Extremal Type II $\mathbb{Z}_4$-codes constructed from binary doubly even self-dual codes of length $40$

TL;DR

This paper shows that all binary doubly even self-dual codes of length 40 can be derived from extremal Type II  -codes over , revealing a large number of such codes and their construction.

Contribution

It establishes a construction method linking binary self-dual codes to extremal Type II  -codes, and quantifies the number of inequivalent codes.

Findings

All binary doubly even self-dual codes of length 40 are residue codes of extremal Type II  -codes.

At least 94,356 inequivalent extremal Type II  -codes of length 40 exist.

The construction provides a comprehensive classification of these codes.

Abstract

In this note, we demonstrate that every binary doubly even self-dual code of length $40$ can be realized as the residue code of some extremal Type II $\mathbb{Z}_4$-code. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II $\mathbb{Z}_4$-codes of length $40$.

Equations

$$\tilde{I_{n}}=\left(\begin{array}[]{cccccc}1&\cdots&1\\ 0&&\\ \vdots&I_{n-1}&\\ 0&&\\ \end{array}\right).$$

$$G_{1}=\left(\begin{array}[]{cc}A_{1}&\tilde{I_{n}}\\ \end{array}\right),$$

$$\left(\begin{array}[]{cccccc}&A_{1}&&\tilde{I_{n}}+2A_{2}\\ \end{array}\right),$$

$$A_4(\mathcal{C})=\frac{1}{2}\{(x_1,\dots ,x_n)\in {\mathbb{Z}}^n\mid (x_1\mathrm{mod}4,\dots ,x_n\mathrm{mod}4)\in \mathcal{C}\}$$

Full text

1 Introduction

Let $\mathbb{Z}_{4}\ (=\{0,1,2,3\})$ denote the ring of integers modulo $4$. A $\mathbb{Z}_{4}$-code $\mathcal{C}$ of length $n$ is a $\mathbb{Z}_{4}$-submodule of $\mathbb{Z}_{4}^{n}$. Two $\mathbb{Z}_{4}$-codes are equivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. A code $\mathcal{C}$ is self-dual if $\mathcal{C}=\mathcal{C}^{\perp}$, where the dual code $\mathcal{C}^{\perp}$ is defined as $\{x\in\mathbb{Z}_{4}^{n}\mid x\cdot y=0$ for all $y\in\mathcal{C}\}$ under the standard inner product $x\cdot y$. The Euclidean weight of a codeword $x=(x_{1},\ldots,x_{n})$ of $\mathcal{C}$ is $n_{1}(x)+4n_{2}(x)+n_{3}(x)$, where $n_{\alpha}(x)$ denotes the number of components $i$ with $x_{i}=\alpha$ $(\alpha=1,2,3)$. A $\mathbb{Z}_{4}$-code $\mathcal{C}$ is Type II if $\mathcal{C}$ is self-dual and the Euclidean weights of all codewords of $\mathcal{C}$ are divisible by 8 [2] and [8]. This is a remarkable class of self-dual $\mathbb{Z}_{4}$-codes related to even unimodular lattices. A Type II $\mathbb{Z}_{4}$-code of length $n$ exists if and only if $n\equiv 0\pmod{8}$. The minimum Euclidean weight $d_{E}$ of $\mathcal{C}$ is the smallest Euclidean weight among all nonzero codewords of $\mathcal{C}$. The minimum Euclidean weight $d_{E}$ of a Type II $\mathbb{Z}_{4}$-code of length $n$ is bounded by $d_{E}\leq 8\lfloor n/24\rfloor+8$ [2]. A Type II $\mathbb{Z}_{4}$-code meeting this bound with equality is called extremal.

The residue code $\mathcal{C}^{(1)}$ of a $\mathbb{Z}_{4}$-code $\mathcal{C}$ is the binary code $\mathcal{C}^{(1)}=\{c\bmod 2\mid c\in\mathcal{C}\}$. If $\mathcal{C}$ is self-dual, then $\mathcal{C}^{(1)}$ is a binary doubly even code [5]. If $\mathcal{C}$ is Type II, then $\mathcal{C}^{(1)}$ contains the all-ones vector $\mathbf{1}$ [8]. It follows that there is a Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ for a given binary doubly even code $B$ containing $\mathbf{1}$ (see [9]). However, it is not known in general whether there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ or not.

A binary doubly even self-dual code of length $n$ exists if and only if $n\equiv 0\pmod{8}$, and the minimum weight $d$ of a binary doubly even self-dual code of length $n$ is bounded by $d\leq 4\lfloor n/24\rfloor+4$. A binary doubly even self-dual code meeting this bound with equality is called extremal. Two binary codes $B$ and $B^{\prime}$ are equivalent if $B$ can be obtained from $B^{\prime}$ by permuting the coordinates. The classification of binary doubly even self-dual codes has been done for lengths up to $40$ (see [1]). For every binary doubly even self-dual code $B$ of length $24$, there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ [4, Postscript] (see also [7]). In addition, for every binary doubly even self-dual code $B$ of length $32$, there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ [6].

In this note, this work is extended to length $40$. We demonstrate that there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ for every binary doubly even self-dual code $B$ of length $40$. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II $\mathbb{Z}_{4}$-codes of length $40$. In addition, our result implies that there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ for every binary doubly even self-dual code $B$ of length $n\in\{8,16,24,32,40\}$. Also, there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ for every binary extremal doubly even self-dual code $B$ of length $n\in\{8,16,24,32,40,48\}$.

All computer calculations in this note were done by Magma [3].

2 Extremal Type II $\mathbb{Z}_{4}$-codes of length 40

2.1 Construction method

We review the method for constructing Type II $\mathbb{Z}_{4}$-codes, which was given in [9]. Let $B$ be a binary doubly even self-dual code of length $n$. Let $I_{n}$ denote the identity matrix of order $n$ and let

[TABLE]

Without loss of generality, we may assume that $B$ has generator matrix of the following form:

[TABLE]

where $A_{1}$ is an $n/2\times n/2$ matrix which has the property that the first row is $\mathbf{1}$. Then we have a generator matrix of a Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ as follows:

[TABLE]

where $A_{2}$ is an $n/2\times n/2$ $(1,0)$-matrix and we regard the matrices as matrices over $\mathbb{Z}_{4}$. Here, we can choose freely the entries above the diagonal elements and the $(1,1)$-entry of $A_{2}$, and the rest is completely determined from the property that $\mathcal{C}$ is Type II. Since any Type II $\mathbb{Z}_{4}$-code is equivalent to some Type II $\mathbb{Z}_{4}$-code containing $\mathbf{1}$ [8], without loss of generality, we may assume that the first row of $A_{2}$ is the zero vector. This reduces our search space for finding extremal Type II $\mathbb{Z}_{4}$-codes. It is the aim of this work to find a $20\times 20$ $(1,0)$-matrix $A_{2}$ such that the matrix of form (2) generates an extremal Type II $\mathbb{Z}_{4}$-code from a generator matrix of form (1) for a given binary doubly even self-dual code of length $40$.

2.2 Extremal Type II $\mathbb{Z}_{4}$-codes of length 40

There are $94343$ inequivalent binary doubly even self-dual codes of length $40$ [1]. Let $B$ be one of the $94343$ binary codes. Without loss of generality, we may assume that $B$ has generator matrix of form (1). In the above method, we explicitly found a $20\times 20$ $(1,0)$-matrix $A_{2}$ such that the matrix of form (2) generates an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$. Note that $\mathcal{C}^{(1)}=B$. This was done for all the $94343$ binary doubly even self-dual codes. Hence, we have the following:

Proposition 1**.**

Let $B$ be a binary doubly even self-dual code of length $40$. Then there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$.

Remark 2*.*

The extremality of the code was verified as follows. Let $\mathcal{C}$ be a Type II $\mathbb{Z}_{4}$-code of length $40$. The following lattice

[TABLE]

has minimum norm $4$ if and only if $\mathcal{C}$ is extremal [2]. Instead of calculating the minimum Euclidean weight of $\mathcal{C}$, we calculated the minimum norm of $A_{4}(\mathcal{C})$. This speeded up the calculations by Magma [3] considerably. As an example, for some five extremal Type II $\mathbb{Z}_{4}$-codes, the calculations for the minimum Euclidean weights took about $1223$ minutes, but the calculations for the minimum norms took about $3$ seconds only, using a single core of a PC Intel i7 4 core processor.

By the above proposition, $94343$ extremal Type II $\mathbb{Z}_{4}$-codes are constructed from the $94343$ inequivalent binary doubly even self-dual codes of length $40$. The $94343$ extremal Type II $\mathbb{Z}_{4}$-codes are inequivalent, since their residue codes are inequivalent. Generator matrices for the $94343$ codes can be written in the form \left(\begin{array}[]{cc}I_{20}&M\\ \end{array}\right), where $M$ can be obtained electronically from http://www.math.is.tohoku.ac.jp/~mharada/Paper/Z4-40-II.txt.

For $m=7,8,\ldots,19$, an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ of length $40$ such that the residue code $\mathcal{C}^{(1)}$ has dimension $m$ is known [6]. Hence, we have the following:

Corollary 3**.**

There are at least $94356$ inequivalent extremal Type II $\mathbb{Z}_{4}$-codes of length $40$.

As described above, for every binary doubly even self-dual code $B$ of length $24$ (resp. $32$), there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$ [4, Postscript] (resp. [6]). Hence, we have the following:

Corollary 4**.**

Suppose that $n\in\{8,16,24,32,40\}$. Let $B$ be a binary doubly even self-dual code of length $n$. Then there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$.

It is known that the binary extended quadratic residue code $QR_{48}$ of length $48$ is the unique binary extremal doubly even self-dual code of that length. The binary code $QR_{48}$ is the residue code of the extended lifted quadratic residue $\mathbb{Z}_{4}$-code of length $48$, which is an extremal Type II $\mathbb{Z}_{4}$-code [2]. Hence, we have the following:

Corollary 5**.**

Suppose that $n\in\{8,16,24,32,40,48\}$. Let $B$ be a binary extremal doubly even self-dual code of length $n$. Then there is an extremal Type II $\mathbb{Z}_{4}$-code $\mathcal{C}$ with $\mathcal{C}^{(1)}=B$.

In this note, $94343$ inequivalent extremal Type II $\mathbb{Z}_{4}$-codes $\mathcal{C}_{i}$ of length $40$ were constructed explicitly ($i=1,2,\ldots,94343$). It is a worthwhile problem to determine whether extremal even unimodular lattices $A_{4}(\mathcal{C}_{i})$ ($i=1,2,\ldots,94343$) are isomorphic or not.

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