Some subfield codes from MDS codes

Can Xiang; Jinquan Luo

doi:10.3934/amc.2021023

Article Contents

2023, Volume 17, Issue 4: 815-827. Doi: 10.3934/amc.2021023

This issue Previous Article Five-weight codes from three-valued correlation of M-sequences Next Article On additive MDS codes over small fields

Some subfield codes from MDS codes

Can Xiang^1, and
Jinquan Luo^2, ,

1.
College of Mathematics and Informatics, South China Agricultural University, Guangzhou, Guangdong 510642, China
2.
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, Hubei 430079, China

^* Corresponding author: Jinquan Luo
^* Corresponding author: Jinquan Luo

Received: November 2020

Revised: March 2021

Early access: July 5, 2021

Published online: July 5, 2021

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Abstract

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal.

Keywords:

Linear code,
subfield code,
optimal.

Mathematics Subject Classification: Primary: 94A24; Secondary: 94B15.

Citation:

Full Text(HTML)

Table 1. The weight distribution of $ {\mathcal{C}}_{ {\mathcal{M}}}^{(2)} $ for $ m\geq 5 $ odd

Weight	Multiplicity
$ 0 $	$ 1 $
$ 2^{m} $	$ 1 $
$ 2^{m-1} $	$ 9\cdot 2^{3 m-4} + 5 \cdot 2^{m-2} - 2^{2 m-2}-2 $
$ 2^{m-1}+1 $	$ 4^{m-2} (-2 + 9\cdot 2^m) $
$ (2^m \pm 2^{\frac{m+1}{2}})/2 $	$ 2^{m-3} (-16 + 5\cdot 2^{2 m} + 2^{2 + m})/3 $
$ (2^m \pm 2^{\frac{m+1}{2}})/2+1 $	$ 2^{2 m-3} (2 + 5\cdot 2^m)/3 $
$ (2^m \pm 2^{\frac{m+3}{2}})/2 $	$ 2^{ m-5} (-2 + 2^m)^2 /3 $
$ (2^m \pm 2^{\frac{m+3}{2}})/2+1 $	$ 2^{2 m-5} (-2 + 2^m)/3 $

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Table 2. The weight distribution of $ {\mathcal{C}}_{ {\mathcal{M}}}^{(2)} $ for $ m\geq 6 $ and $ m\equiv 2 \pmod 4 $

Weight	Multiplicity
$ 0 $	$ 1 $
$ 2^{m-1} $	$ -2 + 17 \cdot 2^{m-3} + 29\cdot 2^{ 3 m-6} - 2^{2m-5} \cdot 23 $
$ 2^{m - 1} + 1 $	$ 4^{m-3} (29\cdot 2^m -20) $
$ (2^m \pm 2^{\frac{m}{2}})/2 $	$ 2^m (-16 + 3 \cdot 4^m + 2^m \cdot 7) /15 $
$ (2^m \pm 2^{\frac{m}{2}})/2+1 $	$ 4^m (1 + 2^m)/5 $
$ (2^m \pm 2^{\frac{m+2}{2}})/2 $	$ 2^{2 m-5} (-10 + 7 \cdot 2^m) /3 $
$ (2^m\pm 2^{\frac{m+2}{2}})/2+1 $	$ 2^{2 m-5} (-4 + 7 \cdot 2^m) /3 $
$ (2^m \pm 2^{\frac{m+4}{2}})/2 $	$ 2^{m-7} (8 + 4^m + 2^{1 + m} (-3)) /15 $
$ (2^m \pm 2^{\frac{m+4}{2}})/2+1 $	$ 2^{2 m-7} (-4 + 2^m) /15 $
$ 2^m $	$ 1 $

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