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Some two-weight codes invariant under the $3$ -fold covers of the Mathieu groups $M_{22}$ and $Aut (M_{22})$

Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa

E-mail Address: bernardo.rodrigues@up.ac.za

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https://doi.org/10.1142/S0219498825500793Cited 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

Using an approach from finite group representation theory we construct quaternary non-projective codes with parameters ${[693, 6, 480]}_{4}, {[1386, 6, 1008]}_{4}, {[2016, 6, 1488]}_{4}$ , quaternary projective codes with parameters ${[231, 6, 160]}_{4}, {[462, 6, 336]}_{4}$ and ${[672, 6, 496]}_{4}$ and binary projective codes with parameters ${[693, 12, 320]}_{2}, {[1386, 12, 672]}_{2}, {[2016, 12, 992]}_{2}$ as examples of two-weight codes on which a finite almost quasisimple group of sporadic type acts transitively as permutation groups of automorphisms. In particular, we show that these codes are invariant under the $3$ -fold covers $\hat{3} M_{22}$ and $\hat{3} M_{22} : 2$ , respectively, of the Mathieu groups $M_{22}$ and $M_{22} : 2$ . Employing a known construction of strongly regular graphs from projective two-weight codes we obtain from the binary projective (respectively, quaternary projective) two-weight codes with parameters those given above, the strongly regular graphs with parameters $(4096, 693, 152, 110), (4096, 1386, 482, 462),$ and $(4096, 2016, 992, 992),$ respectively. The latter graph can be viewed as a $2$ - $(4096, 2016, 992)$ -symmetric design with the symmetric difference property whose residual and derived designs with respect to a block give rise to binary self-complementary codes meeting the Grey–Rankin bound with equality.

Communicated by A.-L. Horlemann

Dedicated to Joachim Rosenthal on the occasion of his 60th birthday

Keywords:

AMSC: 0B05, 20D08, 94B05

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