Directed Strongly Regular Graphs and Their Codes

Directed Strongly Regular Graphs (DSRG) were introduced by Duval as a generalization of strongly regular graphs (SRG’s) [4]. As observed in [8] a special case of these are the doubly regular tournaments or equivalently, the skew Hadamard matrices. As the latter already lead to many interesting codes [10] it is natural to consider the more general case of codes constructed from the adjacency matrix of a DSRG. Another motivation is that the codes of SRG’s have been investigated in [2, 7], and the spectra of SRG and DSRG present many similarities, as evidenced by Section 3.3. In this paper, we investigate the dimensions of these codes by using three techniques: orthogonality of codes, determinant factorization, a classical technique in design theory, and last but not least, spectrum of the adjacency matrices in the spirit of [2, 7]. Examples of codes illustrating the bounds, along with their minimum distances are given. While the codes constructed are not, in general, optimal error correcting codes in the distance vs dimension sense,