Ternary codes from the strongly regular (45, 12, 3, 3) graphs and orbit matrices of 2-(45, 12, 3) designs

Abstract

The enumeration of strongly regular graphs with parameters (45, 12, 3, 3) has been completed, and it is known that there are 78 non-isomorphic strongly regular (45, 12, 3, 3) graphs. A strongly regular graph with these parameters is a symmetric (45, 12, 3) design having a polarity with no absolute points. In this paper we examine the ternary codes obtained from the adjacency (resp. incidence) matrices of these graphs (resp. designs), and those of their corresponding derived and residual designs. Further, we give a generalization of a result of Harada and Tonchev on the construction of non-binary self-orthogonal codes from orbit matrices of block designs under an action of a fixed-point-free automorphism of prime order. Using the generalized result we present a complete classification of self-orthogonal ternary codes of lengths 12, 13, 14, and 15, obtained from non-fixed parts of orbit matrices of symmetric (45, 12, 3) designs admitting an automorphism of order 3. Several of the codes obtained are optimal or near optimal for the given length and dimension. We show in addition that the dual codes of the strongly regular (45, 12, 3, 3) graphs admit majority logic decoding.