Optimal binary linear codes from posets of the disjoint union of two chains

Yansheng Wu; Jong Yoon Hyun; Qin Yue

doi:10.3934/amc.2022042

Article Contents

2022, Volume 16, Issue 4: 1051-1058. Doi: 10.3934/amc.2022042

This issue Previous Article On construction of lightweight MDS matrices Next Article Some shortened codes from linear codes constructed by defining sets

Optimal binary linear codes from posets of the disjoint union of two chains

1.
School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
2.
State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an 710071, China
3.
Konkuk University, Glocal Campus, 268 Chungwon-daero Chungju-si Chungcheongbuk-do 27478, Republic of Korea
4.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

^*Corresponding author: Yansheng Wu
^*Corresponding author: Yansheng Wu

Received: February 2022

Revised: April 2022

Early access: June 2022

Published: November 2022

This work was supported by the National Natural Science Foundation of China (Nos. 12101326, 62172219), the Natural Science Foundation of Jiangsu Province (No. BK20210575), the Natural Science Research Project of Colleges and Universities in Jiangsu Province (No. 21KJB110005), the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)(NRF-2017R1D1A1B05030707), and the Foundation of State Key Laboratory of Integrated Services Networks under Grant ISN23-22

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Abstract

Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.

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Mathematics Subject Classification: Primary: 94B05.

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Figure 1. $\mathbb{P} = ([m,n], \leq)$

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Table 1. Theorem 3.1 (Ⅰ)

Weight	Frequency
$ 2^{n-1}-i+s(0\le s< i) $	$ 2^{n-i}{i\choose s} $
$ 2^{n-1} $	$ 2^{n-i}-1 $

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Table 2. Theorem 3.1 (Ⅱ)

Weight	Frequency
$ 2^{n-1}-(j-m)+t(0\le t< j-m) $	$ 2^{n-(j-m)}{j-m\choose t} $
$ 2^{n-1} $	$ 2^{n-(j-m)}-1 $

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Table 3. Theorem 3.1 (Ⅲ)

Weight	Frequency
$\begin{array}{*{20}{c}} {{2^{n - 1}} + s + t + 2st - (s + 1)(j - m) - (t + 1)i}\\ {0 \le s \le i,0 \le t \le j - m,}\\ {(s,t) \ne (i,j - m)} \end{array}$	$ 2^{n-i-(j-m)}{i\choose s}{j-m\choose t} $
$ 2^{n-1} $	$ 2^{n-i-(j-m)}-1 $

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