Abstract
There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose codewords all have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \(\mathbb {F}_4.\) The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.
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17 January 2023
A Correction to this paper has been published: https://doi.org/10.1007/s10623-022-01170-9
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The original online version of this article was revised: the typos and the incorrect results has been corrected
This research is supported by National Natural Science Foundation of China (12071001), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).
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Shi, M., Wang, S., Kim, JL. et al. Self-orthogonal codes over a non-unital ring and combinatorial matrices. Des. Codes Cryptogr. 91, 677–689 (2023). https://doi.org/10.1007/s10623-021-00948-7
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DOI: https://doi.org/10.1007/s10623-021-00948-7