Self-orthogonal codes from the strongly regular graphs on up to 40 vertices

Dean Crnković; Marija  Maksimović; Bernardo Gabriel  Rodrigues; Sanja Rukavina

doi:10.3934/amc.2016026

Article Contents

2016, Volume 10, Issue 3: 555-582. Doi: 10.3934/amc.2016026

This issue Previous Article On the existence of Hadamard difference sets in groups of order 400 Next Article There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$

Self-orthogonal codes from the strongly regular graphs on up to 40 vertices

1.
Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka
2.
School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Durban 4000

Received: May 2015

Revised: June 2016

Published: August 2016

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Abstract

This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$. We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$, $(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements. That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.

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Mathematics Subject Classification: Primary: 05E30, 94B05; Secondary: 05B05, 20D45.

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