The $ b $-weight distribution for MDS codes

Canze Zhu; Qunying Liao

doi:10.3934/amc.2023028

Article Contents

2024, Volume 18, Issue 6: 1893-1910. Doi: 10.3934/amc.2023028

This issue Previous Article Constacyclic and quasi-twisted codes over $ \mathbb{Z}_{q}[u]/\langle u^{2}-1\rangle $ and new $ \mathbb{Z}_4 $-linear codes Next Article Correction of the article "On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property"

The $ b $-weight distribution for MDS codes

Canze Zhu^1, , and
Qunying Liao^2,

1.
School of Computer Science and Technology, Dongguan University of Technology, Dongguan, China, 523000
2.
College of Mathematical Science, Sichuan Normal University, Chengdu, China, 610066

^*Corresponding author: Canze Zhu
^*Corresponding author: Canze Zhu

Received: September 2022

Revised: July 2023

Early access: August 18, 2023

Published online: August 18, 2023

The second author is supported by [National Natural Science Foundation of China (12071321).].

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Abstract

For a positive integer $ b\ge2 $, the $ b $-symbol code is a new coding framework proposed to combat $ b $-errors in $ b $-symbol read channels. Especially, the $ 2 $-symbol code is called the symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $ b $-symbol code. Recently, for any MDS code $ \mathcal{C} $, Ma and Luo determined the symbol-pair weight distribution of $ \mathcal{C} $. In this paper, by calculating the number of solutions for some equations and utilizing some shortened codes of $ \mathcal{C} $, we give the connection between the $ b $-weight distribution and the number of codewords in shortened codes of $ \mathcal{C} $ with special shape. Note that shortened codes of $ \mathcal{C} $ are also MDS codes, the number of these codewords with special shape are also determined by the shorten method and the number of $ r $-combinations of the multiset problem. From the above calculation, the $ b $-weight distribution of $ \mathcal{C} $ is determined. Our result generalizes the corresponding result of Ma and Luo [15].

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Mathematics Subject Classification: Primary: 94B05; Secondary: 94A24.

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