# Construction of extremal Type II $\mathbb{Z}_{2k}$-codes

Masaaki Harada  Research Center for Pure and Applied Mathematics, Graduate
School of Information Sciences, Tohoku University, Sendai 980–8579, Japan.
email<EMAIL_ADDRESS>
###### Abstract

We give methods for constructing many self-dual $\mathbb{Z}_{m}$-codes and
Type II $\mathbb{Z}_{2k}$-codes of length $2n$ starting from a given self-dual
$\mathbb{Z}_{m}$-code and Type II $\mathbb{Z}_{2k}$-code of length $2n$,
respectively. As an application, we construct extremal Type II
$\mathbb{Z}_{2k}$-codes of length $24$ for $k=4,5,\ldots,20$ and extremal Type
II $\mathbb{Z}_{2k}$-codes of length $32$ for $k=4,5,\ldots,10$. We also
construct new extremal Type II $\mathbb{Z}_{4}$-codes of lengths $56$ and
$64$.

##  1 Introduction

A $\mathbb{Z}_{m}$-code of length $n$ is a $\mathbb{Z}_{m}$-submodule of
$\mathbb{Z}_{m}^{n}$, where $m$ is a positive integer with $m\geq 2$ and
$\mathbb{Z}_{m}$ denotes the ring of integers modulo $m$. A
$\mathbb{Z}_{m}$-code $C$ of length $n$ is self-dual if $C=C^{\perp}$, where
$C^{\perp}$ denotes the dual code of $C$. Self-dual codes are one of the most
interesting classes of codes. This interest is justified by many combinatorial
objects and algebraic objects related to self-dual codes (see e.g. [26] and
the references given therein).

Many methods for constructing self-dual codes are known. For example, starting
from a given self-dual code with generator matrix
$\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$, by transforming $A$,
some methods for constructing many self-dual codes are known, where $I_{n}$
denotes the identity matrix of order $n$ (see e.g. [14], [19], [23] and [27]).
In this paper, we give a new method for constructing many self-dual
$\mathbb{Z}_{m}$-codes starting from a given self-dual $\mathbb{Z}_{m}$-code
with generator matrix $\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$ by
transforming $A$. For self-dual codes over finite fields, the corresponding
result can be found in [23].

Now let us consider the case $m=2k$. A binary doubly even self-dual code is
often called Type II. Self-dual $\mathbb{Z}_{2k}$-codes with the property that
all Euclidean weights are multiples of $4k$ are called _Type II_
$\mathbb{Z}_{2k}$-codes (see [2] and [22] for $k=2$ and [1] for $k\geq 3$).
Type II $\mathbb{Z}_{2k}$-codes are a remarkable class of self-dual codes
related to even unimodular lattices. For example, by Construction A, Type II
$\mathbb{Z}_{2k}$-codes give even unimodular lattices. There is a Type II
$\mathbb{Z}_{2k}$-code of length $n$ if and only if $n$ is divisible by eight
[1]. For Type II $\mathbb{Z}_{2k}$-codes $C$ of length $n$, the following
upper bound on the minimum Euclidean weight

$d_{E}(C)\leq 4k\left\lfloor\frac{n}{24}\right\rfloor+4k$ (1)

holds if $k=1$ [24], if $k=2$ [2] (see also [22]) and if $k\in\\{3,4,5,6\\}$
[21]. Also, the bound (1) holds under the assumption that $\lfloor
n/24\rfloor\leq k-2$ for arbitrary $k$ [1]. When the bound (1) holds, we say
that a Type II $\mathbb{Z}_{2k}$-code meeting (1) with equality is _extremal_.
There have been significant researches on constructing extremal Type II
$\mathbb{Z}_{2k}$-codes for $k=1$ and $2$ (see e.g. [2], [14], [19], [22],
[26], [27] and the references given therein). In this paper, we also give a
new method for constructing many Type II $\mathbb{Z}_{2k}$-codes starting from
a given Type II $\mathbb{Z}_{2k}$-code with generator matrix
$\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$ by transforming $A$. This
generalizes the methods given in [14] and [23]. By four-negacirculant codes
and by the new method, we construct extremal Type II $\mathbb{Z}_{2k}$-codes
of length $24$ for $k=4,5,\ldots,20$ and extremal Type II
$\mathbb{Z}_{2k}$-codes of length $32$ for $k=4,5,\ldots,10$. We also
construct new extremal Type II $\mathbb{Z}_{4}$-codes of lengths $56$ and
$64$.

The paper is organized as follows. In Section 2, we give the definitions and
basic facts on self-dual $\mathbb{Z}_{m}$-codes and Type II
$\mathbb{Z}_{2k}$-codes used throughout this paper. In Section 3, we give a
method for constructing self-dual $\mathbb{Z}_{m}$-codes (resp. Type II
$\mathbb{Z}_{2k}$-codes) of length $2n$ starting from a given self-dual
$\mathbb{Z}_{m}$-code (resp. Type II $\mathbb{Z}_{2k}$-code) of length $2n$
having generator matrix $\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$,
by transforming $A$ (Theorem 3.1 (resp. Theorem 3.4)). In Section 4, we
construct extremal Type II $\mathbb{Z}_{2k}$-codes of length $24$, by four-
negacirculant codes and by Theorem 3.4 for $k=4,5,\ldots,20$. Consequently, it
is shown that if $k\in\\{2,3,\ldots,20\\}$ then there is an extremal Type II
$\mathbb{Z}_{2k}$-code $C$ such that $C^{(2)}\cong B$ for every binary doubly
even self-dual code $B$ of length $24$, where $C^{(2)}$ denotes the binary
part of $C$ and $C\cong D$ means that $C$ and $D$ are equivalent. In Section
5, we construct extremal Type II $\mathbb{Z}_{2k}$-codes of length $32$, by
four-negacirculant codes and by Theorem 3.4 for $k=4,5,\ldots,10$.
Consequently, it is shown that if $k\in\\{2,3,\ldots,10\\}$ then there is an
extremal Type II $\mathbb{Z}_{2k}$-code $C$ such that $C^{(2)}\cong B$ for
every binary extremal doubly even self-dual code $B$ of length $32$. In
Section 6, by four-negacirculant codes and by Theorem 3.4, we construct new
extremal Type II $\mathbb{Z}_{4}$-codes of lengths $56$ and $64$.

All computer calculations in this paper were done with the help of Magma [3].

##  2 Preliminaries

In this section, we give the definitions and basic facts on self-dual
$\mathbb{Z}_{m}$-codes and Type II $\mathbb{Z}_{2k}$-codes used throughout
this paper.

Let $\mathbb{Z}_{m}$ be the ring of integers modulo $m$, where $m$ is a
positive integer with $m\geq 2$. A $\mathbb{Z}_{m}$-code $C$ of length $n$ is
a $\mathbb{Z}_{m}$-submodule of $\mathbb{Z}_{m}^{n}$. An element of $C$ is
called a codeword. A generator matrix of $C$ is a matrix whose rows generate
$C$. We define the inner product of $x=(x_{1},x_{2},\ldots,x_{n})$ and
$y=(y_{1},y_{2},\ldots,y_{n})\in\mathbb{Z}_{m}^{n}$ by

$\langle x,y\rangle=x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}.$

The dual code $C^{\perp}$ of $C$ is defined as

$C^{\perp}=\\{x\in\mathbb{Z}_{m}^{n}\mid\langle x,y\rangle=0\text{ for all
}y\in C\\}.$

A code $C$ is self-dual if $C=C^{\perp}$. Two $\mathbb{Z}_{m}$-codes $C$ and
$C^{\prime}$ are equivalent, denoted $C\cong C^{\prime}$, if one can be
obtained from the other by permuting the coordinates and (if necessary)
changing the signs of certain coordinates.

We now consider self-dual $\mathbb{Z}_{2k}$-codes. In this paper, we take the
set $\mathbb{Z}_{2k}$ to be either $\\{0,1,\ldots,2k-1\\}$ or $\\{0,\pm
1,\ldots,\pm(k-1),k\\}$, using whichever form is more convenient. We define a
map $\rho$ from $\mathbb{Z}_{2k}$ to $\mathbb{Z}$ as follows

$\rho(i)=\begin{cases}0,\ldots,k&\text{ if }i=0,\ldots,k,\text{
respectively,}\\\ 1-k,\ldots,-1&\text{ if }i=k+1,\ldots,2k-1,\text{
respectively.}\end{cases}$ (2)

The Euclidean weight $\operatorname{wt}_{E}(v)$ of a vector
$v=(v_{1},v_{2},\ldots,v_{n})\in\mathbb{Z}_{2k}^{n}$ is

$\sum_{i=1}^{k}i^{2}n_{i}(v)+\sum_{i=k+1}^{2k-1}(2k-i)^{2}n_{i}(v),$

where

$n_{i}(v)=|\\{j\in\\{1,2,\ldots,n\\}\mid\rho(v_{j})\equiv i\pmod{2k}\\}|\
(i=0,1,\ldots,2k-1).$

A binary doubly even self-dual code is often called Type II. Using Euclidean
weights, the notion of binary Type II codes has been generalized as follows.
Self-dual $\mathbb{Z}_{2k}$-codes with the property that all Euclidean weights
are multiples of $4k$ are called _Type II_ $\mathbb{Z}_{2k}$-codes (see [2]
and [22] for $k=2$ and [1] for $k\geq 3$). It was shown in [1] that there is a
Type II $\mathbb{Z}_{2k}$-code of length $n$ if and only if $n$ is divisible
by eight.

Throughout this paper, $I_{n}$ denotes the identity matrix of order $n$, and
$A^{T}$ denotes the transpose of a matrix $A$. The following gives a criteria
for self-dual $\mathbb{Z}_{2k}$-codes and Type II $\mathbb{Z}_{2k}$-codes.

######  Lemma 2.1.

  * (i)

Let $C$ be a $\mathbb{Z}_{2k}$-code of length $2n$ having generator matrix of
form $\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$, where $A$ is an
$n\times n$ matrix. If $AA^{T}=-I_{n}$, then $C$ is self-dual.

  * (ii)

Let $C$ be a self-dual $\mathbb{Z}_{2k}$-code having generator matrix $G$. If
each row of $G$ has Euclidean weight divisible by $4k$, then $C$ is Type II.

###### Proof.

(i) is trivial. (ii) follows from [1, Lemma 2.2]. ∎

The minimum Euclidean weight $d_{E}(C)$ of a Type II $\mathbb{Z}_{2k}$-code
$C$ is the smallest Euclidean weight among all nonzero codewords of $C$. For
Type II $\mathbb{Z}_{2k}$-codes of length $n$, the upper bound (1) on the
minimum Euclidean weight $d_{E}(C)$ holds if $k=1$ [24], if $k=2$ [2] and [22]
and if $k\in\\{3,4,5,6\\}$ [21]. Also, the bound (1) holds under the
assumption that $\lfloor n/24\rfloor\leq k-2$ for arbitrary $k$ [1]. When the
bound (1) holds, we say that a Type II $\mathbb{Z}_{2k}$-code meeting (1) with
equality is _extremal_.

The binary part $C^{(2)}$ of a Type II $\mathbb{Z}_{2k}$-code $C$ of length
$n$ is defined as

$\\{(\rho(c_{1})\\!\\!\\!\pmod{2},\rho(c_{2})\\!\\!\\!\pmod{2},\ldots,\rho(c_{n})\\!\\!\\!\pmod{2})\mid(c_{1},c_{2},\ldots,c_{n})\in
C\\},$

where $\rho$ is the map given in (2). If a Type II $\mathbb{Z}_{2k}$-code has
generator matrix of form $\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$,
then the binary part is a binary doubly even self-dual code [8] and [9]. The
following is trivial.

######  Lemma 2.2.

Let $C$ and $C^{\prime}$ be Type II $\mathbb{Z}_{2k}$-codes. If $C\cong
C^{\prime}$, then $C^{(2)}\cong{C^{\prime}}^{(2)}$.

An $n\times n$ _negacirculant_ matrix has the following form

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

Let $C$ be the $\mathbb{Z}_{m}$-code of length $4n$ having the following
generator matrix

$\left(\begin{array}[]{ccc@{}c}&{\Large I_{2n}}&&\begin{array}[]{cc}A&B\\\
-B^{T}&A^{T}\end{array}\end{array}\right),$ (3)

where $A$ and $B$ are $n\times n$ negacirculant matrices. Such a code is
called a _four-negacirculant_ code [18]. By Lemma 2.1 (i), if
$AA^{T}+BB^{T}=-I_{n}$, then $C$ is self-dual. Let $C$ be a four-negacirculant
self-dual $\mathbb{Z}_{2k}$-code with generator matrix of form (3). Let
$r_{A}$ and $r_{B}$ denote the first rows of $A$ and $B$, respectively. By
Lemma 2.1 (ii), if
$\operatorname{wt}_{E}(r_{A})+\operatorname{wt}_{E}(r_{B})\equiv-1\pmod{4k}$,
then $C$ is Type II.

We end this section by describing unimodular lattices and Construction A. A
(Euclidean) integral lattice $L\subset\mathbb{R}^{n}$ in dimension $n$ is
unimodular if $L=L^{*}$, where $L^{*}$ is the dual lattice under the standard
inner product $(x,y)$. A unimodular lattice $L$ is even if the norm $(x,x)$ of
every vector $x$ of $L$ is even, and odd otherwise. There is an even
unimodular lattice in dimension $n$ if and only if $n$ is divisible by eight.
The minimum norm $\min(L)$ of a unimodular lattice $L$ is the smallest norm
among all nonzero vectors of $L$. The kissing number of $L$ is the number of
vectors of minimum norm in $L$. We now give a method to construct even
unimodular lattices from Type II $\mathbb{Z}_{2k}$-codes, which is called
Construction A. For a Type II $\mathbb{Z}_{2k}$-code $C$ of length $n$, define
the following lattice

$A_{2k}(C)=\frac{1}{\sqrt{2k}}\\{\rho(C)+2k\mathbb{Z}^{n}\\},$ (4)

where

$\rho(C)=\\{(\rho(c_{1}),\rho(c_{2}),\ldots,\rho(c_{n}))\ |\
(c_{1},c_{2},\ldots,c_{n})\in C\\}.$

It is known that $A_{2k}(C)$ is an even unimodular lattice with minimum norm
$\min\\{d_{E}(C)/2k,2k\\}$, where $d_{E}(C)$ denotes the minimum Euclidean
weight of $C$ [1]. The minimum norms of $A_{2k}(C)$ are used to determine the
minimum Euclidean weights of $C$ in Sections 4, 5 and 6.

##  3 Methods for constructing self-dual $\mathbb{Z}_{m}$-codes

In this section, starting from a given self-dual $\mathbb{Z}_{m}$-code (resp.
Type II $\mathbb{Z}_{2k}$-code) with generator matrix
$\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$, by transforming $A$, we
give a new method for constructing many self-dual $\mathbb{Z}_{m}$-codes
(resp. Type II $\mathbb{Z}_{2k}$-codes).

######  Theorem 3.1.

Let $C$ be a self-dual $\mathbb{Z}_{m}$-code of length $2n$ having generator
matrix $\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$. Let $r_{i}$ be
the $i$-th row of $A$. Let $x$ and $y$ be vectors of $\mathbb{Z}_{m}^{n}$.
Define an $n\times n$ matrix $A(x,y)$, where the $i$-th row $r^{\prime}_{i}$
is given by

$r^{\prime}_{i}=r_{i}+\langle r_{i},y\rangle x-\langle r_{i},x\rangle y.$

Let $C(A,x,y)$ be the $\mathbb{Z}_{m}$-code having the following generator
matrix

$\left(\begin{array}[]{cc}I_{n}&A(x,y)\end{array}\right).$

If $\langle x,x\rangle=\langle y,y\rangle=\langle x,y\rangle=0$, then
$C(A,x,y)$ is a self-dual $\mathbb{Z}_{m}$-code of length $2n$.

###### Proof.

Since $\langle x,x\rangle=\langle y,y\rangle=\langle x,y\rangle=0$, we have

$\begin{split}\langle r^{\prime}_{i},r^{\prime}_{j}\rangle=&\langle
r_{i},r_{j}\rangle+\langle r_{j},y\rangle\langle r_{i},x\rangle-\langle
r_{j},x\rangle\langle r_{i},y\rangle\\\ &+\langle r_{i},y\rangle\langle
x,r_{j}\rangle+\langle r_{i},y\rangle\langle r_{j},y\rangle\langle
x,x\rangle-\langle r_{i},y\rangle\langle r_{j},x\rangle\langle x,y\rangle\\\
&-\langle r_{i},x\rangle\langle y,r_{j}\rangle-\langle r_{i},x\rangle\langle
r_{j},y\rangle\langle y,x\rangle+\langle r_{i},x\rangle\langle
r_{j},x\rangle\langle y,y\rangle\\\ =&\langle r_{i},r_{j}\rangle+\langle
r_{j},y\rangle\langle r_{i},x\rangle-\langle r_{j},x\rangle\langle
r_{i},y\rangle+\langle r_{i},y\rangle\langle x,r_{j}\rangle\\\ &-\langle
r_{i},x\rangle\langle y,r_{j}\rangle\\\ =&\langle
r_{i},r_{j}\rangle.\end{split}$ (5)

This implies that $AA^{T}=A(x,y)A(x,y)^{T}$. By Lemma 2.1 (i), $C(A,x,y)$ is
self-dual. ∎

######  Remark 3.2.

For self-dual codes over finite fields, the corresponding result can be found
in [23].

######  Remark 3.3.

We modify the above theorem as follows. Let $C$ be a $\mathbb{Z}_{m}$-code of
length $n+n^{\prime}$ having generator matrix
$\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$, where $A$ is an $n\times
n^{\prime}$ matrix. Then the code $C(A,x,y)$ having generator matrix
$\left(\begin{array}[]{cc}I_{n}&A(x,y)\end{array}\right)$ defined in Theorem
3.1 satisfies $AA^{T}=A(x,y)A(x,y)^{T}$.

We consider a method for constructing Type II $\mathbb{Z}_{2k}$-codes.

######  Theorem 3.4.

Suppose that $C$ is a Type II $\mathbb{Z}_{2k}$-code of length $2n$ having
generator matrix $\left(\begin{array}[]{cc}I_{n}&A\end{array}\right)$. Suppose
that $x$ and $y$ are vectors of $\mathbb{Z}_{2k}^{n}$ satisfying the condition
$\operatorname{wt}_{E}(x)\equiv\operatorname{wt}_{E}(y)\equiv 0\pmod{4k}$ and
$\langle x,y\rangle=0$. Then the code $C(A,x,y)$ having generator matrix
$\left(\begin{array}[]{cc}I_{n}&A(x,y)\end{array}\right)$ defined in Theorem
3.1 is a Type II $\mathbb{Z}_{2k}$-code of length $2n$.

###### Proof.

Let $\rho$ be the map given in (2). For a vector
$v=(v_{1},v_{2},\ldots,v_{n})\in\mathbb{Z}_{2k}$, define the positive integer
$v*v=\sum_{i=1}^{2k-1}i^{2}n_{i}(v)$, where
$n_{i}(v)=|\\{j\in\\{1,2,\ldots,n\\}\mid\rho(v_{j})\equiv i\pmod{2k}\\}|$
$(i=0,1,\ldots,2k-1)$. By [1, Lemma 2.1], $\operatorname{wt}_{E}(v)\equiv
v*v\pmod{4k}$. This implies that $\langle x,x\rangle=\langle y,y\rangle=0$.
Thus, by Theorem 3.1, $C(A,x,y)$ is self-dual. Let $r_{i}$ and
$r^{\prime}_{i}$ denote the $i$-th row of $A$ and $A(x,y)$, respectively. By
Lemma 2.1 (ii), it is sufficient to show that
$\operatorname{wt}_{E}(r^{\prime}_{i})\equiv 4k-1\pmod{4k}$
$(i=1,2,\ldots,n)$. From (5), we have

$\displaystyle\operatorname{wt}_{E}(r^{\prime}_{i})$ $\displaystyle\equiv
r^{\prime}_{i}*r^{\prime}_{i}$ $\displaystyle\equiv r_{i}*r_{i}+x*x+y*y$
$\displaystyle\equiv\operatorname{wt}_{E}(r_{i})+\operatorname{wt}_{E}(x)+\operatorname{wt}_{E}(y)$
$\displaystyle\equiv\operatorname{wt}_{E}(r_{i})$ $\displaystyle\equiv
4k-1\pmod{4k}.$

The result follows. ∎

######  Remark 3.5.

For the case $k=1$, the above theorem can be found in [23]. Theorem 5 in [14]
can be obtained from the above theorem by putting $k=2$, $x=t$ and
$y=(2,2,\ldots,2)$.

##  4 Extremal Type II $\mathbb{Z}_{2k}$-codes of length 24

For $k=4,5,\ldots,20$, in this section, we construct extremal Type II
$\mathbb{Z}_{2k}$-codes of length $24$, by four-negacirculant codes and by
Theorem 3.4.

###  4.1 Motivation and results

For lengths $8$ and $16$, every Type II $\mathbb{Z}_{2k}$-code is extremal. In
other words, the smallest possible length for which there is a nontrivial
extremal Type II $\mathbb{Z}_{2k}$-code is $24$. Meanwhile, an extremal Type
II $\mathbb{Z}_{2k}$-codes of length $24$ are related to the Leech lattice,
which is arguably the most remarkable lattice. In particular, by considering
the existence of $2k$-frames in the Leech lattice, it was shown that there is
an extremal Type II $\mathbb{Z}_{2k}$-code of length $24$ for every positive
integer $k$ with $k\geq 2$ [5] and [12].

There are nine inequivalent binary doubly even self-dual codes of length $24$
[25]. The seven codes are indecomposable and these code are denoted by
$\textrm{A}24,\textrm{B}24,\ldots,\textrm{G}24$ in [25, Table II]. Only the
code $\textrm{G}24$ has minimum weight $8$, that is, $\textrm{G}24$ is
extremal, and it is well known as the extended Golay code. As usual we denote
the two decomposable codes by $e_{8}^{3}$ and $d_{16}e_{8}$. For every binary
doubly even self-dual code $B$ of length $24$, there is an extremal Type II
$\mathbb{Z}_{2k}$-code $C$ such that $C^{(2)}\cong B$ when $k=2$ [14,
Proposition 9] (see [4, Postscript]) and $k=3$ [20, Theorem 3]. In this
section, we establish the following theorem by constructing extremal Type II
$\mathbb{Z}_{2k}$-codes of length $24$ for $k=4,5,\ldots,20$ by using four-
negacirculant codes and by Theorem 3.4 explicitly.

######  Theorem 4.1.

Suppose that $k\in\\{2,3,\ldots,20\\}$. For every binary doubly even self-dual
code $B$ of length $24$, there is an extremal Type II $\mathbb{Z}_{2k}$-code
$C$ such that $C^{(2)}\cong B$.

Although the following lemma is trivial, we give a proof for completeness.

######  Lemma 4.2.

Suppose that $k\geq 3$ and $n\in\\{24,32,40\\}$. Let $C$ be a Type II
$\mathbb{Z}_{2k}$-code of length $n$. Let $A_{2k}(C)$ denote the even
unimodular lattice given in (4). Then $C$ is extremal if and only if
$A_{2k}(C)$ has minimum norm $4$.

###### Proof.

Since $k\geq 3$, a vector of norm $2$ of $A_{2k}(C)$ is written as

$\frac{1}{\sqrt{2k}}(\rho(c_{1}),\rho(c_{2}),\ldots,\rho(c_{n}))$

for some codeword $c=(c_{1},c_{2},\ldots,c_{n})\in C$ such that
$\operatorname{wt}_{E}(c)=4k$, where $\rho$ is the map given in (2). Thus, $C$
contains no codeword of Euclidean weight $4k$ if and only if $A_{2k}(C)$
contains no vector of norm $2$. It is known that the minimum norm $\min(L)$ of
an even unimodular lattice in dimension $n$ is bounded by $\min(L)\leq
2\lfloor\frac{n}{24}\rfloor+2$ (see [7, Chapter 7]). Thus,
$\min(A_{2k}(C))\leq 4$ for $n\in\\{24,32,40\\}$. Hence, $C$ contains no
codeword of Euclidean weight $4k$ if and only if $A_{2k}(C)$ has minimum norm
$4$. The result follows. ∎

By the above lemma, we determined the extremality for all Type II
$\mathbb{Z}_{2k}$-codes found in this section and the next section.

###  4.2 Extremal Type II $\mathbb{Z}_{8}$-codes of length 24

By considering four-negacirculant codes, our computer search found extremal
Type II $\mathbb{Z}_{8}$-codes $C_{8,24,1}$ and $C_{8,24,2}$ of length $24$
(see the end of Section 2 for four-negacirculant codes). The codes
$C_{8,24,1}$ and $C_{8,24,2}$ have generator matrices of form (3), where the
first rows $r_{A}$ and $r_{B}$ of the negacirculant matrices $A$ and $B$ are
as follows

$\displaystyle(r_{A},r_{B})=$
$\displaystyle((5,3,3,3,3,1),(5,5,5,6,7,7))\text{ and }$
$\displaystyle((0,0,0,1,1,3),(0,1,0,3,3,1)),$

respectively. By Theorem 3.4, our computer search found more extremal Type II
$\mathbb{Z}_{8}$-codes $C_{8,24,3},C_{8,24,4},\ldots,C_{8,24,8}$ and
$C_{8,24,9}$ as $C(A,x,y)$ from $M_{8,24,i}$ $(i=1,2,4)$, where $M_{8,24,i}$
denotes the right half of the generator matrix of $C_{8,24,i}$. In Table 1, we
list the matrices $A$ and the vectors $x,y$ in Theorem 3.4.

Table 1: Extremal Type II $\mathbb{Z}_{8}$-codes of length $24$
---
Codes | $A$ | $x$ | $y$
$C_{8,24,3}$ | $M_{8,24,1}$ | $(1,1,1,4,7,3,3,0,2,5,5,6)$ | $(0,0,0,1,1,2,4,6,0,7,2,1)$
$C_{8,24,4}$ | $M_{8,24,1}$ | $(5,4,1,4,1,1,3,1,3,0,7,4)$ | $(0,1,1,1,7,4,6,6,1,3,3,7)$
$C_{8,24,5}$ | $M_{8,24,1}$ | $(6,5,1,4,5,5,6,6,4,6,0,2)$ | $(0,0,0,5,2,2,2,5,6,2,3,1)$
$C_{8,24,6}$ | $M_{8,24,2}$ | $(3,3,3,6,1,0,3,5,1,2,5,4)$ | $(0,0,0,3,1,6,4,5,2,1,4,2)$
$C_{8,24,7}$ | $M_{8,24,2}$ | $(4,3,7,0,0,0,6,2,6,0,1,5)$ | $(0,1,6,1,2,4,5,6,0,0,0,5)$
$C_{8,24,8}$ | $M_{8,24,2}$ | $(4,1,1,7,7,5,2,3,0,3,5,6)$ | $(0,0,2,3,0,2,2,0,3,3,0,3)$
$C_{8,24,9}$ | $M_{8,24,4}$ | $(5,2,2,6,6,2,6,1,5,2,1,0)$ | $(0,0,1,6,5,0,6,0,1,0,3,2)$

###  4.3 Extremal Type II $\mathbb{Z}_{2k}$-codes of length 24 for
$k=5,6,\ldots,20$

For $\mathbb{Z}_{2k}$-codes $(k=5,6,\ldots,20)$, to save space, we list the
results only. Note that the approach is similar to that in the previous
subsection.

For $k=5,6,\ldots,20$, by considering four-negacirculant codes, our computer
search found extremal Type II $\mathbb{Z}_{2k}$-codes $C_{2k,24,i}$ of length
$24$ $(i=1,2,\ldots,i_{k})$, where $i_{k}=3$ if $k\in\\{5,6,\ldots,10\\}$ and
$i_{k}=4$ if $k\in\\{11,12,\ldots,20\\}$. The codes have generator matrices of
form (3), where the first rows $r_{A}$ and $r_{B}$ of the negacirculant
matrices $A$ and $B$ are listed in Table 7. By Theorem 3.4, our computer
search found more extremal Type II $\mathbb{Z}_{2k}$-codes $C_{2k,24,j}$
$(j=i_{k}+1,i_{k}+2,\ldots,9)$ as $C(A,x,y)$ from $M_{2k,24,i}$, where
$M_{2k,24,i}$ denotes the right half of the generator matrix of $C_{2k,24,i}$
$(i=1,2,\ldots,9)$. In Tables 8, 9 and 10, we list the matrices $A$ and the
vectors $x,y$ in Theorem 3.4.

###  4.4 Binary parts of $C_{2k,24,1},C_{2k,24,2},\ldots,C_{2k,24,9}$ for
$k=4,5,\ldots,20$

For a given binary doubly even self-dual code $B$ of length $24$ and a given
$k$, we list in Table 2 the number $i$ such that $C^{(2)}_{2k,24,i}\cong B$.
From Table 2, we have Theorem 4.1 combining with known results in [14] and
[20].

######  Remark 4.3.

  * (i)

For $B=e_{8}^{3}$ and $\textrm{G}24$, an extremal Type II
$\mathbb{Z}_{8}$-code $C$ of length $24$ such that $C^{(2)}\cong B$ can be
found in [11] and [13], respectively.

  * (ii)

Extremal Type II $\mathbb{Z}_{10}$-codes $C$ of length $24$ such that
$C^{(2)}\cong e_{8}^{3}$ can be found in [11].

  * (iii)

For $B=\textrm{E}24$ and $e_{8}^{3}$, an extremal Type II
$\mathbb{Z}_{12}$-code $C$ of length $24$ such that $C^{(2)}\cong B$ can be
found in [11].

  * (iv)

An extremal Type II $\mathbb{Z}_{14}$-code $C$ of length $24$ such that
$C^{(2)}\cong e_{8}^{3}$ can be found in [11].

  * (v)

An extremal Type II $\mathbb{Z}_{16}$-code $C$ of length $24$ such that
$C^{(2)}\cong\textrm{E}24$ can be found in [11].

  * (vi)

An extremal Type II $\mathbb{Z}_{22}$-code $C$ of length $24$ such that
$C^{(2)}\cong\textrm{E}24$ can be found in [12]. For $B=\textrm{E}24$ and
$\textrm{G}24$, an extremal Type II $\mathbb{Z}_{22}$-code $C$ of length $24$
such that $C^{(2)}\cong B$ can be found in [11].

In addition, by Lemma 2.2, we have the following corollary.

######  Corollary 4.4.

Suppose that $k\in\\{2,3,\ldots,20\\}$. Then there are at least nine
inequivalent extremal Type II $\mathbb{Z}_{2k}$-codes of length $24$.

Table 2: Binary parts of $C_{2k,24,i}$ $(k=4,5,\ldots,20)$
---
$k$ | $\textrm{A}24$ | $\textrm{B}24$ | $\textrm{C}24$ | $\textrm{D}24$ | $\textrm{E}24$ | $\textrm{F}24$ | $\textrm{G}24$ | $e_{8}^{3}$ | $d_{16}e_{8}$
4 | 3 | 8 | 5 | 6 | 1 | 2 | 7 | 4 | 9
5 | 3 | 6 | 5 | 4 | 2 | 9 | 7 | 1 | 8
6 | 3 | 5 | 6 | 4 | 2 | 7 | 9 | 1 | 8
7 | 3 | 4 | 7 | 6 | 2 | 8 | 9 | 1 | 5
8 | 4 | 7 | 6 | 5 | 2 | 3 | 8 | 1 | 9
9 | 3 | 5 | 4 | 8 | 2 | 7 | 9 | 1 | 6
10 | 3 | 5 | 7 | 4 | 2 | 8 | 9 | 1 | 6
11 | 3 | 5 | 8 | 7 | 2 | 4 | 9 | 1 | 6
12 | 2 | 8 | 6 | 7 | 3 | 4 | 9 | 1 | 5
13 | 4 | 5 | 8 | 6 | 3 | 2 | 9 | 1 | 7
14 | 2 | 7 | 9 | 5 | 4 | 3 | 8 | 1 | 6
15 | 2 | 5 | 7 | 6 | 4 | 3 | 9 | 1 | 8
16 | 2 | 5 | 7 | 8 | 4 | 3 | 9 | 1 | 6
17 | 2 | 7 | 5 | 8 | 3 | 4 | 9 | 1 | 6
18 | 2 | 5 | 8 | 6 | 3 | 4 | 9 | 1 | 7
19 | 2 | 6 | 5 | 8 | 4 | 3 | 9 | 1 | 7
20 | 2 | 7 | 5 | 8 | 4 | 3 | 9 | 1 | 6

We end this section with the following question.

######  Question 1.

Suppose that $k\geq 21$. For every binary doubly even self-dual code $B$ of
length $24$, is there an extremal Type II $\mathbb{Z}_{2k}$-code $C$ such that
$C^{(2)}\cong B$.

##  5 Extremal Type II $\mathbb{Z}_{2k}$-codes of length 32

For $k=4,5,\ldots,10$, in this section, we construct extremal Type II
$\mathbb{Z}_{2k}$-codes of length $32$, by four-negacirculant codes and by
Theorem 3.4.

There are five inequivalent binary extremal doubly even self-dual codes of
length $32$ [6]. The five codes are denoted by
$\textrm{C}81,\textrm{C}82,\ldots,\textrm{C}85$ in [6, Table A]. For every
binary extremal doubly even self-dual code $B$ of length $32$, there is an
extremal Type II $\mathbb{Z}_{2k}$-code $C$ such that $C^{(2)}\cong B$ when
$k=2$ [10, Theorem 5] and $k=3$ [15, Proposition 3].

In this section, by constructing extremal Type II $\mathbb{Z}_{2k}$-codes of
length $32$ ($k=4,5,\ldots,10$) by using four-negacirculant codes and by
Theorem 3.4, we explicitly establish the following theorem.

######  Theorem 5.1.

Suppose that $k\in\\{2,3,\ldots,10\\}$. For every binary extremal doubly even
self-dual code $B$ of length $32$, there is an extremal Type II
$\mathbb{Z}_{2k}$-code $C$ such that $C^{(2)}\cong B$.

Table 3: Extremal Type II four-negacirculant $\mathbb{Z}_{2k}$-codes of length
$32$ $(k=4,5,\ldots,10)$
---
Codes | $r_{A}$ | $r_{B}$
$C_{8,32,1}$ | $(7,4,3,6,3,5,6,3)$ | $(5,6,0,2,0,4,7,4)$
$C_{8,32,2}$ | $(7,4,3,6,4,5,2,5)$ | $(6,5,2,2,0,7,1,2)$
$C_{10,32,1}$ | $(5,1,4,5,1,8,0,4)$ | $(4,6,0,0,2,3,1,5)$
$C_{10,32,2}$ | $(5,1,4,5,4,0,5,0)$ | $(6,6,5,2,8,4,9,7)$
$C_{12,32,1}$ | $(2,5,11,3,1,7,6,10)$ | $(5,10,5,6,8,6,0,4)$
$C_{12,32,2}$ | $(2,5,11,3,3,4,10,2)$ | $(0,6,10,1,2,0,5,1)$
$C_{14,32,1}$ | $(3,11,6,1,3,10,13,12)$ | $(4,10,3,0,7,12,6,8)$
$C_{14,32,2}$ | $(3,11,6,1,2,2,11,8)$ | $(12,0,2,13,1,10,6,9)$
$C_{16,32,1}$ | $(2,13,4,2,11,2,0,15)$ | $(4,15,9,0,2,0,15,11)$
$C_{16,32,2}$ | $(2,13,4,2,14,7,7,0)$ | $(6,12,7,7,5,1,6,10)$
$C_{18,32,1}$ | $(14,17,15,11,1,3,14,2)$ | $(12,10,3,4,5,6,4,4)$
$C_{18,32,2}$ | $(14,17,15,11,5,16,16,14)$ | $(8,16,5,6,0,3,15,14)$
$C_{20,32,1}$ | $(2,0,8,9,15,5,11,7)$ | $(10,0,15,8,5,4,8,16)$
$C_{20,32,2}$ | $(2,0,8,9,1,14,12,1)$ | $(6,18,0,1,1,19,17,16)$

The approach is similar to that in the previous section. To save space, we
list the results only. By considering four-negacirculant codes, our computer
search found extremal Type II $\mathbb{Z}_{2k}$-codes $C_{2k,32,1}$ and
$C_{2k,32,2}$ of length $32$ $(k=4,5,\ldots,10)$. The codes have generator
matrices of form (3), where the first rows $r_{A}$ and $r_{B}$ of the
negacirculant matrices $A$ and $B$ are listed in Table 3. By Theorem 3.4, our
computer search found more extremal Type II $\mathbb{Z}_{2k}$-codes
$C_{2k,32,3}$, $C_{2k,32,4}$ and $C_{2k,32,5}$ as $C(A,x,y)$ from
$M_{2k,32,i}$, where $M_{2k,32,i}$ denotes the right half of the generator
matrix of $C_{2k,32,i}$ $(i=1,2,k=4,5,\ldots,10)$. In Table 11, we list the
matrices $A$ and the vectors $x,y$ in Theorem 3.4. For a given binary extremal
doubly even self-dual code $B$ of length $32$ and a given $k$, we list in
Table 4 the number $i$ such that $C^{(2)}_{2k,32,i}\cong B$
$(k=4,5,\ldots,10)$. From Table 4, we have Theorem 5.1 combining with known
results in [10] and [15]. In addition, by Lemma 2.2, we have the following
corollary.

######  Corollary 5.2.

Suppose that $k\in\\{2,3,\ldots,10\\}$. Then there are at least five
inequivalent extremal Type II $\mathbb{Z}_{2k}$-codes of length $32$.

Table 4: Binary parts of $C_{2k,32,i}$ $(k=4,5,\ldots,10)$
---
$k$ | $\textrm{C}81$ | $\textrm{C}82$ | $\textrm{C}83$ | $\textrm{C}84$ | $\textrm{C}85$
4 | 2 | 4 | 1 | 3 | 5
5 | 2 | 4 | 1 | 5 | 3
6 | 2 | 5 | 1 | 4 | 3
7 | 2 | 4 | 1 | 3 | 5
8 | 2 | 4 | 1 | 5 | 3
9 | 2 | 4 | 1 | 5 | 3
10 | 2 | 5 | 1 | 3 | 4

We end this section with the following question.

######  Question 2.

Suppose that $k\geq 11$. For every binary extremal doubly even self-dual code
$B$ of length $32$, is there an extremal Type II $\mathbb{Z}_{2k}$-code $C$
such that $C^{(2)}\cong B$.

##  6 Extremal Type II $\mathbb{Z}_{4}$-codes of lengths 56 and 64

In this section, by four-negacirculant codes and by Theorem 3.4, we construct
new extremal Type II $\mathbb{Z}_{4}$-codes of lengths $56$ and $64$.

###  6.1 Extremality

We describe how to determine the extremality of a given Type II
$\mathbb{Z}_{4}$-code of lengths $48,56$ and $64$. Although the following
lemma is trivial, we give a proof for completeness.

######  Lemma 6.1.

Suppose that $n\in\\{48,56,64\\}$. Let $C$ be a Type II $\mathbb{Z}_{4}$-code
of length $n$. Let $A_{4}(C)$ denote the even unimodular lattice given in (4).
Then $C$ is extremal if and only if $A_{4}(C)$ has minimum norm $4$ and
kissing number $2n$.

###### Proof.

It is trivial that $C$ has a codeword of Euclidean weight $8$ if and only if
$A_{4}(C)$ has a vector of norm $2$. If $C$ has a codeword of Euclidean weight
$16$, then $A_{4}(C)$ has a vector of norm $4$. In addition, $A_{4}(C)$ has
$2n$ vectors of norm $4$, which have the following form

$(\pm 2,0,0,\ldots,0),(0,\pm 2,0,\ldots,0),\ldots,(0,0,\ldots,0,\pm 2).$

Hence, $A_{4}(C)$ has minimum norm $4$ and kissing number $2n$ if and only if
$C$ has no codeword of Euclidean weight $16$. ∎

By the above lemma, we determined the extremality for all Type II
$\mathbb{Z}_{4}$-codes found in this section.

###  6.2 Extremal Type II $\mathbb{Z}_{4}$-codes of length 64

The first example of an extremal Type II $\mathbb{Z}_{4}$-code of length $64$
was found in [16]. Note that the binary part of the code is not self-dual.

By considering four-negacirculant codes, our computer search found four
extremal Type II $\mathbb{Z}_{4}$-codes $C_{4,64,i}$ $(i=1,2,3,4)$ of length
$64$. These codes have generator matrices of form (3), where the first rows
$r_{A}$ and $r_{B}$ of the negacirculant matrices $A$ and $B$ are listed in
Table 5.

Table 5: Extremal Type II four-negacirculant $\mathbb{Z}_{4}$-codes of length
$64$
---
Codes | $r_{A}$ | $r_{B}$
$C_{4,64,1}$ | $(1,0,1,1,3,3,2,3,3,2,2,3,3,2,3,2)$ | $(2,0,1,1,3,1,2,3,3,2,3,1,2,0,3,0)$
$C_{4,64,2}$ | $(3,0,3,2,3,0,3,2,0,3,0,1,2,2,0,0)$ | $(0,2,0,0,1,2,3,2,1,2,0,0,3,0,2,1)$
$C_{4,64,3}$ | $(3,0,0,0,2,0,1,0,2,3,2,2,3,0,1,1)$ | $(1,2,1,3,1,3,2,0,0,3,3,2,1,0,1,2)$
$C_{4,64,4}$ | $(0,0,3,3,3,1,1,1,0,1,0,3,0,3,3,2)$ | $(1,0,1,2,1,2,2,0,3,1,2,3,1,0,3,3)$

In addition, by Theorem 3.4, our computer search found one more extremal Type
II $\mathbb{Z}_{4}$-code $C_{4,64,5}$ as $C(M_{4,64,2},x,y)$, where
$M_{4,64,2}$ denotes the right half of the generator matrix of $C_{4,64,2}$,
and

$\displaystyle
x=(1,1,3,0,1,3,1,0,3,2,3,0,2,2,1,1,1,2,1,3,2,2,1,0,0,0,0,0,0,0,1,1),$
$\displaystyle
y=(1,3,2,0,0,3,2,2,0,2,0,3,1,2,1,0,2,0,2,2,0,2,2,3,1,1,1,1,2,1,2,2).$

By Gleason’s theorem (see [24]), the weight enumerator of a binary doubly even
self-dual code of length $64$ and minimum weight at least $8$ is written as

$W_{64}(a)=1+ay^{8}+(2976+20a)y^{12}+(454956+2a)y^{16}+\cdots,$

where $a$ is the number of codewords of weight $8$. Our computer search
verified that the binary parts $C_{4,64,i}^{(2)}$ $(i=1,2,3,4,5)$ have weight
enumerators $W_{64}(16)$, $W_{64}(64)$, $W_{64}(0)$, $W_{64}(0)$,
$W_{64}(19)$, respectively. In addition, our computer search verified that the
binary parts $C_{4,64,3}^{(2)}$ and $C_{4,64,4}^{(2)}$ are inequivalent. By
Lemma 2.2, we have the following proposition.

######  Proposition 6.2.

There are at least six inequivalent extremal Type II $\mathbb{Z}_{4}$-codes of
length $64$.

###  6.3 Extremal Type II $\mathbb{Z}_{4}$-codes of length 56

The first example of an extremal Type II $\mathbb{Z}_{4}$-code of length $56$
was found in [16]. Note that the binary part of the code is not self-dual. Two
more extremal Type II $\mathbb{Z}_{4}$-codes of length $56$ were constructed
by considering double circulant codes in [17]. The two codes are denoted by
$\mathcal{D}_{56,1}$ and $\mathcal{C}_{56}$ in [17]. Note that the binary
parts of these codes are binary extremal doubly even self-dual codes.

By considering four-negacirculant codes, our computer search found three
extremal Type II $\mathbb{Z}_{4}$-codes $C_{4,56,i}$ $(i=1,2,3)$ of length
$56$. These codes have generator matrices of form (3), where the first rows
$r_{A}$ and $r_{B}$ of the negacirculant matrices $A$ and $B$ are listed in
Table 6.

By Gleason’s theorem (see [24]), the weight enumerator of a binary doubly even
self-dual code of length $56$ and minimum weight at least $8$ is written as

$W_{56}(a)=1+ay^{8}+(8190+6a)y^{12}+(622314-83a)y^{16}+\cdots,$

where $a$ is the number of codewords of weight $8$. Our computer search
verified that the binary parts $C_{4,56,i}^{(2)}$ $(i=1,2,3)$ have weight
enumerators $W_{56}(35)$, $W_{56}(42)$, $W_{56}(0)$, respectively. In
addition, our computer search verified that $\mathcal{D}_{56,1}^{(2)}$,
$\mathcal{C}_{56}^{(2)}$ and $C_{4,56,3}^{(2)}$ are inequivalent. By Lemma
2.2, we have the following proposition.

######  Proposition 6.3.

There are at least six inequivalent extremal Type II $\mathbb{Z}_{4}$-codes of
length $56$.

Table 6: Extremal Type II four-negacirculant $\mathbb{Z}_{4}$-codes of length
$56$
---
Codes | $r_{A}$ | $r_{B}$
$C_{4,56,1}$ | $(2,2,0,1,2,0,2,0,2,2,1,1,2,1)$ | $(0,0,3,3,2,0,3,1,3,1,1,0,0,2)$
$C_{4,56,2}$ | $(2,2,2,0,1,0,1,0,1,2,3,0,3,1)$ | $(2,3,0,0,0,1,3,2,1,2,0,2,2,1)$
$C_{4,56,3}$ | $(1,2,2,0,3,2,2,3,1,1,1,3,1,0)$ | $(2,1,3,1,1,2,2,2,0,1,3,3,0,0)$

By an approach is similar to that used in the previous subsection, we tried to
construct a new extremal Type II $\mathbb{Z}_{4}$-code of length $56$ by
Theorem 3.4. However, our extensive search failed to construct such a code.

Acknowledgments. This work was supported by JSPS KAKENHI Grant Number
19H01802. The author would like to thank the anonymous reviewers for useful
comments.

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Table 7: Extremal Type II four-negacirculant $\mathbb{Z}_{2k}$-codes of length
$24$ $(k=5,12,\ldots,20)$
---
Codes | $r_{A}$ | $r_{B}$ | Codes | $r_{A}$ | $r_{B}$
$C_{10,24,1}$ | $(9,8,2,5,4,2)$ | $(3,8,6,0,6,0)$ | $C_{26,24,4}$ | $(14,13,16,2,8,20)$ | $(7,7,7,11,11,9)$
$C_{10,24,2}$ | $(1,7,3,1,9,3)$ | $(7,5,3,1,1,8)$ | $C_{28,24,1}$ | $(15,8,10,7,6,8)$ | $(22,16,9,16,22,14)$
$C_{10,24,3}$ | $(5,8,1,6,5,0)$ | $(5,7,2,5,2,1)$ | $C_{28,24,2}$ | $(2,23,20,27,24,1)$ | $(22,3,0,3,21,25)$
$C_{12,24,1}$ | $(6,4,8,8,5,6)$ | $(10,9,0,2,5,10)$ | $C_{28,24,3}$ | $(13,15,1,18,18,20)$ | $(3,7,5,22,9,22)$
$C_{12,24,2}$ | $(3,7,9,7,11,11)$ | $(9,9,2,7,11,1)$ | $C_{28,24,4}$ | $(17,5,3,13,25,14)$ | $(21,1,5,1,27,25)$
$C_{12,24,3}$ | $(6,9,8,7,4,3)$ | $(1,0,1,6,9,3)$ | $C_{30,24,1}$ | $(1,8,28,9,12,18)$ | $(6,8,4,2,1,20)$
$C_{14,24,1}$ | $(2,6,11,4,0,5)$ | $(6,4,6,0,8,11)$ | $C_{30,24,2}$ | $(29,28,23,13,21,14)$ | $(22,11,4,3,0,27)$
$C_{14,24,2}$ | $(11,9,13,7,9,5)$ | $(10,5,9,7,3,7)$ | $C_{30,24,3}$ | $(29,28,23,13,21,14)$ | $(22,11,7,21,18,0)$
$C_{14,24,3}$ | $(7,8,11,3,11,12)$ | $(13,10,1,6,5,0)$ | $C_{30,24,4}$ | $(18,19,29,5,23,11)$ | $(27,1,1,3,27,23)$
$C_{16,24,1}$ | $(13,4,2,11,4,4)$ | $(15,8,14,0,8,14)$ | $C_{32,24,1}$ | $(2,16,16,4,29,2)$ | $(18,6,1,2,0,7)$
$C_{16,24,2}$ | $(1,3,13,15,5,14)$ | $(5,9,3,7,9,5)$ | $C_{32,24,2}$ | $(16,5,16,21,22,5)$ | $(27,27,0,29,2,7)$
$C_{16,24,3}$ | $(8,9,8,3,15,1)$ | $(6,0,2,3,15,9)$ | $C_{32,24,3}$ | $(14,11,19,17,30,0)$ | $(29,13,2,27,6,1)$
$C_{18,24,1}$ | $(3,2,8,15,8,14)$ | $(16,5,2,14,0,0)$ | $C_{32,24,4}$ | $(15,3,31,23,29,27)$ | $(27,10,13,3,19,3)$
$C_{18,24,2}$ | $(15,3,8,17,13,5)$ | $(17,3,17,1,3,5)$ | $C_{34,24,1}$ | $(16,19,30,22,25,14)$ | $(0,3,4,32,2,0)$
$C_{18,24,3}$ | $(5,9,13,17,5,7)$ | $(4,10,2,8,2,1)$ | $C_{34,24,2}$ | $(33,13,25,6,33,8)$ | $(11,12,1,12,33,16)$
$C_{20,24,1}$ | $(17,10,16,5,10,4)$ | $(12,11,12,8,14,8)$ | $C_{34,24,3}$ | $(4,25,3,27,15,13)$ | $(3,29,9,5,31,27)$
$C_{20,24,2}$ | $(13,3,17,17,3,17)$ | $(5,2,15,5,15,1)$ | $C_{34,24,4}$ | $(18,7,9,17,10,18)$ | $(21,10,3,24,23,13)$
$C_{20,24,3}$ | $(1,9,8,17,2,13)$ | $(5,6,1,4,13,8)$ | $C_{36,24,1}$ | $(20,7,24,22,11,6)$ | $(2,8,0,7,4,24)$
$C_{22,24,1}$ | $(11,0,10,12,10,14)$ | $(17,2,0,19,6,10)$ | $C_{36,24,2}$ | $(2,11,34,35,6,1)$ | $(11,2,1,25,23,8)$
$C_{22,24,2}$ | $(7,21,11,5,2,3)$ | $(21,7,1,7,15,13)$ | $C_{36,24,3}$ | $(15,15,25,7,3,10)$ | $(35,5,1,33,13,19)$
$C_{22,24,3}$ | $(16,9,14,11,9,21)$ | $(16,17,4,5,18,5)$ | $C_{36,24,4}$ | $(21,34,13,17,17,25)$ | $(31,20,2,31,22,22)$
$C_{22,24,4}$ | $(1,10,7,19,5,19)$ | $(12,15,10,6,11,4)$ | $C_{38,24,1}$ | $(16,36,16,32,25,26)$ | $(16,29,0,22,25,14)$
$C_{24,24,1}$ | $(7,4,6,17,8,18)$ | $(9,8,4,10,4,12)$ | $C_{38,24,2}$ | $(32,33,18,1,20,1)$ | $(18,7,20,19,7,9)$
$C_{24,24,2}$ | $(0,3,16,13,22,13)$ | $(15,5,1,0,17,2)$ | $C_{38,24,3}$ | $(17,17,27,17,26,9)$ | $(2,2,3,34,22,13)$
$C_{24,24,3}$ | $(9,11,5,23,11,19)$ | $(5,19,2,21,3,15)$ | $C_{38,24,4}$ | $(29,3,4,15,17,31)$ | $(11,3,3,33,13,17)$
$C_{24,24,4}$ | $(7,3,16,12,14,3)$ | $(2,11,0,19,23,23)$ | $C_{40,24,1}$ | $(34,26,39,2,38,6)$ | $(26,10,1,0,4,23)$
$C_{26,24,1}$ | $(17,2,14,8,8,10)$ | $(10,15,0,0,1,10)$ | $C_{40,24,2}$ | $(25,38,31,19,37,8)$ | $(7,20,1,0,9,22)$
$C_{26,24,2}$ | $(15,20,5,15,17,8)$ | $(14,0,21,11,1,12)$ | $C_{40,24,3}$ | $(5,1,32,15,30,35)$ | $(24,25,3,3,6,28)$
$C_{26,24,3}$ | $(1,7,14,23,5,21)$ | $(3,25,1,11,11,13)$ | $C_{40,24,4}$ | $(37,13,15,19,27,15)$ | $(3,15,5,31,31,10)$
Table 8: Extremal Type II $\mathbb{Z}_{2k}$-codes of length $24$
$(k=5,6,\ldots,9)$
---
Codes | $A$ | $x$ | $y$
$C_{10,24,4}$ | $M_{10,24,1}$ | $(7,1,6,2,4,1,0,3,5,1,7,7)$ | $(0,0,0,0,1,5,9,2,4,0,3,2)$
$C_{10,24,5}$ | $M_{10,24,1}$ | $(6,5,6,0,3,6,3,5,4,0,8,2)$ | $(0,0,0,4,0,4,4,6,9,5,7,1)$
$C_{10,24,6}$ | $M_{10,24,1}$ | $(1,7,1,1,0,6,0,4,5,7,1,9)$ | $(0,0,0,0,3,0,8,1,5,8,4,9)$
$C_{10,24,7}$ | $M_{10,24,4}$ | $(1,9,5,9,8,5,5,3,5,2,2,4)$ | $(0,0,0,0,8,2,3,1,2,1,6,1)$
$C_{10,24,8}$ | $M_{10,24,6}$ | $(9,6,0,6,0,4,7,6,7,8,2,7)$ | $(0,0,0,5,4,6,7,2,1,3,4,8)$
$C_{10,24,9}$ | $M_{10,24,7}$ | $(5,4,3,8,8,6,7,0,0,4,0,9)$ | $(0,0,0,0,0,0,1,6,5,5,8,7)$
$C_{12,24,4}$ | $M_{12,24,1}$ | $(5,7,9,5,6,7,7,1,0,8,1,10)$ | $(0,0,0,0,1,5,1,8,10,2,6,9)$
$C_{12,24,5}$ | $M_{12,24,1}$ | $(11,4,10,8,7,8,2,3,0,0,1,10)$ | $(0,0,0,0,10,3,11,10,5,11,6,4)$
$C_{12,24,6}$ | $M_{12,24,2}$ | $(8,4,3,9,8,10,0,5,2,8,7,10)$ | $(0,0,0,0,9,2,0,0,7,3,8,3)$
$C_{12,24,7}$ | $M_{12,24,4}$ | $(7,2,11,8,2,0,2,10,6,1,5,0)$ | $(0,0,0,1,3,10,2,4,6,11,3,4)$
$C_{12,24,8}$ | $M_{12,24,5}$ | $(3,4,5,1,7,8,4,7,3,4,7,9)$ | $(0,0,0,3,6,7,7,6,1,8,10,8)$
$C_{12,24,9}$ | $M_{12,24,7}$ | $(5,6,6,10,1,2,5,9,1,5,5,11)$ | $(0,0,1,0,6,7,8,7,0,11,6,10)$
$C_{14,24,4}$ | $M_{14,24,1}$ | $(7,8,6,6,11,2,6,8,2,1,8,5)$ | $(0,0,0,0,9,10,13,0,6,7,11,12)$
$C_{14,24,5}$ | $M_{14,24,1}$ | $(0,9,6,7,12,8,5,7,4,12,0,8)$ | $(0,0,0,0,11,10,1,3,2,2,9,10)$
$C_{14,24,6}$ | $M_{14,24,3}$ | $(4,5,13,0,6,7,1,2,1,3,1,5)$ | $(0,0,0,0,7,8,4,13,10,7,7,6)$
$C_{14,24,7}$ | $M_{14,24,3}$ | $(4,11,12,13,8,13,10,2,1,4,8,0)$ | $(0,0,0,0,6,9,11,0,7,6,2,3)$
$C_{14,24,8}$ | $M_{14,24,6}$ | $(6,13,6,11,7,0,5,6,1,1,3,7)$ | $(0,0,0,0,12,5,4,11,11,4,13,2)$
$C_{14,24,9}$ | $M_{14,24,8}$ | $(6,6,10,1,2,6,8,3,4,5,13,6)$ | $(0,0,0,1,3,3,2,7,7,11,11,1)$
$C_{16,24,4}$ | $M_{16,24,1}$ | $(1,0,7,7,5,9,7,9,2,13,10,8)$ | $(0,0,0,0,7,12,6,3,15,10,2,13)$
$C_{16,24,5}$ | $M_{16,24,1}$ | $(11,14,13,10,5,11,12,5,1,8,9,13)$ | $(0,0,0,3,0,7,9,7,14,0,12,12)$
$C_{16,24,6}$ | $M_{16,24,1}$ | $(4,11,3,10,13,7,0,1,1,2,7,11)$ | $(0,0,0,0,10,14,9,2,0,5,5,7)$
$C_{16,24,7}$ | $M_{16,24,1}$ | $(14,14,15,10,2,7,15,11,2,0,0,8)$ | $(0,0,0,0,1,9,1,6,15,6,10,8)$
$C_{16,24,8}$ | $M_{16,24,3}$ | $(2,6,2,14,13,10,7,10,3,7,4,10)$ | $(0,0,0,0,15,5,14,11,8,1,6,2)$
$C_{16,24,9}$ | $M_{16,24,7}$ | $(10,14,9,14,3,14,7,1,5,13,9,7)$ | $(0,0,0,0,9,4,3,9,12,14,1,12)$
$C_{18,24,4}$ | $M_{18,24,2}$ | $(2,14,14,12,17,5,7,17,1,3,7,3)$ | $(0,0,0,0,6,0,3,8,13,3,1,0)$
$C_{18,24,5}$ | $M_{18,24,3}$ | $(8,4,13,3,13,4,2,10,0,5,2,0)$ | $(0,0,0,0,13,13,7,7,11,17,3,3)$
$C_{18,24,6}$ | $M_{18,24,3}$ | $(3,15,11,15,7,16,1,8,1,14,8,11)$ | $(0,0,0,0,6,13,12,9,12,5,11,0)$
$C_{18,24,7}$ | $M_{18,24,5}$ | $(11,17,9,14,12,3,11,1,0,17,6,3)$ | $(0,0,0,0,8,11,0,13,12,1,10,11)$
$C_{18,24,8}$ | $M_{18,24,5}$ | $(9,16,0,14,10,1,16,10,0,3,3,6)$ | $(0,0,0,0,13,5,10,14,5,5,6,6)$
$C_{18,24,9}$ | $M_{18,24,7}$ | $(4,3,0,9,0,6,0,3,4,2,3,12)$ | $(0,0,0,0,4,3,0,1,4,7,11,2)$
Table 9: Extremal Type II $\mathbb{Z}_{2k}$-codes of length $24$
$(k=10,12,\ldots,15)$
---
Codes | $A$ | $x$ | $y$
$C_{20,24,4}$ | $M_{20,24,1}$ | $(19,7,14,7,1,7,4,10,0,3,17,9)$ | $(0,0,0,0,14,1,10,11,15,0,14,19)$
$C_{20,24,5}$ | $M_{20,24,1}$ | $(16,10,16,5,7,10,15,2,0,0,17,16)$ | $(0,0,0,0,10,1,1,9,14,18,19,4)$
$C_{20,24,6}$ | $M_{20,24,1}$ | $(2,1,15,10,17,11,0,18,4,2,10,4)$ | $(0,0,0,0,5,15,9,14,2,17,8,14)$
$C_{20,24,7}$ | $M_{20,24,2}$ | $(11,0,17,19,0,1,0,4,0,4,0,14)$ | $(0,0,0,0,14,10,19,0,9,6,5,9)$
$C_{20,24,8}$ | $M_{20,24,3}$ | $(8,0,12,0,1,17,1,18,2,1,14,4)$ | $(0,0,0,0,0,6,10,5,5,10,5,7)$
$C_{20,24,9}$ | $M_{20,24,4}$ | $(13,6,19,6,7,3,6,4,0,2,0,2)$ | $(0,0,0,1,8,14,5,8,9,3,2,14)$
$C_{22,24,5}$ | $M_{22,24,1}$ | $(18,20,15,19,8,18,1,6,0,2,4,7)$ | $(8,12,15,21,3,20,9,17,1,14,9,15)$
$C_{22,24,6}$ | $M_{22,24,1}$ | $(20,10,4,2,16,3,0,19,2,12,11,9)$ | $(18,3,2,18,1,13,12,10,14,14,19,14)$
$C_{22,24,7}$ | $M_{22,24,3}$ | $(21,12,11,6,20,12,1,0,0,1,18,4)$ | $(13,11,12,6,0,10,7,9,6,4,8,8)$
$C_{22,24,8}$ | $M_{22,24,3}$ | $(19,14,3,4,15,15,7,6,0,9,17,19)$ | $(5,7,17,12,1,0,3,3,4,15,8,15)$
$C_{22,24,9}$ | $M_{22,24,4}$ | $(7,18,20,4,17,17,8,15,0,8,20,16)$ | $(2,6,20,0,1,18,4,11,3,1,0,10)$
$C_{24,24,5}$ | $M_{24,24,1}$ | $(18,21,21,18,23,9,0,0,0,0,10,8)$ | $(10,5,1,8,8,20,12,23,4,6,21,10)$
$C_{24,24,6}$ | $M_{24,24,1}$ | $(9,3,23,3,23,14,0,0,1,1,9,10)$ | $(0,18,21,4,2,20,14,4,17,7,5,16)$
$C_{24,24,7}$ | $M_{24,24,2}$ | $(13,21,17,12,7,11,0,0,0,5,9,19)$ | $(10,5,21,0,19,12,0,16,20,6,3,10)$
$C_{24,24,8}$ | $M_{24,24,2}$ | $(21,16,20,22,9,13,0,0,0,2,2,15)$ | $(5,13,8,15,5,14,3,8,7,1,9,10)$
$C_{24,24,9}$ | $M_{24,24,4}$ | $(4,10,6,3,22,3,0,0,0,0,21,21)$ | $(6,16,15,6,20,7,20,8,17,0,18,9)$
$C_{26,24,5}$ | $M_{26,24,1}$ | $(2,11,8,18,7,22,0,0,0,0,7,7)$ | $(13,24,16,18,2,17,22,12,16,6,2,5,3)$
$C_{26,24,6}$ | $M_{26,24,2}$ | $(9,18,16,17,23,6,0,0,0,0,19,16)$ | $(16,20,12,23,21,11,3,23,1,1,20,17)$
$C_{26,24,7}$ | $M_{26,24,3}$ | $(15,9,2,14,11,2,9,18,0,16,24,24)$ | $(5,9,9,21,23,13,5,13,9,13,7,25)$
$C_{26,24,8}$ | $M_{26,24,5}$ | $(20,18,16,17,4,3,0,0,0,5,7,20)$ | $(21,1,12,4,25,24,2,16,22,8,14,25)$
$C_{26,24,9}$ | $M_{26,24,6}$ | $(20,21,2,15,15,21,0,0,0,24,12,14)$ | $(8,7,17,12,13,4,15,14,1,7,15,9)$
$C_{28,24,5}$ | $M_{28,24,1}$ | $(21,21,4,10,2,8,0,0,0,1,21,2)$ | $(2,27,26,23,23,12,23,27,11,25,20,5)$
$C_{28,24,6}$ | $M_{28,24,1}$ | $(20,19,24,13,25,12,20,4,0,10,8,13)$ | $(13,17,10,15,8,22,20,18,11,18,24,2)$
$C_{28,24,7}$ | $M_{28,24,2}$ | $(16,26,10,6,20,27,0,0,0,3,23,17)$ | $(0,13,0,21,22,27,26,3,17,27,25,7)$
$C_{28,24,8}$ | $M_{28,24,3}$ | $(23,20,12,18,14,26,0,0,0,3,3,11)$ | $(4,2,27,18,20,22,18,25,14,3,17,4)$
$C_{28,24,9}$ | $M_{28,24,4}$ | $(3,16,16,15,22,7,0,0,0,0,8,13)$ | $(16,16,22,22,15,14,5,14,20,1,25,18)$
$C_{30,24,5}$ | $M_{30,24,1}$ | $(29,29,27,4,2,17,8,29,0,3,11,5)$ | $(2,12,9,13,26,21,2,7,13,29,19,11)$
$C_{30,24,6}$ | $M_{30,24,2}$ | $(16,18,23,3,2,13,0,0,0,0,5,8)$ | $(16,4,14,26,27,24,2,1,23,28,18,17)$
$C_{30,24,7}$ | $M_{30,24,2}$ | $(19,18,21,18,7,22,0,0,0,1,26,10)$ | $(14,15,28,28,26,6,9,16,7,4,4,29)$
$C_{30,24,8}$ | $M_{30,24,2}$ | $(29,9,9,4,27,2,3,25,0,11,22,3)$ | $(3,10,29,22,19,19,22,13,25,1,6,3)$
$C_{30,24,9}$ | $M_{30,24,6}$ | $(3,1,16,3,29,26,0,0,0,14,6,4)$ | $(6,15,25,9,16,24,1,23,21,17,15,26)$
Table 10: Extremal Type II $\mathbb{Z}_{2k}$-codes of length $24$
$(k=16,17,\ldots,20)$
---
Codes | $A$ | $x$ | $y$
$C_{32,24,5}$ | $M_{32,24,1}$ | $(18,10,25,10,3,22,23,6,0,0,28,23)$ | $(15,8,10,23,30,10,24,4,27,20,29,0)$
$C_{32,24,6}$ | $M_{32,24,1}$ | $(14,4,9,10,22,3,31,18,0,7,14,12)$ | $(30,4,29,22,3,31,16,16,31,12,8,10)$
$C_{32,24,7}$ | $M_{32,24,2}$ | $(21,23,19,23,3,1,23,15,0,8,8,0)$ | $(9,5,19,4,23,24,29,13,24,9,29,0)$
$C_{32,24,8}$ | $M_{32,24,2}$ | $(27,27,7,25,0,25,28,31,0,0,9,5)$ | $(2,13,27,27,14,16,30,6,18,4,12,9)$
$C_{32,24,9}$ | $M_{32,24,3}$ | $(22,24,21,19,31,31,0,4,0,8,22,18)$ | $(8,18,29,12,3,10,20,0,14,28,21,5)$
$C_{34,24,5}$ | $M_{34,24,1}$ | $(3,26,20,2,1,9,20,32,0,2,18,33)$ | $(14,31,26,5,0,32,21,25,24,10,28,10)$
$C_{34,24,6}$ | $M_{34,24,1}$ | $(3,2,32,30,30,12,5,22,0,11,20,1)$ | $(10,10,15,19,2,8,8,26,32,13,20,3)$
$C_{34,24,7}$ | $M_{34,24,2}$ | $(32,25,16,3,16,22,27,29,0,0,12,18)$ | $(31,16,17,3,25,17,29,8,30,9,32,13)$
$C_{34,24,8}$ | $M_{34,24,4}$ | $(15,20,3,23,6,23,6,6,0,12,20,28)$ | $(6,26,25,22,4,7,28,6,15,14,30,23)$
$C_{34,24,9}$ | $M_{34,24,4}$ | $(13,10,26,7,11,19,6,24,1,1,23,13)$ | $(13,13,7,12,31,26,7,31,7,22,13,8)$
$C_{36,24,5}$ | $M_{36,24,1}$ | $(35,16,10,11,0,33,20,23,0,2,2,4)$ | $(35,25,34,0,22,11,35,0,35,11,3,19)$
$C_{36,24,6}$ | $M_{36,24,1}$ | $(28,15,24,19,13,16,8,21,0,0,12,2)$ | $(3,9,5,33,7,24,23,3,2,21,22,32)$
$C_{36,24,7}$ | $M_{36,24,1}$ | $(34,25,25,32,20,4,27,20,0,0,31,20)$ | $(8,8,0,26,31,24,33,4,34,35,11,26)$
$C_{36,24,8}$ | $M_{36,24,3}$ | $(19,29,26,9,30,6,30,18,0,1,2,22)$ | $(3,29,31,29,20,3,13,10,28,33,10,31)$
$C_{36,24,9}$ | $M_{36,24,4}$ | $(4,28,15,7,33,26,13,19,1,1,2,21)$ | $(33,9,20,4,6,32,17,24,15,12,22,10)$
$C_{38,24,5}$ | $M_{38,24,1}$ | $(18,5,17,23,6,5,29,1,0,0,23,21)$ | $(3,33,36,1,15,11,8,12,19,17,10,33)$
$C_{38,24,6}$ | $M_{38,24,1}$ | $(1,22,16,12,19,14,31,19,0,0,36,24)$ | $(31,5,17,28,10,21,18,11,35,25,5,20)$
$C_{38,24,7}$ | $M_{38,24,1}$ | $(34,16,34,7,37,2,17,19,0,0,6,32)$ | $(21,4,28,35,26,34,9,30,36,11,0,26)$
$C_{38,24,8}$ | $M_{38,24,3}$ | $(2,22,1,1,37,5,18,9,0,1,19,35)$ | $(30,11,12,23,11,32,24,17,29,35,19,33)$
$C_{38,24,9}$ | $M_{38,24,3}$ | $(4,28,24,35,37,13,3,34,0,0,28,12)$ | $(30,18,10,10,13,32,15,36,7,6,23,32)$
$C_{40,24,5}$ | $M_{40,24,1}$ | $(11,26,7,16,35,33,17,9,0,1,7,2)$ | $(24,36,27,6,1,1,34,18,10,7,16,34)$
$C_{40,24,6}$ | $M_{40,24,1}$ | $(12,2,17,30,8,28,22,39,0,0,9,17)$ | $(32,38,35,4,28,9,30,39,19,30,12,20)$
$C_{40,24,7}$ | $M_{40,24,2}$ | $(19,20,22,15,5,23,15,9,0,1,10,33)$ | $(3,1,29,34,19,21,1,1,28,15,28,6)$
$C_{40,24,8}$ | $M_{40,24,3}$ | $(6,39,12,39,21,27,1,32,1,1,15,34)$ | $(35,29,21,13,5,37,28,14,23,1,28,34)$
$C_{40,24,9}$ | $M_{40,24,3}$ | $(23,16,0,28,5,9,22,22,0,2,32,27)$ | $(7,29,11,38,30,21,8,12,26,10,22,24)$
Table 11: Extremal Type II $\mathbb{Z}_{2k}$-codes of length $32$
$(k=4,5,\ldots,10)$
---
Codes | $A$ | $x$ | $y$
$C_{8,32,3}$ | $M_{8,32,1}$ | $(0,0,7,6,4,1,1,3,5,0,3,0,0,1,3,2)$ | $(0,0,0,0,3,7,2,2,7,1,0,4,2,4,6,2)$
$C_{8,32,4}$ | $M_{8,32,1}$ | $(5,5,0,5,0,1,6,3,1,3,3,0,1,1,1,1)$ | $(0,0,6,3,4,7,1,6,4,3,4,5,5,7,1,4)$
$C_{8,32,5}$ | $M_{8,32,2}$ | $(6,0,2,4,4,1,3,6,4,5,1,0,0,0,0,4)$ | $(0,0,0,0,0,1,3,7,5,1,7,4,0,1,4,5)$
$C_{10,32,3}$ | $M_{10,32,1}$ | $(4,1,3,6,5,4,2,6,6,3,0,6,1,1,3,5)$ | $(0,0,0,0,2,7,8,7,4,1,0,8,0,6,6,9)$
$C_{10,32,4}$ | $M_{10,32,1}$ | $(3,3,8,8,2,9,0,4,7,3,1,4,0,1,1,4)$ | $(5,2,3,6,1,1,2,9,4,1,7,5,2,2,0,0)$
$C_{10,32,5}$ | $M_{10,32,2}$ | $(3,9,9,8,2,6,4,1,5,8,6,3,0,0,3,5)$ | $(0,0,0,1,1,1,8,8,2,5,4,1,3,3,9,2)$
$C_{12,32,3}$ | $M_{12,32,1}$ | $(6,7,1,0,8,7,4,3,2,8,0,11,0,1,3,3)$ | $(0,0,0,0,8,4,1,8,2,8,4,7,1,2,10,5)$
$C_{12,32,4}$ | $M_{12,32,1}$ | $(10,3,9,4,3,11,6,11,11,2,0,0,0,1,1,2)$ | $(2,10,11,10,1,0,0,7,5,7,0,1,0,7,7,10)$
$C_{12,32,5}$ | $M_{12,32,1}$ | $(1,8,4,3,7,1,2,11,6,0,1,7,0,0,3,0)$ | $(0,11,3,5,7,9,6,9,11,3,11,5,0,1,4,5)$
$C_{14,32,3}$ | $M_{14,32,1}$ | $(0,2,13,11,1,7,4,3,7,3,12,6,0,1,0,6)$ | $(0,11,11,11,11,8,11,13,8,11,9,7,0,11,7,7)$
$C_{14,32,4}$ | $M_{14,32,1}$ | $(11,12,1,1,3,9,11,7,13,5,12,13,0,5,8,5)$ | $(8,3,13,13,9,13,4,1,5,0,13,3,3,3,7,2)$
$C_{14,32,5}$ | $M_{14,32,2}$ | $(10,6,7,5,10,2,5,0,5,4,13,3,0,0,1,1)$ | $(1,4,7,7,10,11,5,11,4,0,11,12,0,1,2,10)$
$C_{16,32,3}$ | $M_{16,32,1}$ | $(15,0,3,3,12,5,11,0,15,5,14,6,0,0,0,3)$ | $(1,2,9,9,13,15,5,12,11,7,3,14,3,2,11,13)$
$C_{16,32,4}$ | $M_{16,32,1}$ | $(3,11,1,0,12,3,10,10,10,11,6,12,1,1,3,0)$ | $(15,8,14,1,1,13,2,2,2,6,5,11,0,1,3,6)$
$C_{16,32,5}$ | $M_{16,32,2}$ | $(1,11,15,3,11,15,8,5,10,8,12,0,0,0,2,7)$ | $(2,14,3,10,2,11,11,0,11,13,1,1,0,1,0,12)$
$C_{18,32,3}$ | $M_{18,32,1}$ | $(9,4,16,10,2,7,4,7,2,3,2,5,1,0,3,5)$ | $(13,0,9,4,17,16,2,0,0,12,8,2,0,14,3,10)$
$C_{18,32,4}$ | $M_{18,32,1}$ | $(6,12,5,7,5,6,6,7,16,15,5,9,1,0,2,4)$ | $(3,6,6,4,13,12,2,13,7,12,4,1,3,1,4,9)$
$C_{18,32,5}$ | $M_{18,32,2}$ | $(15,17,15,12,10,2,5,7,6,6,12,7,0,1,1,2)$ | $(12,16,1,17,13,16,10,14,7,9,4,9,1,10,0,13)$
$C_{20,32,3}$ | $M_{20,32,1}$ | $(1,5,8,10,14,18,4,18,17,11,18,16,0,0,2,6)$ | $(6,12,15,14,6,2,5,16,6,18,6,10,3,8,5,18)$
$C_{20,32,4}$ | $M_{20,32,1}$ | $(6,6,15,6,6,3,1,18,1,19,11,10,0,0,3,5)$ | $(8,5,5,9,4,18,9,4,0,5,7,15,2,15,14,18)$
$C_{20,32,5}$ | $M_{20,32,1}$ | $(5,13,11,19,5,2,18,8,6,9,19,12,0,0,6,3)$ | $(19,14,13,18,2,14,3,2,10,12,12,8,19,16,12,2)$