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Self-orthogonal codes over a non-unital ring from two class association schemes

Research Group of Algebraic Structures and Applications, Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

E-mail Address: analahmadi@kau.edu.sa

Corresponding author.

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Asmaa Melaibari

https://orcid.org/0000-0002-1485-2584

Research Group of Algebraic Structures and Applications, Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Department of Mathematics, College of Science, University of Jeddah, Jeddah 21493, Saudi Arabia

E-mail Address: amelaibari@uj.edu.sa

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, and

Patrick Solé

https://orcid.org/0000-0002-4078-8301

I2M, Aix Marseille University, CNRS, Centrale Marseille, Marseilles, France

E-mail Address: sole@enst.fr

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https://doi.org/10.1142/S0219498825501269Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue on Recent Advances in Coding Theory and Cryptography In honor of Joachim Rosenthal
Guest Editors: Elisa Gorla, Anna-Lena Horlemann, Roxana Smarandache and Jens Zumbrägel

Abstract

There is a non-unital ring $I$ $I$ of order 4 defined by generators and relations as $I = 〈 a, b | 2 a = 2 b = 0, a^{2} = b, a b = 0 〉 .$ $I = 〈 a, b | 2 a = 2 b = 0, a^{2} = b, a b = 0 〉 .$ In this paper, we present special constructions of linear codes over $I$ $I$ from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in $I$ $I$ are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given.

Communicated by Jens Zumbraegel

Keywords:

AMSC: 94B05, 16D10, 05E30

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