Self-orthogonal codes from orbit matrices of 2-designs

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

“…We also examine the ternary hulls of the residual designs. Moreover, we study ternary codes spanned by adjacency matrices of strongly regular graphs with parameters (36, 15,6,6), (36,14,4,6) and (35,18,9,9). Together with the results of W. H. Haemers, R. Peeters and J. M. van Rijckevorsel [13], that completes the classification of self-orthogonal codes spanned by the adjacency matrices and orbit matrices of the strongly regular graphs on up to 40 vertices.…”

“…6. The ternary codes of the residual 2- (27,9,4) designs According to [20, Table II.1, p. 38] there are at least 2.45×10 8 2-(27, 9, 4) designs. This number is not feasible for any classification purposes.…”

“…The use of orbit matrices of block designs for constructing self-orthogonal codes was initiated by Harada and Tonchev in [15]. Recently, in [9] studies were carried to further examine an interplay between codes and orbit matrices of block designs. It was established in particular that under certain conditions the codes spanned by the non-fixed parts of the orbits matrices of symmetric designs are self-orthogonal.…”

“…The paper is organized as follows: after a brief description of our terminology and some background results, in Sections 3 and 4 we present a method of constructing self-orthogonal codes from orbit matrices of strongly regular graphs, and apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular (40, 12,2,4) and (40,27,18,18) graphs. Sections 5 and 6 deal with the ternary codes of the (40, 27,18) designs and those of 2- (27,9,4) designs obtained as residual designs of the (40, 13,4) designs (complementary designs of the symmetric (40, 27, 18) designs) and Section 7 examines the ternary hulls of the 2- (27,9,4) designs. Finally, in Sections 8, 9 and 10 we study self-orthogonal codes related to strongly regular graphs with parameters (36, 15,6,6), (36,14,4,6) and (35,18,9,9).…”

“…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.2010 Mathematics Subject Classification: Primary: 05E30, 94B05; Secondary: 05B05, 20D45.…”