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

Abstract:One of the main problems in the theory of strongly regular graphs (SRGs) is constructing and classifying SRGs with given parameters. Strongly regular graphs with parameters (37,18,8,9), (41,20,9,10), (45,22,10,11), (49, 24,11,12), (49,18,7,6) and (50,21,8,9) are the only strongly regular graphs on up to 50 vertices that still have to be classified. In this paper, we give the enumeration of SRGs with these parameters having S 3 as an automorphism group. The construction of SRGs in this paper is a step in the classification of SRGs on up to 50 vertices.

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All strongly regular graphs having at most 36 vertices have been enumerated. Hence, the first open case is enumeration of the SRGs with parameters (37, 18, 8, 9). In this paper we show that there are exactly forty SRGs with parameters (37, 18, 8, 9) having nontrivial automorphisms. Comparing the constructed graphs with previously known SRGs with these parameters we conclude that six of the SRGs with parameters (37, 18, 8, 9) constructed in this paper are new, and that up to isomorphism there are at least 6766 strongly regular graphs with parameters (37, 18, 8, 9).

Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters (45,22,10,11), (49,24,11,12), and (50,21,8,9) that have Z6 as the automorphism group and constructed regular two-graphs from SRGs (45,22,10,11), SRGs (49,24,11,12), and SRGs (50,21,8,9) that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG (45,22,10,11) and 51 new regular two-graphs on 50 vertices leading to 398 new SRG (49,24,11,12).