New doubly even self-dual codes having minimum weight 20

TL;DR

This paper constructs new doubly even self-dual codes with minimum weight 20 for lengths 112, 120, and 128, revealing the existence of multiple inequivalent extremal codes at length 112.

Contribution

It introduces new extremal doubly even self-dual codes of specific lengths, expanding the known classifications.

Findings

Existence of at least three inequivalent extremal codes of length 112.

Construction of new codes with minimum weight 20 for lengths 120 and 128.

Enhanced understanding of the structure of self-dual codes.

Abstract

In this note, we construct new doubly even self-dual codes having minimum weight $20$ for lengths $112$, $120$ and $128$. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length $112$.

Tables

Table 1. Table 1: Weight distribution of C 112 subscript 𝐶 112 C_{112}

$i$	$A_i$	$i$	$A_i$
$0,112$	1	$38,74$	31676520067584
$18,94$	8512	$40,72$	109690203298312
$20,92$	186060	$42,70$	325630986391040
$22,90$	3239936	$44,68$	831288282918576
$24,88$	47551798	$46,66$	1829637194737408
$26,86$	561437184	$48,64$	3479230392288469
$28,84$	5424089452	$50,62$	5725819388994432
$30,82$	43459872064	$52,60$	8165553897114152
$32,80$	291008417322	$54,58$	10099951175046656
$34,78$	1639219687168	$56$	10841051388476292
$36,76$	7813559379696

Table 2. Table 2: m ​ ( H 112 ) , m ​ ( D 112 ) 𝑚 subscript 𝐻 112 𝑚 subscript 𝐷 112 m(H_{112}),m(D_{112}) and m ​ ( E 112 ) 𝑚 subscript 𝐸 112 m(E_{112})

$m\left(H_{112}\right)$
10613, 10649, 10661, 10703, 10709, 10715, 10721, 10727, 10733, 10739, 10745,
10769, 10775, 10781, 10787, 10799, 10805, 10811, 10823, 10829, 10835, 10841,
10847, 10853, 10859, 10865, 10871, 10883, 10895, 10901, 10907, 10913, 10919,
10925, 10931, 10937, 10943, 10949, 10967, 10973, 10985, 10991, 10997, 11009,
11021, 11033, 11045, 11057, 11063, 11069, 11093, 11099, 11117, 63525
$m\left(D_{112}\right)$
10618, 10663, 10672, 10702, 10708, 10717, 10735, 10750, 10765, 10768, 10771,
10777, 10783, 10786, 10789, 10801, 10810, 10819, 10831, 10834, 10837, 10840,
10843, 10846, 10849, 10852, 10858, 10861, 10864, 10867, 10873, 10882, 10885,
10900, 10903, 10906, 10909, 10912, 10918, 10921, 10924, 10927, 10930, 10936,
10945, 10954, 10957, 10978, 10984, 10987, 11002, 11011, 11023, 11041, 11044,
11056, 11065, 11080, 11086, 11098, 11110, 63525
$m\left(E_{112}\right)$
10581, 10620, 10641, 10653, 10659, 10668, 10674, 10689, 10698, 10701, 10704,
10707, 10719, 10728, 10734, 10749, 10758, 10761, 10764, 10770, 10776, 10779,
10782, 10785, 10791, 10794, 10797, 10806, 10809, 10812, 10815, 10818, 10821,
10824, 10827, 10830, 10833, 10842, 10848, 10851, 10854, 10860, 10863, 10866,
10872, 10875, 10878, 10881, 10884, 10890, 10893, 10896, 10899, 10902, 10905,
10911, 10914, 10917, 10923, 10926, 10929, 10932, 10935, 10938, 10941, 10950,
10953, 10959, 10965, 10968, 10971, 10977, 10980, 10989, 10992, 10995, 11001,
11013, 11016, 11025, 11028, 11040, 11046, 11049, 11052, 11055, 11073, 11085,
11103, 11151, 63525

Table 3. Table 3: Weight enumerators for length 120 120 120

Numbers of codewords of weight $20$
93180, 93936, 94512, 95136, 95202, 95376, 95496, 95532, 95826, 95946, 95952, 96012, 96096, 96126,
96156, 96216, 96240, 96312, 96336, 96360, 96366, 96372, 96486, 96540, 96576, 96666, 96690, 96720,
96762, 96780, 96816, 96840, 96846, 96876, 96906, 96912, 96936, 96996, 97026, 97056, 97092, 97116,
97176, 97230, 97260, 97266, 97272, 97296, 97326, 97356, 97422, 97446, 97452, 97476, 97566, 97572,
97590, 97596, 97626, 97632, 97656, 97716, 97746, 97770, 97776, 97782, 97836, 97842, 97866, 97890,
97896, 97926, 97950, 97962, 97986, 98016, 98040, 98076, 98130, 98136, 98166, 98196, 98220, 98226,
98250, 98256, 98262, 98286, 98292, 98316, 98346, 98412, 98466, 98496, 98502, 98526, 98532, 98556,
98562, 98580, 98586, 98610, 98616, 98622, 98640, 98646, 98670, 98676, 98682, 98700, 98706, 98712,
98730, 98742, 98772, 98796, 98802, 98826, 98832, 98856, 98886, 98910, 98916, 98940, 98952, 98976,
99000, 99036, 99066, 99090, 99096, 99120, 99126, 99156, 99162, 99180, 99186, 99210, 99216, 99222,
99240, 99246, 99252, 99270, 99282, 99306, 99312, 99330, 99336, 99342, 99372, 99390, 99396, 99402,
99432, 99450, 99456, 99486, 99516, 99540, 99546, 99576, 99612, 99666, 99672, 99690, 99696, 99702,
99720, 99726, 99750, 99756, 99786, 99792, 99810, 99816, 99846, 99876, 99906, 99936, 99942, 99966,
99972, 99996, 100026, 100032, 100062, 100086, 100110, 100116, 100122, 100140, 100146, 100170,
100176, 100182, 100200, 100206, 100212, 100236, 100242, 100260, 100266, 100290, 100296, 100350,
100356, 100380, 100446, 100452, 100476, 100482, 100500, 100506, 100512, 100536, 100542, 100560,
100566, 100590, 100596, 100626, 100650, 100656, 100662, 100680, 100686, 100716, 100722, 100746,
100752, 100770, 100776, 100782, 100800, 100806, 100842, 100860, 100872, 100896, 100902, 100920,
100926, 100956, 100980, 100986, 100992, 101046, 101052, 101070, 101076, 101082, 101106, 101112,
101130, 101136, 101142, 101160, 101166, 101196, 101202, 101226, 101232, 101250, 101256, 101280,
101286, 101316, 101376, 101382, 101400, 101406, 101412, 101436, 101442, 101472, 101496, 101526,
101532, 101550, 101556, 101586, 101616, 101622, 101640, 101646, 101652, 101670, 101676, 101700,
101706, 101730, 101736, 101760, 101766, 101772, 101790, 101796, 101802, 101820, 101826, 101850,
101856, 101862, 101880, 101892, 101910, 101916, 101940, 101946, 101952, 101970, 101976, 101982,
102000, 102006, 102030, 102036, 102042, 102066, 102072, 102096, 102120, 102126, 102150, 102156,
102180, 102186, 102210, 102216, 102240, 102246, 102252, 102270, 102312, 102336, 102342, 102360,
102366, 102372, 102402, 102420, 102426, 102456, 102480, 102486, 102492, 102516, 102540, 102546,
102570, 102576, 102582, 102606, 102636, 102660, 102666, 102672, 102690, 102696, 102702, 102726,
102732, 102750, 102756, 102780, 102786, 102792, 102816, 102840, 102846, 102870, 102876, 102906,
102930, 102936, 102942, 102966, 102972, 102996, 103002, 103020, 103026, 103032, 103050, 103056,
103080, 103086, 103092, 103116, 103140, 103146, 103176, 103182, 103206, 103236, 103266, 103272,
103296, 103320, 103326, 103332, 103356, 103380, 103386, 103410, 103416, 103422, 103452, 103500,
103506, 103530, 103560, 103566, 103590, 103596, 103632, 103650, 103656, 103686, 103692, 103710,
103716, 103722, 103740, 103746, 103752, 103770, 103776, 103800, 103806, 103830, 103836, 103860,
103896, 103932, 103962, 103986, 104022, 104046, 104076, 104106, 104166, 104220, 104226, 104232,
104256, 104286, 104316, 104346, 104436, 104442, 104496, 104502, 104532, 104556, 104580, 104592,
104616, 104622, 104646, 104652, 104676, 104736, 104772, 104796, 104820, 104880, 104886, 104892,
104910, 104916, 104970, 104982, 105066, 105096, 105156, 105336, 105396, 105426, 105456, 105510,
105546, 105576, 105636, 105666, 105696, 105762, 105966, 106152, 106236, 106266, 106290, 106386,
106626, 106662, 106812, 106836, 107220, 107406, 108486, 108600

Table 4. Table 4: Doubly even self-dual codes of length 120 120 120 and minimum weight 20 20 20

$r_A$	$r_B$
$\left(100000111110000011111010001101\right)$	$\left(010000000010010010110101001111\right)$
$\left(100001100101001111000100110010\right)$	$\left(100101100010011100001110011100\right)$
$\left(100000011001010111001110101001\right)$	$\left(111101110101111111001100111010\right)$
$\left(100001001111010001100000000011\right)$	$\left(010110111001100010000111011101\right)$
$\left(100001011011001010010110000010\right)$	$\left(001010001100000100000010001000\right)$
$\left(100000110010110111100100111110\right)$	$\left(100111000011110011101010001010\right)$
$\left(100000000000111000011000111101\right)$	$\left(111000101010011000111000111011\right)$
$\left(100001001111110101101101110111\right)$	$\left(011001000000100100101101110100\right)$
$\left(100000001100001100111011101110\right)$	$\left(111111010001001110011000000000\right)$
$\left(100000101111001000010100100110\right)$	$\left(001101011010111010101111011110\right)$

Table 5. Table 5: Weight enumerators for length 128 128 128

Numbers of codewords of weight $20$
21376, 21824, 22016, 22400, 22464, 22880, 22944, 23008, 23104, 23136, 23232,
23296, 23328, 23360, 23392, 23520, 23552, 23616, 23648, 23680, 23808, 23936,
24000, 24032, 24064, 24096, 24128, 24160, 24192, 24224, 24256, 24288, 24320,
24352, 24384, 24416, 24448, 24480, 24512, 24544, 24576, 24640, 24672, 24704,
24736, 24768, 24800, 24832, 24864, 24896, 24928, 24960, 24992, 25024, 25056,
25088, 25120, 25152, 25184, 25216, 25248, 25280, 25312, 25344, 25376, 25408,
25440, 25472, 25504, 25536, 25568, 25600, 25632, 25664, 25696, 25728, 25760,
25824, 25856, 25888, 25920, 25952, 25984, 26016, 26048, 26080, 26112, 26144,
26176, 26208, 26240, 26272, 26304, 26336, 26368, 26400, 26432, 26464, 26496,
26528, 26560, 26592, 26624, 26656, 26688, 26720, 26752, 26784, 26816, 26848,
26880, 26912, 26944, 26976, 27008, 27040, 27072, 27104, 27136, 27168, 27200,
27232, 27264, 27296, 27328, 27360, 27392, 27424, 27456, 27488, 27520, 27584,
27616, 27648, 27680, 27712, 27744, 27776, 27808, 27840, 27872, 27904, 27936,
27968, 28000, 28032, 28064, 28096, 28128, 28160, 28192, 28224, 28256, 28288,
28320, 28352, 28384, 28416, 28448, 28480, 28512, 28544, 28576, 28608, 28640,
28736, 28768, 28800, 28832, 28864, 28896, 28928, 28992, 29024, 29056, 29088,
29120, 29152, 29216, 29248, 29312, 29344, 29376, 29536, 29600, 29632, 29696,
29760, 29792, 29824, 29856, 29888, 30048, 30144, 30176, 30208, 30240, 30304,
30368, 31584

Table 6. Table 6: Doubly even self-dual codes of length 128 128 128 and minimum weight 20 20 20

$r_A$	$r_B$
$\left(10000111111110100010000110111100\right)$	$\left(00101010111100011101101011110100\right)$
$\left(10000111010101110101100011100110\right)$	$\left(10001000101110011011111011100110\right)$
$\left(10000010011000000001101010010001\right)$	$\left(01010001111001100001001011000010\right)$
$\left(10000111000111101111100111100111\right)$	$\left(11010000000111100110110101111111\right)$
$\left(10000010001100010100011010000111\right)$	$\left(01110100011000110110100110000110\right)$
$\left(10000111010101110010100010001010\right)$	$\left(01011110001110011011111011100111\right)$
$\left(10000100011111000010100010100010\right)$	$\left(10001101000000111010111010010101\right)$
$\left(10000101110011001010100011011010\right)$	$\left(00111111111010001111100110111000\right)$
$\left(10000000110010100000011111111000\right)$	$\left(11110000010011001110110110110101\right)$
$\left(10000010010000110011101110110100\right)$	$\left(10101000001000010101011011100001\right)$

Equations

$$d\le 4\lfloor \frac{n}{24}\rfloor +4,$$

$$W_C=\sum _{j=0}^{\lfloor n/8\rfloor }{a}_j(1+y^2{)}^{n/2-4j}(y^2(1-y^2{)}^2{)}^j,$$

$$W_C=\sum _{j=0}^{\lfloor n/24\rfloor }{a}_j(1+14y^4+y^8{)}^{n/8-3j}(y^4(1-y^4{)}^4{)}^j,$$

$$W_S=\sum _{j=0}^{\lfloor n/8\rfloor }(-1{)}^j{a}_j{2}^{n/2-6j}{y}^{n/2-4j}(1-y^4{)}^{2j}.$$

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

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

$$r_A$$

$$r_B$$

$$W_{112}^C=$$

$$+(1745240+1024c-64b-17a)y^{22}$$

$$+(44404374+65536d-10240c-160b-36a)y^{24}$$

$$+(572977944-4194304e-1048576d+33792c+960b+135a)y^{26}$$

$$+\cdots ,$$

$$W_{112}^S=$$

$$+(-2600e+276d+22c+b)y^{16}$$

$$+(14950e-2024d-231c-20b-4a)y^{20}+\cdots ,$$

$$A_{18}=8512,d(S)=16,B_{16}=728.$$

$$a=-90664,b=728,c=d=e=0.$$

$$⟨(C_{112}\cap ⟨x{⟩}^{\perp }),x⟩,$$

$$\{1,2,3,9,10,12,13,16,24,61,66,69,82,89,92,96,97,108,111,112\}.$$

$$r_A$$

$$r_B$$

$$m(C)=\{m_{i,j}\mid i,j\in \{1,2,\dots ,112\}\}.$$

$$1+a{y}^{20}+(39703755-20a)y^{24}+(6101289120+190a)y^{28}$$

$$+(475644139425-1140a)y^{32}+(18824510698240+4845a)y^{36}$$

$$+(397450513031544-15504a)y^{40}+(4630512364732800+38760a)y^{44}$$

$$+(30531599026535880-77520a)y^{48}+(116023977311397120+125970a)y^{52}$$

$$+(257257766776517715-167960a)y^{56}$$

$$+(335200280030755776+184756a)y^{60}+\cdots ,$$

$$1+a{y}^{20}+(13228320-6a)y^{24}+(2940970496-89a)y^{28}$$

$$+(320411086380+1500a)y^{32}+(18072021808640-10925a)y^{36}$$

$$+(552523816524960+51186a)y^{40}+(9491115264030720-173451a)y^{44}$$

$$+(94116072808107840+449616a)y^{48}+(549827773219608576-920550a)y^{52}$$

$$+(1920594735166941760+1518100a)y^{56}$$

$$+(4051982995220321280-2040714a)y^{60}$$

$$+(5193576851944293670+2250664a)y^{64}+\cdots ,$$

Full text

1 Introduction

Self-dual codes are an important class of linear codes for both theoretical and practical reasons (see [10]). It is a fundamental problem to determine the largest minimum weights among self-dual codes of that length and to construct self-dual codes with the largest minimum weight.

Let $\mathbb{F}_{2}$ denote the finite field of order $2$. Codes over $\mathbb{F}_{2}$ are called binary and all codes in this note are binary. 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 is at least one codeword of weight $\equiv 2\pmod{4}$. It is known that a self-dual code of length $n$ exists if and only if $n$ is even, and a doubly even self-dual code of length $n$ exists if and only if $n$ is divisible by eight. The minimum weight $d$ of a doubly even self-dual code of length $n$ is bounded by

[TABLE]

[9]. A doubly even self-dual code meeting the bound is called extremal.

In this note, we study the existence of doubly even self-dual codes having minimum weight $20$. By (1), if there is a doubly even self-dual code of length $n$ and minimum weight $20$, then $n\geq 96$. For length $96$, it is unknown whether there is an extremal doubly even self-dual code. For length $104$, the extended quadratic residue code is the only known extremal doubly even self-dual code (see [10]). For length $112$, the first extremal doubly even self-dual code was found in [7]. For lengths $120$ and $128$, it is unknown whether there is an extremal doubly even self-dual code. The first doubly even self-dual code of length $120$ and minimum weight $20$ was found in [5]. Then $25$ more doubly even self-dual codes of length $120$ and minimum weight $20$ were found in [11]. The existence of a doubly even self-dual code of length $128$ and minimum weight $20$ is known [4]. In this note, we construct new doubly even self-dual codes having minimum weight $20$ for lengths $112$, $120$ and $128$. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length $112$.

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

2 Preliminaries

Let $C$ be a code of length $n$. The elements of $C$ are called codewords and the weight $\operatorname{wt}(x)$ of a codeword $x$ is the number of non-zero coordinates. The support of a codeword $x=(x_{1},x_{2},\ldots,x_{n})$ is $\{i\mid x_{i}=1\}$. We denote the support of $x$ by $\operatorname{supp}(x)$. The minimum non-zero weight of all codewords in $C$ is called the minimum weight of $C$. Let $A_{i}$ be the number of codewords of weight $i$ in $C$. The weight enumerator $W_{C}$ of $C$ is given by $\sum_{i=0}^{n}A_{i}y^{i}$. By Gleason’s theorem [6] (see also [9]), the weight enumerator $W_{C}$ of a self-dual code of length $n$ is written as:

[TABLE]

for some integers $a_{j}$. In addition, a doubly even self-dual code of length $n$ exists then $n$ is divisible by eight, and the weight enumerator $W_{C}$ of a doubly even self-dual code of length $n$ is written as:

[TABLE]

for some integers $a_{j}$. Then Mallows and Sloane [9] established the upper bound (1) on the minimum weights of doubly even self-dual codes.

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 $C_{0}^{\perp}\setminus C$ [3]. 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 $S=C_{1}\cup C_{3}$. If $C$ is a singly even self-dual code of length divisible by $8$, then $C$ has two doubly even self-dual neighbors, namely $C_{0}\cup C_{1}$ and $C_{0}\cup C_{3}$ (see [2]). Let $B_{i}$ be the number of vectors of weight $i$ in $S$. The weight enumerator $W_{S}$ of $S$ is given by $\sum_{i=d(S)}^{n-d(S)}B_{i}y^{i}$, respectively, where $d(S)$ denotes the minimum weight of $S$. If $W_{C}$ is written as in (2), then $W_{S}$ can be written as follows [3, Theorem 5]:

[TABLE]

Two self-dual codes $C$ and $C^{\prime}$ of length $n$ are said to be neighbors if $\dim(C\cap C^{\prime})=n/2-1$. Two codes are equivalent if one can be obtained from the other by permuting the coordinates. An automorphism of a code $C$ is a permutation of the coordinates of $C$ which preserves $C$. The set consisting of all automorphisms of $C$ is called the automorphism group of $C$.

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 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 matrix $A$. 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 (5) are called four-circulant [8]. In this note, we found a singly even self-dual four-circulant code of length $112$ and minimum weight $18$ and doubly even self-dual four-circulant codes of length $n$ and minimum weight $20$ for $n=112,120,128$, by a non-exhaustive search. An exhaustive search is beyond our current computer resources.

3 New extremal doubly even self-dual codes of length 112

3.1 A singly even self-dual code

of length 112 and minimum weight 18

By a non-exhaustive search, we found a singly even self-dual four-circulant code $C_{112}$ of length $112$ and minimum weight $18$. The first rows $r_{A}$ and $r_{B}$ of $A$ and $B$ in the generator matrix (5) of $C_{112}$ are as follows:

[TABLE]

respectively.

Let $C$ be a singly even self-dual code of length $112$ and minimum weight $18$. Let $S$ be the shadow of $C$. From (2) and (4), the possible weight enumerators of $C$ and $S$ are determined as follows:

[TABLE]

respectively, where $a,b,c,d,e$ are integers. In order to determine the weight enumerator of $C_{112}$, we found that

[TABLE]

This gives

[TABLE]

The weight distribution of $C_{112}$ is listed in Table 1. Note that singly even self-dual codes of length $112$ and minimum weight $18$ with weight enumerators corresponding to $e=1$ were found in [7].

3.2 New extremal doubly even self-dual codes of length 112

If $C$ is a singly even self-dual code of length divisible by $8$, then $C$ has two doubly even self-dual neighbors (see Section 2). We verified that the two doubly even self-dual neighbors of $C_{112}$ have minimum weights $20$ and $16$. We denote the extremal doubly even self-dual neighbor by $D_{112}$. The code $D_{112}$ is constructed as

[TABLE]

where the support $\operatorname{supp}(x)$ of $x$ is

[TABLE]

Moreover, by a non-exhaustive search, we found an extremal doubly even self-dual four-circulant code $E_{112}$. The first rows $r_{A}$ and $r_{B}$ of $A$ and $B$ in the generator matrix (5) of $E_{112}$ are as follows:

[TABLE]

respectively.

We denote the known extremal doubly even self-dual code in [7] by $H_{112}$. In order to distinguish $H_{112},D_{112}$ and $E_{112}$, we consider the following invariant. Let $C$ be an extremal doubly even self-dual code of length $112$. Let $M(C)$ be the matrix with rows composed of the codewords of weight $20$ in $C$, where the $(1,0)$-matrix $M(C)$ is regarded as a matrix over $\mathbb{Z}$. Let $m_{i,j}$ denote the $(i,j)$-entry of the matrix $M(C)^{T}M(C)$. Then define

[TABLE]

In Table 2, we list all elements of $m(C)$ for $C=H_{112},D_{112}$ and $E_{112}$. Table 2 shows that the three codes are inequivalent.

Proposition 1**.**

There are at least three inequivalent extremal doubly even self-dual codes of length $112$.

Remark 2*.*

The code $H_{112}$ has automorphism group of order $112$ [7]. We verified that the codes $D_{112}$ and $E_{112}$ have automorphism group of order $112$.

4 New doubly even self-dual codes

of length 120 and minimum weight 20

From (3), the possible weight enumerator of a doubly even self-dual code of length $120$ and minimum weight $20$ is determined as follows:

[TABLE]

where $a$ is the number of codewords of weight $20$.

The first doubly even self-dual code of length $120$ and minimum weight $20$ was found in [5]. Then $25$ more doubly even self-dual codes of length $120$ and minimum weight $20$ were found in [11]. From [11, Table 1], these codes have different weight enumerators. By a non-exhaustive search, we found $500$ doubly even self-dual four-circulant codes of length $120$ and minimum weight $20$. The numbers $a$ of codewords of weight $20$ in these codes are listed in Table 3. It follows that these codes and the $26$ codes in [5] and [11] have distinct weight enumerators. Hence, we have the following:

Proposition 3**.**

There are at least $526$ inequivalent doubly even self-dual codes of length $120$ and minimum weight $20$.

Our feeling is that the number of inequivalent doubly even self-dual codes of length $120$ and minimum weight $20$ might be even bigger.

The first rows $r_{A}$ and $r_{B}$ of $A$ and $B$ in the generator matrices (5) of the $500$ codes are listed in http://www.math.is.tohoku.ac.jp/~mharada/Paper/120-d20.txt. As an example, we list the first rows $r_{A}$ and $r_{B}$ for ten codes in Table 4.

5 New doubly even self-dual codes

of length 128 and minimum weight 20

From (3), the possible weight enumerator of a doubly even self-dual code of length $128$ and minimum weight $20$ is determined as follows:

[TABLE]

where $a$ is the number of codewords of weight $20$.

The existence of a doubly even self-dual code of length $128$ and minimum weight $20$ is known [4]. By a non-exhaustive search, we found $200$ doubly even self-dual four-circulant codes of length $128$ and minimum weight $20$. These codes have distinct weight enumerators, where the numbers $a$ of codewords of weight $20$ are listed in Table 5.

Proposition 4**.**

There are at least $200$ inequivalent doubly even self-dual codes of length $128$ and minimum weight $20$.

Our feeling is that the number of inequivalent doubly even self-dual codes of length $128$ and minimum weight $20$ might be even bigger.

The first rows $r_{A}$ and $r_{B}$ of $A$ and $B$ in the generator matrices (5) of the $200$ codes are listed in http://www.math.is.tohoku.ac.jp/~mharada/Paper/128-d20.txt. As an example, we list the first rows $r_{A}$ and $r_{B}$ for ten codes in Table 6.

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