37,40,44,48,52,55,56,58,60\}$	$\beta =0$
$H_{60,5}$	$G_{60,5}$	$\{1,31,32,33,35,36,37,42,43,44,46,49,56,58\}$	$\beta =0$
$H_{60,6}$	$G_{60,6}$	$\{1,31,32,35,38,40,41,43,44,45,51,56,58,59\}$	$\beta =0$
$H_{60,7}$	$G_{60,7}$	$\{1,3,5,32,37,38,41,42,50,51,53,56,57,59\}$	$\beta =0$
$H_{60,8}$	$G_{60,8}$	$\{1,2,3,5,6,33,34,35,36,38,42,55,56,57\}$	$\beta =0$
$H_{60,9}$	$G_{60,9}$	$\{1,2,6,33,35,39,41,42,43,44,46,47,55,56\}$	$\beta =0$
$H_{60,10}$	$G_{60,10}$	$\{2,30,32,33,36,41,43,44,48,49,51,56,59,60\}$	$\beta =0$
$H_{60,11}$	$G_{60,11}$	$\{2,3,32,39,40,42,43,44,45,48,51,52,56,60\}$	$\beta =0$
$H_{60,12}$	$G_{60,12}$	$\{2,3,32,33,34,35,36,38,45,47,51,55,56,60\}$	$\beta =0$
$H_{60,13}$	$G_{60,13}$	$\{1,30,35,37,41,43,44,45,46,47,48,50,54,56\}$	$\beta =0$
$J_{60,1}$	$H_{60,1}$	$\{4,6,36,41,43,48,49,51,53,55,56,59\}$	$\beta =10$
$J_{60,2}$	$H_{60,3}$	$\{1,3,4,30,32,36,37,38,51,52,55,56\}$	$\beta =0$
$J_{60,3}$	$H_{60,3}$	$\{1,31,34,36,39,40,41,44,45,55,56,59\}$	$\beta =0$
$J_{60,4}$	$H_{60,4}$	$\{2,5,33,34,37,38,39,42,44,50,57,59\}$	$\beta =0$
$J_{60,5}$	$H_{60,4}$	$\{3,6,7,30,34,36,39,42,48,50,53,58\}$	$\beta =6$
$K_{60,1}$	$J_{60,1}$	$\{1,3,4,6,36,40,41,46,47,51,54,57\}$	$\beta =10$
$K_{60,2}$	$J_{60,4}$	$\{5,7,34,36,37,40,41,42,47,49,50,60\}$	$\beta =6$
$L_{60,1}$	$K_{60,1}$	$\{2,3,32,33,37,38,41,46,51,52,57,58\}$	$\beta =10$
$L_{60,2}$	$K_{60,2}$	$\{2,3,32,34,37,38,41,44,47,51,53,58\}$	$\beta =6$

Table 6. Table 6: Extremal self-dual [ 58 , 29 , 10 ] 58 29 10 [58,29,10] codes C 58 , i subscript 𝐶 58 𝑖 C_{58,i}

Code	$(i,j)$	Code	$(i,j)$	Code	$(i,j)$
$C_{58,1}$	$(2,36)$	$C_{58,7}$	$(12,31)$	$C_{58,13}$	$(22,41)$
$C_{58,2}$	$(2,41)$	$C_{58,8}$	$(12,36)$	$C_{58,14}$	$(22,48)$
$C_{58,3}$	$(2,58)$	$C_{58,9}$	$(12,53)$	$C_{58,15}$	$(22,58)$
$C_{58,4}$	$(7,31)$	$C_{58,10}$	$(17,36)$	$C_{58,16}$	$(27,31)$
$C_{58,5}$	$(7,41)$	$C_{58,11}$	$(17,53)$	$C_{58,17}$	$(27,48)$
$C_{58,6}$	$(7,48)$	$C_{58,12}$	$(17,58)$	$C_{58,18}$	$(27,53)$

Table 7. Table 7: Extremal self-dual [ 58 , 29 , 10 ] 58 29 10 [58,29,10] neighbors

$C$	$D$	$\mathrm{supp}\left(x\right)$	$W$
$D_{58,1}$	$C_{58,1}$	$\{3,4,28,30,33,41,43,44,52,53,55,56\}$	$(2,102)$
$D_{58,2}$	$C_{58,1}$	$\{2,3,5,28,33,34,35,41,42,44,45,50\}$	$(2,108)$
$D_{58,3}$	$C_{58,3}$	$\{1,4,6,7,8,39,40,41,42,43,47,52\}$	$(2,28)$
$E_{58,1}$	$D_{58,2}$	$\{1,3,6,7,34,35,40,47,49,51\}$	$(2,106)$
$E_{58,2}$	$D_{58,3}$	$\{1,6,10,28,31,32,33,40,53,54\}$	$(0,24)$
$E_{58,3}$	$D_{58,3}$	$\{2,6,7,8,31,34,38,45,51,57\}$	$(1,24)$
$E_{58,4}$	$D_{58,3}$	$\{6,8,28,30,39,40,41,46,57,58\}$	$(1,30)$
$E_{58,5}$	$D_{58,3}$	$\{6,32,33,38,41,42,44,46,47,57\}$	$(2,16)$
$E_{58,6}$	$D_{58,3}$	$\{2,5,6,33,36,39,42,52,53,56\}$	$(2,20)$
$E_{58,7}$	$D_{58,3}$	$\{5,6,8,36,38,41,48,51,52,55\}$	$(2,24)$
$E_{58,8}$	$D_{58,3}$	$\{3,6,7,12,32,33,34,43,47,54\}$	$(2,26)$
$E_{58,9}$	$D_{58,3}$	$\{1,8,13,35,36,44,47,50,53,55\}$	$(2,30)$
$F_{58,1}$	$E_{58,6}$	$\{1,4,36,38,39,41,43,45,49,58\}$	$(0,14)$
$F_{58,2}$	$E_{58,5}$	$\{1,5,6,7,8,11,29,31,44,46\}$	$(1,16)$
$F_{58,3}$	$E_{58,5}$	$\{1,2,30,32,35,44,46,47,53,58\}$	$(1,18)$
$F_{58,4}$	$E_{58,5}$	$\{4,6,9,34,41,42,45,50,51,56\}$	$(1,20)$
$F_{58,5}$	$E_{58,5}$	$\{4,5,37,41,47,49,50,55,56,58\}$	$(1,22)$
$F_{58,6}$	$E_{58,5}$	$\{1,4,6,7,8,35,39,41,42,43\}$	$(2,8)$
$F_{58,7}$	$E_{58,5}$	$\{6,7,12,15,41,43,46,47,49,56\}$	$(2,12)$
$F_{58,8}$	$E_{58,5}$	$\{1,6,8,29,34,39,47,50,54,55\}$	$(2,18)$
$F_{58,9}$	$E_{58,5}$	$\{3,35,36,38,39,42,47,49,50,56\}$	$(2,22)$
$G_{58,1}$	$F_{58,1}$	$\{2,7,11,29,31,32,33,48,49,52\}$	$(0,8)$
$G_{58,2}$	$F_{58,7}$	$\{3,9,32,38,47,48,49,51,52,55\}$	$(0,4)$
$G_{58,3}$	$F_{58,7}$	$\{1,8,12,30,33,40,42,49,50,55\}$	$(2,14)$
$H_{58}$	$G_{58,2}$	$\{5,6,7,9,32,44,46,47,49,58\}$	$(0,6)$

Equations

$$W_{60,1}$$

$$W_{60,2}$$

$$W_{58,1}$$

$$W_{58,2}$$

$$\beta =0\text{ and }$$

$$\beta =1\text{ and }$$

$$\beta =2\text{ and }$$

$$\beta =0\text{ and }$$

$$\beta =1\text{ and }$$

$$\beta =2\text{ and }$$

$$\left(\begin{array}[]{ccccc}r_{0}&r_{1}&r_{2}&\cdots&r_{n-1}\\ r_{n-1}&r_{0}&r_{1}&\cdots&r_{n-2}\\ \vdots&\vdots&\vdots&&\vdots\\ r_{1}&r_{2}&r_{3}&\cdots&r_{0}\end{array}\right),$$

$$\left(\begin{array}[]{ccc@{}c}&{\Large I_{2n}}&&\begin{array}[]{cc}A&B\\ B^{T}&A^{T}\end{array}\end{array}\right),$$

$$n_i=\{\begin{array}{ll}3& \text{if }i=1,\\ 1& \text{if }i=2,4,10,12,\\ 0& \text{otherwise.}\end{array}$$

$$⟨(D\cap ⟨x{⟩}^{\perp }),x⟩,$$

$$1+2683y^{12}+32832y^{14}+280017y^{16}+1719808y^{18}+7800120y^{20}$$

$$+26380032y^{22}+67167368y^{24}+130134528y^{26}+193185267y^{28}$$

$$+220336512y^{30}+\cdots +y^{60}.$$

$$H_{60,2}\cong C_{60,4},H_{60,5}\cong C_{60,1},H_{60,6}\cong C_{60,3},H_{60,7}\cong C_{60,8},H_{60,8}\cong C_{60,7},$$

$$H_{60,9}\cong C_{60,2},H_{60,11}\cong H_{60,3},H_{60,12}\cong H_{60,10},H_{60,13}\cong H_{60,4},H_{60,10}\cong D_{60,2},$$

$$C_{60,i}(i=1,2,\dots ,13),D_{60,i}(i=1,2,\dots ,7),E_{60,i}(i=1,2,3,4),F_{60},$$

$$H_{60,i}(i=1,3,4),J_{60,i}(i=1,2,3,4,5),K_{60,i}(i=1,2),L_{60,i}(i=1,2).$$

$$1+2939y^{12}+31296y^{14}+282321y^{16}+1723904y^{18}+7784760y^{20}$$

$$+26386176y^{22}+67197064y^{24}+130097664y^{26}+193168371y^{28}$$

$$+220392832y^{30}+\cdots +y^{60}.$$

$$C_{58,1}\cong C_{58,i}(i=2,4,5,7,8,11,12,14,15,17,18),$$

$$C_{58,3}\cong C_{58,i}(i=6,9,10,13,16),$$

$$⟨(D\cap ⟨x{⟩}^{\perp }),x⟩,$$

$$(n,d)=(58,10),(74,14)\text{ and }(98,18).$$

$$y+\gamma y^9+(23918-10\gamma )y^{13}+\cdots .$$

$$165-2\gamma =\gamma .$$

$$1+2044y^{14}+159067y^{16}+520782y^{18}+\cdots ,$$

$$y+2044y^{13}+679849y^{17}+44010824y^{21}+\cdots ,$$

$$1+22116y^{18}+2016048y^{20}+7181104y^{22}+\cdots ,$$

$$y+22116y^{17}+9197152y^{21}+964758896y^{25}+\cdots ,$$

Full text

Keywords: extremal self-dual code; weight enumerator; neighbor.

MSC 2010 Codes: 94B05

1 Introduction

Let $C$ be a (binary) singly even self-dual code. All codes in this note are binary. Let $C_{0}$ denote the subcode of $C$ consisting of codewords having weight $\equiv 0\pmod{4}$. 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 [3] 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. By considering shadows, the largest possible minimum weights of singly even self-dual codes of lengths up to $72$ are given in [3, Table I]. In this note, we say that a singly even self-dual code with the largest possible minimum weight given in [3, Table I] is extremal.

The possible weight enumerators of extremal singly even self-dual codes are given in [3] for lengths up to $64$ and length $72$ (see also [7] for length $60$). It is a fundamental problem to find which weight enumerators actually occur for the possible weight enumerators (see [3]). The possible weight enumerators of extremal singly even self-dual $[60,30,12]$ codes are known as follows:

[TABLE]

where $\beta$ is an integer. If there is an extremal singly even self-dual $[60,30,12]$ code with weight enumerator $W_{60,1}$, then $\beta\in\{0,1,2,\ldots,8,10\}$ [8]. For $\beta=0,1,5,7$ and $10$, an extremal singly even self-dual code with weight enumerator $W_{60,1}$ was found in [12], [2], [13], [5] and [7], respectively. An extremal singly even self-dual code with weight enumerator $W_{60,2}$ was found in [3].

One of the main aims of this note is to show the following:

Proposition 1**.**

There is an extremal singly even self-dual $[60,30,12]$ code with weight enumerator $W_{60,1}$ for $\beta=2,6$.

These codes are constructed from four-circulant singly even self-dual $[60,30,d]$ codes for $d=10$ and $12$ by considering self-dual neighbors. It remains to determine whether there is an extremal singly even self-dual $[60,30,12]$ code with weight enumerator $W_{60,1}$ for $\beta=3,4,8$.

The possible weight enumerators of extremal singly even self-dual $[58,29,10]$ codes are known as follows:

[TABLE]

where $\beta,\gamma$ are integers [3]. If there is an extremal singly even self-dual $[58,29,10]$ code with weight enumerator $W_{58,2}$, then $\beta\in\{0,1,2\}$ [8]. An extremal singly even self-dual code with weight enumerator $W_{58,1}$ is known for $\gamma=55$ [11]. An extremal singly even self-dual code with weight enumerator $W_{58,2}$ is known for

[TABLE]

(see [9], [10], [13]).

The following proposition is one of the main results of this note.

Proposition 2**.**

There is an extremal singly even self-dual $[58,29,10]$ code with weight enumerator $W_{58,2}$ for

[TABLE]

These codes are constructed from extremal singly even self-dual $[60,30,12]$ codes constructed in this note by subtracting and their self-dual neighbors. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight $1$ (Proposition 5). As a consequence, it is shown that $\gamma=55$ for the possible weight enumerator $W_{58,1}$ (Corollary 6). All self-dual codes in this note are singly even. From now on, we omit the term singly even.

All computer calculations in this note were done with the help of Magma [1].

2 Extremal four-circulant

self-dual $[60,30,12]$ codes

An $n\times n$ circulant matrix has the following form:

[TABLE]

so that each successive row is a cyclic shift of the previous one. Let $A$ and $B$ be $n\times n$ circulant matrices. Let $C$ be a $[4n,2n]$ code with generator matrix of the following form:

[TABLE]

where $I_{n}$ denotes the identity matrix of order $n$ and $A^{T}$ denotes the transpose of $A$. It is easy to see that $C$ is self-dual if $AA^{T}+BB^{T}=I_{n}$. The codes with generator matrices of the form (1) are called four-circulant.

In this section, we give a classification of extremal four-circulant self-dual $[60,30,12]$ codes. Two codes are equivalent if one can be obtained from the other by a permutation of coordinates. Our exhaustive search found all distinct extremal four-circulant self-dual $[60,30,12]$ codes, which must be checked further for equivalence to complete the classification. This was done by considering all pairs of $15\times 15$ circulant matrices $A$ and $B$ satisfying the condition that $AA^{T}+BB^{T}=I_{15}$, the sum of the weights of the first rows of $A$ and $B$ is congruent to $1\pmod{4}$ and the sum of the weights is greater than or equal to $13$. Since a cyclic shift of the first rows gives an equivalent code, we may assume without loss of generality that the last entry of the first row of $B$ is $1$. Then our computer search shows that the above distinct extremal four-circulant self-dual $[60,30,12]$ codes are divided into $13$ inequivalent codes.

Proposition 3**.**

Up to equivalence, there are $13$ extremal four-circulant self-dual $[60,30,12]$ codes.

We denote the $13$ codes by $C_{60,i}$ $(i=1,2,\ldots,13)$. For the $13$ codes $C_{60,i}$ $(i=1,2,\ldots,13)$, the first rows $r_{A}$ (resp. $r_{B}$) of the circulant matrices $A$ (resp. $B$) in generator matrices (1) are listed in Table 1. We verified that the codes $C_{60,i}$ have weight enumerator $W_{60,1}$, where $\beta$ are also listed in Table 1.

3 Extremal self-dual $[60,30,12]$ neighbors

Two self-dual codes $C$ and $C^{\prime}$ of length $n$ are said to be neighbors if $\dim(C\cap C^{\prime})=n/2-1$. Any self-dual code of length $n$ can be reached from any other by taking successive neighbors (see [3]). It is known that a self-dual code $C$ of length $n$ has $2(2^{n/2-1}-1)$ self-dual neighbors. These neighbors are constructed by finding $2^{n/2-1}-1$ subcodes of codimension $1$ in $C$ containing the all-one vector. A computer program written in Magma, which was used to find self-dual neighbors, can be obtained electronically from http://www.math.is.tohoku.ac.jp/~mharada/Paper/neighbor.txt. In this section, we construct extremal self-dual $[60,30,12]$ codes by considering self-dual neighbors.

For $i=1,2,\ldots,13$, by finding all $2(2^{29}-1)$ self-dual neighbors of $C_{60,i}$, we determined the equivalence classes among extremal self-dual neighbors of $C_{60,i}$. Our computer search shows that the code $C_{60,i}$ has $n_{i}$ inequivalent extremal self-dual neighbors, which are equivalent to none of the $13$ codes $C_{60,j}$, where $n_{i}$ are given by

[TABLE]

We denote the $7$ extremal self-dual codes by $D_{60,i}$ $(i=1,2,\ldots,7)$. These codes $C=D_{60,i}$ are constructed as

[TABLE]

where $D$ and the support $\operatorname{supp}(x)$ of $x$ are listed in Table 2. We verified that the codes $D_{60,i}$ have weight enumerator $W_{60,1}$, where $W$ in Table 2 indicates the values $\beta$ in the weight enumerator $W_{60,1}$. The code $D_{60,3}$ has the following weight enumerator:

[TABLE]

We verified that there is no pair of equivalent codes among the $13$ codes $C_{60,i}$ and the $7$ codes $D_{60,i}$.

We continue the search to find extremal self-dual codes by considering self-dual neighbors. We found all inequivalent extremal self-dual neighbors $E_{60,i_{1}}$ of $D_{60,i_{2}}$, which are equivalent to none of the extremal self-dual codes previously obtained in this note. For the codes $E_{60,i_{1}}=\langle(D\cap\langle x\rangle^{\perp}),x\rangle$, $D$ and $\operatorname{supp}(x)$ are listed in Table 3. In the table, $W$ indicates the values $\beta$ in the weight enumerator $W_{60,1}$. By continuing this process, we found all inequivalent extremal self-dual neighbors of $E_{60,i}$, which are equivalent to none of the extremal self-dual codes previously obtained in this note. Finally, we verified that there is no extremal self-dual neighbor of $F_{60}$, which are equivalent to none of the extremal self-dual codes previously obtained in this note.

4 Extremal four-circulant

self-dual $[60,30,10]$ codes and self-dual neighbors

Using an approach similar to that given in Section 2, our exhaustive search found all distinct four-circulant self-dual $[60,30,10]$ codes. Then our computer search shows that the distinct four-circulant self-dual $[60,30,10]$ codes are divided into $113$ inequivalent codes.

Proposition 4**.**

Up to equivalence, there are $113$ four-circulant self-dual $[60,30,10]$ codes.

We denote the $113$ codes by $G_{60,i}$ $(i=1,2,\ldots,113)$. For the $13$ codes $G_{60,i}$ $(i=1,2,\ldots,13)$, the first rows $r_{A}$ (resp. $r_{B}$) of the circulant matrices $A$ (resp. $B$) in generator matrices (1) are listed in Table 4. The first rows for the all codes can be obtained from http://www.math.is.tohoku.ac.jp/~mharada/Paper/60-4cir-d10.txt.

In addition, we found extremal self-dual $[60,30,12]$ codes by considering self-dual neighbors of $G_{62,i}$ $(i=1,2,\ldots,113)$. Using a method similar to that given in [4], we completed the classification of extremal self-dual $[60,30,12]$ neighbors of $G_{62,i}$ $(i=1,2,\ldots,113)$. Our computer search shows that there is an extremal self-dual $[60,30,12]$ neighbor $H_{60,i}$ for $i=1,2,\ldots,13$ and that there is no extremal self-dual $[60,30,12]$ neighbor for $i=14,15,\ldots,113$. The codes $H_{60,i}$ are constructed as $\langle(D\cap\langle x\rangle^{\perp}),x\rangle$, where $D$ and $\operatorname{supp}(x)$ are listed in Table 5 and $W$ indicates the values $\beta$ in the weight enumerator $W_{60,1}$. We verified that there are the following equivalent codes among $C_{60,i_{1}},D_{60,i_{2}},E_{60,i_{3}},F_{60},H_{60,i_{4}}$:

[TABLE]

where $C\cong D$ means that $C$ and $D$ are equivalent.

Similar to Section 3, by continuing this process, we completed a classification of extremal self-dual neighbors $J_{60,i}$ (resp.  $K_{60,i}$, $L_{60,i}$), which are equivalent to none of the extremal self-dual codes previously obtained in this note, of $H_{60,j}$ (resp. $J_{60,j}$, $K_{60,j}$). Finally, we verified that there is no extremal self-dual neighbor of $L_{60,i}$ $(i=1,2)$, which are equivalent to none of the $37$ codes in Tables 1, 2, 3 and 5. We remark that there is no pair of equivalent codes among the following $37$ codes:

[TABLE]

The codes $D_{60,3}$ and $J_{60,5}$ (see Tables 2 and 5) establish Proposition 1. The code $J_{60,5}$ has the following weight enumerator:

[TABLE]

5 Extremal self-dual $[58,29,10]$ codes

An extremal self-dual $[60,30,12]$ code gives an extremal self-dual $[58,29,10]$ code by subtracting two coordinates. We found all the extremal self-dual $[58,29,10]$ codes by subtracting from the $37$ inequivalent extremal self-dual $[60,30,12]$ codes given in Sections 2, 3 and 4. The only extremal self-dual $[60,30,12]$ code $D_{60,3}$ gives $18$ extremal self-dual $[58,29,10]$ codes $C_{58,i}$ $(i=1,2,\ldots,18)$ with weight enumerator for which no extremal self-dual code was previously known to exist. More precisely, the codes by subtracting $i$ and $j$ have weight enumerator $W_{58,2}$ for $\beta=2$ and $\gamma=104$, where $(i,j)$ are listed in Table 6. We verified that there are the following equivalent codes:

[TABLE]

where $C_{58,1}$ and $C_{58,3}$ are inequivalent.

Similar to Sections 3 and 4, we continue the search to find extremal self-dual $[58,29,10]$ codes with weight enumerator for which no extremal self-dual code was previously known to exist, by considering self-dual neighbors of $C_{58,i}$ $(i=1,3)$. These codes $C=D_{58,i}$ are constructed as

[TABLE]

where $D$ and $\operatorname{supp}(x)$ are listed in Table 7. We verified that the codes $D_{58,i}$ have weight enumerator $W_{58,2}$, where $W$ in Table 7 indicates the values $(\beta,\gamma)$ in the weight enumerator $W_{58,2}$. By continuing this process, we found more extremal self-dual $[58,29,10]$ codes with weight enumerator for which no extremal self-dual code was previously known to exist. The results are listed in Table 7. From Tables 6 and 7, we have Proposition 2.

6 Weight enumerator $W_{58,1}$

In this section, we give a remark on the possible weight enumerator $W_{58,1}$. First, we discuss a general case including $W_{58,1}$.

Proposition 5**.**

Let $C$ be a self-dual $[n,n/2,d]$ code with shadow $S$ of minimum weight $1$. Let $A_{i}$ and $B_{i}$ denote the numbers of vectors of weight $i$ in $C$ and $S$, respectively. Suppose that $n\equiv 2\pmod{8}$ and $d\equiv 2\pmod{4}$. Then $B_{d-1}=A_{d}$.

Proof.

Let $x$ be the vector of weight $1$ and let $y$ be a vector of weight $d-1$ in $S$. Since $x+y\in C$, $x+y$ has weight $d$. Thus, $B_{d-1}\leq A_{d}$.

Now let $c$ be a codeword of weight $d$ in $C$ and let $x$ be the vector of weight $1$ in $S$. Then we have $x+c\in S$. From the assumption that $n\equiv 2\pmod{8}$, the weight of $x+c$ is congruent to $1\pmod{4}$ by Theorem 5 in [3]. Hence, from the assumption that $d\equiv 2\pmod{4}$, $x+c$ has weight $d-1$. Thus, $B_{d-1}\geq A_{d}$. The result follows. ∎

For example, Proposition 5 can be applied to the following parameters:

[TABLE]

• •

$(n,d)=(58,10)$:

The possible weight enumerator of the shadow of an extremal self-dual $[58,29,10]$ code with weight enumerator $W_{58,1}$ is as follows [3]:

[TABLE]

By Proposition 5, we have

[TABLE]

Since there is an extremal self-dual $[58,29,10]$ code with weight enumerator $W_{58,1}$ for $\gamma=55$ [11], we have the following:

Corollary 6**.**

There is an extremal self-dual $[58,29,10]$ code with weight enumerator $W_{58,1}$ if and only if $\gamma=55$.

• •

$(n,d)=(74,14)$:

The weight enumerator $W_{2}$ in [6, p. 2039] is the possible weight enumerator of an extremal self-dual $[74,37,14]$ code with shadow of minimum weight $1$. By Proposition 5, we have $\alpha=-135$ for $W_{2}$ in [6, p. 2039]. The weight enumerators of such a code and its shadow are as follows:

[TABLE]

respectively. It is still unknown whether there is an extremal self-dual $[74,37,14]$ code (with shadow of minimum weight $1$).

• •

$(n,d)=(98,18)$:

The weight enumerator $W_{3}$ in [6, p. 2041] is the unique weight enumerator for an extremal self-dual $[98,49,18]$ code with shadow of minimum weight $1$. The weight enumerators of such a code and its shadow are as follows:

[TABLE]

respectively. It is still unknown whether there is an extremal self-dual $[98,49,18]$ code (with shadow of minimum weight $1$).

Acknowledgment. This work was supported by JSPS KAKENHI Grant Number 15H03633. The author would like to thank the anonymous reviewers for the useful comments.