---
abstract: 'By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length $24m$ form a self-orthogonal $5$-design.  In this paper, we study the codes constructed from self-orthogonal $5$-designs with the same parameters as the above $5$-designs.  We give some parameters of a self-orthogonal $5$-design whose existence is equivalent to that of an extremal doubly even self-dual code of length $24m$ for $m=3,4,5,6$. If $m \in \{1,\ldots,6\}$, $k \in \{m+1,\ldots,5m-1\}$ and $(m,k) \ne (6,18)$, then it is shown that an extremal doubly even self-dual code of length $24m$ is generated by codewords of weight $4k$.'
author:
- 'Masaaki Harada[^1]'
title: ' On a 5-design related to a putative extremal doubly even self-dual code of length a multiple of 24'
---

[**Keywords**]{} self-orthogonal $t$-design, extremal doubly even self-dual code, weight enumerator

[**Mathematics Subject Classification**]{} 94B05, 05B30

Introduction {#sec:1}
============

A doubly even self-dual code of length $n$ exists if and only if $n$ is divisible by $8$. The minimum weight $d(C)$ of a doubly even self-dual code $C$ of length $n$ is bounded above by $d(C) \le 4 \lfloor n/24\rfloor+4$ [@MS73].  A doubly even self-dual code meeting the bound is called [*extremal*]{}.  In case that $n \equiv 0 \pmod{24}$, the only known extremal doubly even self-dual codes are the extended Golay code and the extended quadratic residue code of length $48$. The existence of an extremal doubly even self-dual code of length $72$ is a long-standing open question [@Sloane72].  A $t$-$(v,k,\lambda)$ design is called [*self-orthogonal*]{} if the block intersection numbers have the same parity as the block size $k$ (see [@T86]).  If ${\mathcal{D}}$ is a self-orthogonal $t$-$(v,k,\lambda)$ design with $k$ even, then the code $C({\mathcal{D}})$, which is generated by the rows of an incidence matrix of ${\mathcal{D}}$, is a self-orthogonal code.  By the Assmus and Mattson theorem [@AM], the supports of the codewords of weight $4k \ (\ne 0,24m)$ in an extremal doubly even self-dual code of length $24m$ form a self-orthogonal $5$-design.  We denote the parameters of the design by $5$-$(24m,4k,\lambda_{24m,4k})$. Then, throughout this paper, we denote any self-orthogonal $5$-$(24m,4k,\lambda_{24m,4k})$ design by ${\mathcal{D}}_{24m,4k}$. That is, ${\mathcal{D}}_{24m,4k}$ is a self-orthogonal $5$-design with the same parameters as the self-orthogonal $5$-design formed from the supports of the codewords of weight $4k$ in an extremal doubly even self-dual code of length $24m$. This gives rise to a natural question, namely, is the code $C({\mathcal{D}}_{24m,4k})$ always an extremal doubly even self-dual code? 

It is well known that $C({\mathcal{D}}_{24,8})$ is the extended Golay code (see [@AK-book Theorem 8.6.2]).  It was shown that $C({\mathcal{D}}_{24m,4m+4})$ ($m =2,3,4$) is an extremal doubly even self-dual code [@HMT05; @HKM04; @H05], respectively.  This means that the existence of an extremal doubly even self-dual code of length $24m$ ($m =1,2,3,4$) is equivalent to that of a self-orthogonal $5$-$(24m,4k,\lambda_{24m,4k})$ design, where $(4k,\lambda_{24m,4k})=(8,  1)$, $(12, 8 )$, $(16, 78 )$ and $(20, 816 )$, respectively.  The powerful tool which is used in [@HKM04; @HMT05] is the use of fundamental equations, sometimes called the Mendelsohn equations [@M71] (see also e.g., [@T86]), obtained by counting the number of blocks that meet $S$ in $i$ points for some subset $S$ of the point set.  The approach in [@H05] is also similar to that in [@HKM04; @HMT05] except that Gleason’s theorem (see [@MS73]) is employed to obtain stronger consequences. 

In this paper, we study self-orthogonal $5$-designs $C({\mathcal{D}}_{24m,4k})$ for $k \in \{m+2,\ldots,5m-1\}$, which are related to codewords of weight other than the minimum weight.  More precisely, we consider a problem whether $C({\mathcal{D}}_{24m,4k})$ is an extremal doubly even self-dual code or not for $m \in \{1,\ldots,6\}$ and $k \in \{m+2,\ldots,5m-1\}$. In addition to the above approach done in [@H05; @HKM04; @HMT05], it is useful in this paper to observe weight enumerators of $C({\mathcal{D}}_{24m,4k})$ and its dual codes, and singly even self-dual codes containing $C({\mathcal{D}}_{24m,4k})$ and their shadows.  As a summary, in Table \[Tab:D\][^2], we list some partial answers to the above problem for $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,3m\}$. For the cases $(24m,4k)$ that $C({\mathcal{D}}_{24m,4k})$ is self-dual, we list “Yes” in the second column of Table \[Tab:D\].  When $C({\mathcal{D}}_{24m,4k})$ is self-dual, we list “Yes” in the third column in case that $C({\mathcal{D}}_{24m,4k})$ is extremal.  We also list the possible minimum weights, when $C({\mathcal{D}}_{24m,4k})$ is self-dual but not extremal.  It is shown that $C({{\mathcal{D}}}_{24m,4k}) = C({{\mathcal{D}}}_{24m,24m-4k})$ for $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,3m-1\}$ (Proposition \[prop:CD\]). 

The main results of this paper are the following theorems. 

\[thm:main1\] Suppose that $(24m,k,\lambda)$ is each of the following: $$\begin{aligned}
&
(72, 24, 1406405 ), (72, 32, 238957796 ),
\\ &
(96,36, 28080500448 ), 
(96,44, 1167789832440 ),
\\ &
(120, 56, 5156299310025435),
(144, 68, 21788133027489299328).\end{aligned}$$ Then the existence of a self-orthogonal $5$-$(24m,k,\lambda)$ design is equivalent to that of an extremal doubly even self-dual code of length $24m$.

\[thm:main2\] Suppose that $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,5m-1\}$. If $(m,k) \ne (6,18)$, then an extremal doubly even self-dual code of length $24m$ is generated by codewords of weight $4k$.

For some cases $(m,k)$, the above theorem is already known (see Table \[Tab:D\]).  It is still unknown whether $C({\mathcal{D}}_{144, 72})$ is self-dual or not (see Remark \[rem\]). 

[l|c|c|c]{} & Self-dual & Extremal & Ref.\
$( 24,  8,  1)$ &Yes & Yes & (see [@AK-book])\
$( 24, 12, 48)$ &Yes & Yes &[@T86]\
$( 48, 12, 8 )$ &Yes & Yes &[@HMT05]\
$( 48, 16, 1365 )$ &Yes & Yes &[@D]\
$( 48, 20, 36176 )$ &Yes & Yes &[@D]\
$( 48, 24, 190680 )$&Yes & 8, 12\
$( 72, 16, 78 )$ &Yes & Yes &[@HKM04]\
$( 72, 20, 20064 )$ &Yes & 12, 16 &[@D]\
$( 72, 24, 1406405 )$ &Yes & Yes$^*$\
$( 72, 28, 30888000 )$ &Yes$^*$ & 12, 16 &\
$( 72, 32, 238957796 )$&Yes & Yes$^*$\
$( 72, 36, 693996160 )$&Yes & 12, 16 &[@D]\
$( 96, 20, 816 )$ & Yes & Yes &[@H05]\
$( 96, 24, 257180 )$ &Yes & 16, 20&[@D]\
$( 96, 28, 29975400 )$ &Yes & 12, 20$^*$\
$( 96, 32, 1390528685 )$ &Yes & 12, 16, 20 &[@D]\
$( 96, 36, 28080500448 )$ &Yes & Yes$^*$\
$( 96, 40, 261513764460 )$ &Yes & 12, 16, 20 &[@D]\
$( 96, 44, 1167789832440 )$&Yes & Yes$^*$\
$( 96, 48, 2561776811880 )$&Yes$^*$ & 12, 16, 20 &\
$( 120, 24, 8855 )$ & Yes & 16, 24 &[@CW12]\
$( 120, 28, 3146400 )$ &Yes & 16, 20, 24 &\
$( 120, 32, 502593700 )$ &Yes & 12, 16, 24$^*$&\
$( 120, 36, 37237713920 )$ &Yes$^*$ & 12–24&\
$( 120, 40, 1372275835848 )$ &Yes$^*$ & 12, 24$^*$&\
$( 120, 44, 26386953577600 )$ &Yes$^*$ & 12–24&\
$( 120, 48, 274320081834480 )$ &Yes$^*$ & 12, 24$^*$&\
$( 120, 52, 1582247888524800 )$&Yes$^*$ & 12–24 &\
$( 120, 56, 5156299310025435 )$&Yes & Yes$^*$&\
$( 120, 60, 9606041207517888 )$&Yes$^*$ & 12–24 &\
$( 144, 28, 98280 )$ &Yes & 16, 20, 28 &[@HMM]\
$( 144, 32, 37756202 )$ &Yes & 16–28&\
$( 144, 36, 7479335776 )$ &Yes & 16, 20, 28$^*$&\
$( 144, 40, 765322879032 )$ &Yes & 12–28&\
$( 144, 44, 42785304274536 )$ &Yes & 12, 16, 20, 28$^*$&\
$( 144, 48, 1359454757387265 )$ &Yes & 12–28&\
$( 144, 52, 25319185698144240 )$ &Yes & 12, 16, 28$^*$&\
$( 144, 56, 283096123959568608 )$ &Yes$^*$ & 12–28&\
$( 144, 60, 1935608752827917264 )$ &Yes & 12, 28$^*$&\
$( 144, 64, 8205989047403924124 )$ &Yes & 12–28&\
$( 144, 68, 21788133027489299328 )$&Yes & Yes$^*$&\
$( 144, 72, 36470135955078919440 )$& ? & –\ 

Preliminaries {#sec:Pre}
=============

Self-dual codes and shadows
---------------------------

In this paper, codes mean binary codes.  A code is called [*doubly even*]{} if every codeword has weight a multiple of $4$. A code $C$ is called *self-orthogonal* if $C \subset C^{\perp}$, and $C$ is called *self-dual* if $C = C^{\perp}$, where $C^{\perp}$ is the dual code of $C$ under the standard inner product.  A self-dual code which is not doubly even is called [*singly even*]{}, namely, a singly even self-dual code contains a 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 $d(C)$ of a doubly even self-dual code $C$ of length $n$ is bounded by $d(C)  \le 4  \lfloor n/24 \rfloor + 4$ [@MS73].  A doubly even self-dual code meeting the bound is called [*extremal*]{}.  In case that $n \equiv 0 \pmod{24}$, the only known extremal doubly even self-dual codes are the extended Golay code and the extended quadratic residue code of length $48$. The existence of an extremal doubly even self-dual code of length $72$ is a long-standing open question [@Sloane72]. 

Let $C$ be a singly even self-dual code and let $C_0$ denote the subcode of codewords having weight $\equiv0\pmod4$. Then $C_0$ is a subcode of codimension $1$. The [*shadow*]{} $S$ of $C$ is defined to be $C_0^\perp \setminus C$.  Shadows were introduced by Conway and Sloane [@C-S], in order to provide restrictions on the weight enumerators of singly even self-dual codes (see [@C-S] for fundamental results on shadows).  Let $D$ be a doubly even code of length $n \equiv 0 \pmod 8$. Suppose that $D$ has dimension $n/2-1$ and $D$ contains the all-one vector ${\mathbf{1}}$. Then there are three self-dual codes lying between $D^\perp$ and $D$, one of which is singly even and the others are doubly even (see [@MST]). 

Self-orthogonal designs and Mendelsohn equations
------------------------------------------------

A $t$-$(v,k,\lambda)$ design ${\mathcal{D}}$ is a set $X$ of $v$ points together with a collection of $k$-subsets of $X$ (called blocks) such that every $t$-subset of $X$ is contained in exactly $\lambda$ blocks.  A $t$-design with no repeated block is called [*simple*]{}.  In this paper, designs mean simple designs.  It follows that every $i$-subset of points ($i\le t$) is contained in exactly $\lambda_i =\lambda {\binom{v-i}{t-i}}/{\binom{k-i}{t-i}}$ blocks.  The number $\lambda_1$ is traditionally denoted by $r$, and the total number of blocks is $b=\lambda_0$. A $t$-design can be represented by its (block-point) incidence matrix $A=(a_{ij})$, where $a_{ij}=1$ if the $j$th point is contained in the $i$th block and $a_{ij}=0$ otherwise. 

The [*block intersection numbers*]{} of a $t$-$(v,k,\lambda)$ design are the cardinalities of the intersections of any two distinct blocks.  A $t$-$(v,k,\lambda)$ design is called [*self-orthogonal*]{} if the block intersection numbers have the same parity as the block size $k$ (see [@T86]).  The term self-orthogonal is due to a natural connection between such designs and self-orthogonal codes.  Throughout this paper, we denote the code generated by the rows of an incidence matrix of ${\mathcal{D}}$ by $C({\mathcal{D}})$. If ${\mathcal{D}}$ is a self-orthogonal $t$-$(v,k,\lambda)$ design with $k$ even, then $C({\mathcal{D}})$ is a self-orthogonal code. 

Let ${\mathcal{D}}$ be a $t$-$(v,k,\lambda)$ design.  Let $v\in C({\mathcal{D}})^{\perp}$ be a vector of weight $w>0$. Denote by $n_i$ the number of rows of an incidence matrix of ${\mathcal{D}}$ intersecting exactly $i$ positions of the support of $v$ in ones.  Then we have the system of equations: $$\label{eq:M}
\sum_{i=0}^{\min\{k,w\}}\binom{i}{j}n_{i}
=\lambda_{j}\binom{w}{j}
\quad(j=0,1,\ldots,t).$$ These fundamental equations, which are sometimes called Mendelsohn equations [@M71] (see also [@T86]), are the powerful tool in the study of this paper.  We note that $n_i=0$ if $i$ is odd, $i > k$ or $i > w$.

The following lemma follows immediately. 

\[lem:basic\] Let ${\mathcal{D}}$ be a self-orthogonal $t$-$(v,k,\lambda)$ design with $k \equiv 0 \pmod 4$.

-   If the system of equations [(\[eq:M\])]{} has no solution $(n_0,n_2,\ldots)$ consisting of nonnegative integers for some $w$, then $C({\mathcal{D}})^\perp$ contains no vector of weight $w$.

-   If the system of equations [(\[eq:M\])]{} has no solution $(n_0,n_2,\ldots)$ consisting of nonnegative integers for each $w$ with $0 < w < v$, $w \not \equiv 0 \pmod 4$, then $C({\mathcal{D}})$ is doubly even self-dual. 

The complementary design $\overline{{\mathcal{D}}}$ of a design ${\mathcal{D}}$ is obtained by replacing each block of ${\mathcal{D}}$ by its complement.  The following lemma is used in Section \[sec:min\] to show that $C({{\mathcal{D}}}_{24m,4k}) = C({{\mathcal{D}}}_{24m,24m-4k})$ for $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,3m-1\}$.

\[lem:CD\] Let ${\mathcal{D}}$ be a self-orthogonal $t$-$(v,k,\lambda)$ design with $k$ even.  Suppose that $C({\mathcal{D}})$ is self-dual.  Then $C({\mathcal{D}})=C(\overline{{\mathcal{D}}})$ if ${\mathbf{1}}\in C(\overline{{\mathcal{D}}})$, and $C(\overline{{\mathcal{D}}}) \subset C({\mathcal{D}})$ with $|C({\mathcal{D}}):C(\overline{{\mathcal{D}}})|=2$ otherwise. 

Since $C({\mathcal{D}})$ is self-dual, ${\mathbf{1}}\in C({\mathcal{D}})$. It turns out that $C(\overline{{\mathcal{D}}}) \subseteq C({\mathcal{D}})$ and $\langle C(\overline{{\mathcal{D}}}), {\mathbf{1}}\rangle = C({\mathcal{D}})$. The result follows. 

On the self-duality {#sec:SD}
===================

In this section, we describe how to determine the self-duality given in the second column of Table \[Tab:D\] for the cases denoted by $*$ in Table \[Tab:D\].  For the other cases, the self-duality is determined by Lemma \[lem:basic\] (ii) only. 

\[prop:SD-P1\] The codes $C({\mathcal{D}}_{72,28})$, $C({\mathcal{D}}_{96,48})$, $C({\mathcal{D}}_{120,60})$ and $C({\mathcal{D}}_{120,52})$ are self-dual. 

All cases are similar, and we only give the details for $C({\mathcal{D}}_{72,28})$.

Note that ${\mathcal{D}}_{72,28}$ has the following parameters: $$\begin{gathered}
\lambda_{0} = 4397342400, 
\lambda_{1} = 1710077600, 
\lambda_{2} = 650311200, 
\\
\lambda_{3} = 241544160, 
\lambda_{4} = 87516000, 
\lambda_{5} = 30888000.\end{gathered}$$ Let ${v}\in C({\mathcal{D}}_{72,28})^{\perp}$ be a vector of weight $w >0$. For each $w$ of the cases with $w \equiv 1 \pmod 2$ and $w \le 8$, the system of equations (\[eq:M\]) has no solution.  In addition, for $w=10$, (\[eq:M\]) has the following unique solution: $$\begin{gathered}
n_0 = 41076475, 
n_2 = 1096595775, 
n_4 = 2375199750, 
\\
n_6 = 834337350,
n_8 = 50284575, 
n_{10}= -151525.\end{gathered}$$ Hence, there is no vector of weights $2,4,6,8,10$ in $C({\mathcal{D}}_{72,28})^{\perp}$. The number $\lambda_0$ of blocks satisfies that $2^{32} < \lambda_0 < 2^{33}$. Therefore, $C({\mathcal{D}}_{72,28})^{\perp}$ is an even code such that the minimum weight is at least $12$ and the dimension is at most $39$.

Let $D_{72}$ be a doubly even code of length $72$ satisfying the conditions that $D_{72}$ has dimension $\ell \in \{33,34,35,36\}$, both $D_{72}$ and $D_{72}^\perp$ have minimum weights at least $12$ and ${\mathbf{1}}\in D_{72}$. We denote the weight enumerators of $D_{72}$ and $D_{72}^\perp$ by $W_{D_{72}}$ and $W_{D_{72}^\perp}$, respectively.  In this case, $W_{D_{72}}$ can be written as: $$\begin{aligned}
&
x^{72} + a x^{60} y^{12} + b x^{56} y^{16} + c x^{52} y^{20} 
+ d x^{48} y^{24} + e x^{44} y^{28} 
+ f x^{40} y^{32} 
\\&
+ (2^\ell -2 -2 a - 2 b - 2 c - 2 d - 2 e - 2 f) x^{36} y^{36} 
+\cdots + y^{72},\end{aligned}$$ using nonnegative integers $a,b,c,d,e,f$. Set $W_{D_{72}^\perp} = \sum_{i=0}^{72} B_i x^{72-i}y^i$. By the MacWilliams identity, we have: $$\begin{aligned}
2^{\ell}B_2 =
& 2^6(\chi_{2,\ell} + 36 a + 25 b + 16 c + 9 d + 4 e + f),\\
2^{\ell}B_4 =&2^6(\chi_{4,\ell} + 5640 a + 2450 b + 800 c + 114 d 
  - 56 e - 30 f),\\
2^{\ell}B_6 =& 
2^6(\chi_{6,\ell} + 313060 a + 77385 b + 8976 c - 1223 d + 196 e + 433 f),\\
2^{\ell}B_8 =& 
2^6(\chi_{8,\ell} + 7582080 a + 811360 b - 43520 c - 5280 d  + 1408 e - 4000 f),\\
2^{\ell}B_{10} =& 
2^6(\chi_{10,\ell} + 86892960 a + 887656 b - 372096 c + 100584 d - 17248 e 
\\ &+ 26536 f),\end{aligned}$$ where $(\chi_{2i,33},\chi_{2i,34},\chi_{2i,35})$ are as follows: $$\begin{aligned}
&(-4831838127    , -9663676335     ,  -19327352751), \\
&(84557200770    , 169114369410    ,  338228706690), \\
&(-958309695231  , -1916624273151  ,  -3833253428991), \\
&(7906469297760  , 15812564565600  ,  31624755101280), \\
&(-50582253079512, -101181262793688,  -202379282222040),\end{aligned}$$ for $i=1,2,3,4,5$, respectively. 

The assumptions $B_{2i}=0$ $(i=1,2,3,4,5)$ yield the following: $$b = \alpha_\ell - 12 a,
c = \beta_\ell + 66 a, 
d = \gamma_\ell - 220 a,
e = \delta_\ell + 495 a,
f = \varepsilon_\ell - 792 a,$$ where $$\begin{aligned}
(\alpha_\ell,\beta_\ell,\gamma_\ell,\delta_\ell,\varepsilon_\ell)=&
(30105,2273040,57830955,549766080,2075173947), \\
&(61497,4534992,115706955,1099419840,4150537083),\\
&(124281,9058896,231458955,2198727360,8301263355),\end{aligned}$$ for $\ell=33,34,35$, respectively.  For $\ell=33,34,35$, it follows from $b \ge 0$ that $$e =\delta_\ell + 495a 
\le \delta_\ell + \frac{165}{4} \alpha_\ell <  4397342400 =\lambda_0.$$ Since $C({\mathcal{D}}_{72,28})$ contains at least $4397342400$ codewords of weight $28$, we obtain a contradiction.  Therefore, $C({\mathcal{D}}_{72,28})$ must be self-dual. 

\[prop:SD-P2\] The codes $C({\mathcal{D}}_{120, 36})$, $C({\mathcal{D}}_{120, 40})$, $C({\mathcal{D}}_{120, 44})$, $C({\mathcal{D}}_{120, 48})$ and $C({\mathcal{D}}_{144, 56})$ are self-dual. 

All cases are similar, and we only give the details for $C({\mathcal{D}}_{120, 40})$.

Note that ${\mathcal{D}}_{120, 40}$ has the following parameters: $$\begin{gathered}
\lambda_{0} = 397450513031544, 
\lambda_{1} = 132483504343848, 
\lambda_{2} = 43418963608488, 
\\
\lambda_{3} = 13982378111208,
\lambda_{4} = 4421777693288, 
\lambda_{5} = 1372275835848.\end{gathered}$$ Let ${v}\in C({\mathcal{D}}_{120, 40})^{\perp}$ be a vector of weight $w >0$. For each $w$ of the cases with $w \equiv 1 \pmod 2$ and $w \le 8$, the system of equations (\[eq:M\]) has no solution.  The number $\lambda_0$ of blocks satisfies that $2^{48} < \lambda_0 < 2^{49}$. Hence, $C({\mathcal{D}}_{120, 40})^{\perp}$ is an even code such that the minimum weight is at least $10$ and the dimension is at most $71$.

Let $D_{120}$ be a doubly even code of length $120$ satisfying the conditions that $D_{120}$ has dimension $\ell \in \{49,\ldots,60\}$, $D_{120}$ has minimum weight at least $12$, $D_{120}^\perp$ has minimum weight at least $10$ and ${\mathbf{1}}\in D_{120}$. We show that $\ell \ne 49,50,\ldots,59$ in the following two steps. 

The first step shows that $\ell \ne 49,\ldots,58$. The approach is similar to that given in Proposition \[prop:SD-P1\].  Suppose that $\ell \in \{49,\ldots,58\}$. Then, by considering the possible weight enumerators of $D_{120}$ and $D_{120}^\perp$, one can obtain a contradiction for each $\ell$. Since the situation is more complicated than that for $C({\mathcal{D}}_{72,28})$ considered in Proposition \[prop:SD-P1\], we omit the details to save space.  We remark that this argument does not work to show that $\ell \ne 59$.

The second step shows that $\ell \ne 59$. The approach is to consider singly even self-dual codes containing $D_{120}$. Suppose that $\ell=59$. Since $D_{120}$ contains ${\mathbf{1}}$, there are three self-dual codes lying between $D_{120}^\perp$ and $D_{120}$, one of which is singly even and the others are doubly even (see [@MST]).  We denote the singly even code by $C_{120}$, noting that $D_{120}$ is the subcode $(C_{120})_0$ consisting of codewords of weight $\equiv 0 \pmod 4$ of $C_{120}$. Let $S_{120}$ be the shadow of $C_{120}$. Since the weight of a vector in $S_{120}$ is divisible by four [@C-S] and $D_{120}^{\perp}$ has minimum weight at least $10$, $C_{120}$ and $S_{120}$ have minimum weights at least $10$ and $12$, respectively.  Using [@C-S (10) and (11)], from the condition on the minimum weights, one can determine the possible weight enumerators $\sum_{i=0}^{120} A_i x^{120-i}y^i$ and $\sum_{i=0}^{120} B_i x^{120-i}y^i$ of $C_{120}$ and $S_{120}$, respectively.  In this case, the possible weight enumerators can be written using integers $a,b,c,d,e,f,g,h$. We investigate the number of codewords of weight $40$. In this case, we have that $$\begin{gathered}
A_{40}=
198725556937080 + 32980992a - 28160b - 15504c \\ 
+ 4896d + 161525e - 599494f - 4385880g + 91345008h.\end{gathered}$$ Using the mathematical software [Mathematica]{}, we have verified that $A_{2i} \ge 0$ $(i=5,\ldots,16)$ and $B_{4i} \ge 0$ $(i=3,\ldots,9)$ yield $$A_{40} <397450513031544 = \lambda_0,$$ where $A_{2i}$ $(i=5,\ldots,16)$ and $B_{4i}$ $(i=3,\ldots,9)$ are listed in Tables \[Tab:120-A\] and \[Tab:120-B\], respectively.  Since $C({\mathcal{D}}_{120,40})$ contains at least $397450513031544$ codewords of weight $40$, we obtain a contradiction.  Therefore, $C({\mathcal{D}}_{120,40})$ must be self-dual.  This completes the proof. 

[c|l]{} $i$ &\
&$h$\
[12]{}&$g + 30h$\
[14]{}&$f + 24g + 425h$\
[16]{}&$e + 18f + 264g + 3760h$\
[18]{}&$d + 12e + 139f + 1736g + 23100h$\
[20]{}&$c + 6d + 50e + 564f + 7380g + 103256h$\
[22]{}&$64b - 3d + 28e +  1009f + 19800g + 339180h + 26391755$\
[24]{}&$4096a - 384b - 20c - 88d - 441e - 1218f + 25080g + 789840h$\
[26]{}&$265912320 - 49152a - 64b - 102d - 1288e- 10717f - 35640g + 1096410h$\
[28]{}&$2968094880 + 221184a + 4864b + 190c + 564d+ 364e  - 20424f
     - 238590g - 118980h$\
[30]{}&$29559455744 - 311296a - 6720b + 1210d+ 7800e  + 7631f 
    - 473880g - 4961862h$\
[32]{}&$238259763105 - 946176a - 25984b - 1140c - 1944d + 9971e + 103766f 
    - 182952g - 13088880h$\

[c|l]{} $i$ &\
&$a$\
[16]{}&$17250 - 24a - b$\
[20]{}&$- 315744 + 276a + 22b + c$\
[24]{}&$42581630 - 2024a - 231b - 20c - 64d$\
[28]{}&$6084129120 + 10626a + 1540b + 190c +  1152d + 4096e$\
[32]{}&$475718702550 - 42504a - 7315b - 1140c - 9792d - 65536e - 262144f$\
[36]{}&$18824260734240 + 134596a + 26334b + 4845c +  52224d
     + 491520e + 3670016f + 16777216g$\

\[rem\] If $C({\mathcal{D}}_{144,72})^\perp$ has minimum weight at least $10$, then one can show that $C({\mathcal{D}}_{144,72})$ is self-dual by an argument similar to that described in above. 

For $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,3m-1\}$, the self-duality of $C({\mathcal{D}}_{24m,4k})$ has been verified above.  As a consequence, we have the following:

\[prop:CD\] If $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,3m-1\}$, then $C({{\mathcal{D}}}_{24m,4k}) = C({{\mathcal{D}}}_{24m,24m-4k})$.

It is trivial that ${{\mathcal{D}}}_{24m,24m-4k} = \overline{{{\mathcal{D}}}_{24m,4k}}$. For $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,3m-1\}$, the codes $C({{\mathcal{D}}}_{24m,4k})$ are self-dual (see Table \[Tab:D\]). 

For $(24m,4k) \in \{(72,16),
(72,32),
(120,32),
(144,32), 
(144,64)\}$, since the $5$-design $\overline{{{\mathcal{D}}}_{24m,4k}}$ has odd $r$, ${\mathbf{1}}\in C(\overline{{{\mathcal{D}}}_{24m,4k}})$. Consider the remaining cases.  The system of equations (\[eq:M\]) has no solution $(n_0,n_2,\ldots)$ consisting of nonnegative integers for each odd $w$. By Lemma \[lem:basic\] (i), ${\mathbf{1}}\in C(\overline{{{\mathcal{D}}}_{24m,4k}})$. The result follows from Lemma \[lem:CD\]. 

By the above proposition, for $m \in \{1,\ldots,6\}$ and $k \in \{m+1,\ldots,3m-1\}$, $C({{\mathcal{D}}}_{24m,4k})$ and $C({{\mathcal{D}}}_{24m,24m-4k})$ are self-dual.  In addition, $C({{\mathcal{D}}}_{24m,12m})$ are self-dual for $m \in
\{1,\ldots,5\}$. This completes the proof of Theorem \[thm:main2\]. 

On the minimum weights {#sec:min}
======================

In this section, we describe how to determine the minimum weights given in the third column of Table \[Tab:D\] for the cases denoted by $*$ in Table \[Tab:D\].  For the other cases, the minimum weights are determined by Lemma \[lem:basic\] (i) only.  The result in this section completes the proof of Theorem \[thm:main1\]. 

$(24m,4k)=(72, 24),( 72, 32)$
-----------------------------

Suppose that $4k \in\{24, 32\}$. Let ${v}\in C({\mathcal{D}}_{72,4k})^{\perp}$ be a vector of weight $w >0$. For each $w \in \{4,8\}$, the system of equations (\[eq:M\]) has no solution.  From the result in the previous section, $C({\mathcal{D}}_{72,4k})$ is a doubly even self-dual code.  By Lemma \[lem:basic\] (i), $C({\mathcal{D}}_{72,4k})$ is a doubly even self-dual code of length $72$ and minimum weight at least $12$.

By Gleason’s theorem (see [@MS73]), the weight enumerator of a doubly even self-dual code of length $n$ can be written as: $$\sum_{i=0}^{\lfloor n/24 \rfloor} a_i (x^8+14x^4y^4+y^8)^{n/8-3i}
(x^4y^4(x^4-y^4)^4)^i,$$ using integers $a_i$. Hence, the weight enumerator of $C({\mathcal{D}}_{72,4k})$ can be written as: $$\begin{aligned}
&
x^{72} + \alpha x^{60} y^{12}
+ (249849  - 12 \alpha) x^{56} y^{16}
+ (18106704  + 66 \alpha) x^{52} y^{20}
\\&
+ (462962955  -  220 \alpha) x^{48} y^{24}
+ (4397342400  + 495 \alpha) x^{44} y^{28}
\\&
+ (16602715899 - 792 \alpha) x^{40} y^{32}
+ (25756721120 +  924 \alpha) x^{36} y^{36}
+ \cdots,\end{aligned}$$ using a nonnegative integer $\alpha$. If $\alpha > 0$, then the number of codewords of weight $4k=24$ (resp.  $32$) is less than $462962955$ (resp.  $16602715899$), which is the number of blocks of ${\mathcal{D}}_{72,24}$ (resp.  ${\mathcal{D}}_{72,32}$).  Hence, $\alpha=0$. This means that $C({\mathcal{D}}_{72,4k})$ must be extremal. 

$(24m,4k)=(96,28), ( 96, 36), ( 96, 44)$
----------------------------------------

The numbers of blocks of ${\mathcal{D}}_{96,28}, {\mathcal{D}}_{96,36}$ and ${\mathcal{D}}_{96,44}$ are $$18642839520, 4552866656416 \text{ and }65727011639520,$$ respectively.  If $4k \in\{28, 36,  44\}$, then it follows from (\[eq:M\]) that the doubly even self-dual code $C({\mathcal{D}}_{96,4k})$ has minimum weight at least $12$. The weight enumerator $\sum_{i=0}^{96}A_i x^{96-i}y^i$ of $C({\mathcal{D}}_{96,4k})$ can be written using integers $\alpha,\beta$, where $A_i$ are listed in Table \[Tab:96\].  If there is an integer $i\in \{12,16\}$ with $A_i > 0$, then $$A_{36} =4552866656416 -4368 A_{12} -192412 A_{16} <4552866656416,$$ which is the number of the blocks of ${\mathcal{D}}_{96,36}$. This gives a contradiction.  Hence, $A_{12}=A_{16}=0$, then $\alpha=\beta=0$. This means that $C({\mathcal{D}}_{96,36})$ is extremal.  Similarly, one can easily show that $C({\mathcal{D}}_{96,44})$ is extremal, and that $C({\mathcal{D}}_{96,28})$ is extremal if $d(C({\mathcal{D}}_{96,28})) \ge 16$.

[c|l]{} $i$ &\
12& $\beta$\
16& $\alpha + 30 \beta$\
20& $3217056 - 16 \alpha + 153 \beta $\
24& $369844880   + 120 \alpha - 1712 \beta $\
28& $18642839520  - 560 \alpha - 3084 \beta $\
32& $422069980215  + 1820 \alpha + 69576 \beta $\
36& $4552866656416 - 4368 \alpha - 323452 \beta $\
40& $24292689565680 + 8008 \alpha + 842544 \beta $\
44& $65727011639520 - 11440 \alpha- 1443090 \beta $\
48& $91447669224080 + 12870 \alpha+ 1718068 \beta $\

$(24m,4k)=(120, 32),(120,40),(120,48),(120,56)$
-----------------------------------------------

The numbers of blocks of ${\mathcal{D}}_{120, 32}, {\mathcal{D}}_{120,40},
{\mathcal{D}}_{120,48}$ and ${\mathcal{D}}_{120,56}$ are $$\begin{gathered}
475644139425,
397450513031544, 
\\
30531599026535880 \text{ and  }
257257766776517715,\end{gathered}$$ respectively.  If $4k \in\{32,40,48,56\}$, then it follows from (\[eq:M\]) that the doubly even self-dual code $C({\mathcal{D}}_{120,4k})$ has minimum weight at least $12$. The weight enumerator $W_{120,12}=\sum_{i=0}^{120}A_i x^{120-i}y^i$ of $C({\mathcal{D}}_{120,4k})$ can be written using integers $\alpha,\beta,\gamma$, where $A_i$ are listed in Table \[Tab:120\].  If there is an integer $i\in \{12,16,20\}$ with $A_i > 0$, then $$\begin{aligned}
A_{56} = &
257257766776517715-1130786592 A_{12} - 16300570 A_{16} 
\\ & - 167960 A_{20}  < 257257766776517715,\end{aligned}$$ which gives a contradiction.  Hence, $A_{12}=A_{16}=A_{20}=0$, then $\alpha=\beta=\gamma=0$. This means that $C({\mathcal{D}}_{120,56})$ is extremal.  Similarly, one can easily show that $C({\mathcal{D}}_{120,4k})$ is extremal for $4k=40,48$, and that $C({\mathcal{D}}_{120, 32})$ is extremal if $d(C({\mathcal{D}}_{120, 32})) \ge 20$.

[c|l]{} $i$ &\
12& $\gamma$\
16& $\beta + 72\gamma$\
20& $\alpha + 26\beta + 2004\gamma$\
24& $39703755 -20\alpha + 39\beta + 25272\gamma$\
28& $6101289120 + 190\alpha - 2148\beta + 100866\gamma$\
32& $475644139425 -1140\alpha + 4563\beta - 621288\gamma$\
36& $18824510698240+ 4845\alpha + 71058\beta - 3973756\gamma$\
40& $397450513031544 -15504\alpha - 613259\beta + 18650088\gamma$\
44& $4630512364732800+ 38760\alpha + 2564432\beta + 37650159\gamma$\
48& $30531599026535880 -77520\alpha - 7035366\beta - 434682288\gamma$\
52& $116023977311397120+ 125970\alpha + 13909076\beta + 1412322984\gamma$\
56& $257257766776517715 -167960\alpha - 20667530\beta - 2641019472\gamma$\
60& $335200280030755776+ 184756\alpha + 23538216\beta + 3223090716\gamma$\

$(24m,4k)=(144,36), (144,52),(144,60),(144,68)$
-----------------------------------------------

The numbers of blocks of ${\mathcal{D}}_{144,36}$, ${\mathcal{D}}_{144,52}$, ${\mathcal{D}}_{144,60}$ and ${\mathcal{D}}_{144,68}$ are $$\begin{gathered}
9542972508784, 
4686006803807297232, 
\\
170473729066542803616  \text{ and  }
1005386522059285093728, \end{gathered}$$ respectively.  If $4k \in\{36,52,60,68\}$, then it follows from (\[eq:M\]) that the doubly even self-dual code $C({\mathcal{D}}_{144,4k})$ has minimum weight at least $12$. The weight enumerator $W_{144,12}=\sum_{i=0}^{144}A_i x^{144-i}y^i$ of $C({\mathcal{D}}_{144,4k})$ can be written using integers $\alpha,\beta,\gamma,\delta$, where $A_i$ are listed in Table \[Tab:144\].  If there is an integer $i\in \{12,16,20,24\}$ with $A_i > 0$, then $$\begin{aligned}
A_{68} =& 1005386522059285093728 - 1215686694585 A_{12}
\\& 
-16397532256 A_{16} -246582076 A_{20} -2496144 A_{24}\\ 
<& 1005386522059285093728,\end{aligned}$$ which gives a contradiction.  Hence, $A_{12}=A_{16}=A_{20}=A_{24}=0$, then $\alpha=\beta=\gamma=\delta=0$. This means that $C({\mathcal{D}}_{144,68})$ is extremal.  Similarly, one can easily show that $C({\mathcal{D}}_{144,60})$ is extremal, that $C({\mathcal{D}}_{144,52})$ is extremal if $d(C({\mathcal{D}}_{144,52})) \ge 20$, and that $C({\mathcal{D}}_{144,36})$ is extremal if $d(C({\mathcal{D}}_{144,36})) \ge 24$.

[c|l]{} $i$ &\
12&$\delta$\
16&$\gamma + 114\delta$\
20&$\beta + 68\gamma + 5619\delta$\
24&$\alpha + 22\beta + 1722\gamma + 154820\delta$\
28&$ 481008528-24\alpha - 59\beta + 17684\gamma + 2550861\delta$\
32&$90184804281+276\alpha - 2152\beta + 11515\gamma + 24260742\delta$\
36&$9542972508784-2024\alpha + 13286\beta - 881064\gamma + 102200559\delta$\
40&$559456467836112+10626\alpha + 39788\beta - 982492\gamma - 215159832\delta$\
44&$18950225255363376-42504\alpha - 861482\beta + 30439192\gamma - 3223863171\delta$\
48&$381888573368657355+134596\alpha + 5423416\beta - 58206711\gamma + 568124866\delta$\
52&$4686006803807297232-346104\alpha - 21252317\beta - 458108660\gamma +55774876695\delta$\
56&$35648745873701148864+735471\alpha + 59961226\beta + 3298378982\gamma -82891353732\delta$\
60&$170473729066542803616-1307504\alpha - 129387017\beta - 11030355684\gamma -479267780119\delta$\
64&$517692242136399518331 +1961256\alpha + 220368688\beta + 24037485819\gamma +2310638405958\delta$\
68&$1005386522059285093728-2496144\alpha - 301497244\beta - 37463473392\gamma -4857003070893\delta$\
72&$1253789175212713133280 +2704156\alpha + 334387688\beta + 43291346040\gamma +6110981295024\delta$\

[**Acknowledgments.**]{}  The author would like to thank Tsuyoshi Miezaki for verifying the calculations in the proofs of Propositions \[prop:SD-P1\] and \[prop:SD-P2\], independently.  This work is supported by JSPS KAKENHI Grant Number 23340021. 

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[^1]: Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980–8579, Japan.  email: mharada@m.tohoku.ac.jp.  This work was partially carried out at Yamagata University. 

[^2]: See Sections \[sec:SD\] and \[sec:min\] for the marks $*$ in Table \[Tab:D\].