Directed Strongly Regular Graphs and Their Codes

Abstract

Abstract. The rank over a finite field of the adjacency matrix of a directed strongly regular graph is studied, with some applications to the construction of linear codes. Three techniques are used: code orthogonality, adjacency matrix determinant, and adjacency matrix spectrum.

Directed Strongly Regular Graphs and their Codes: Determinant Factorization for a Tight p-rank

“…The following counterexamples have their adjacency matrices given in [1]. Consider the adjacency matrix given by a tuple…”

“…The adjacency matrix has only zeros and ones in its first row, just as any other row. By the algorithm for the Smith normal form, we can have a 1 in the position A (1,1) and 0 in each of the positions…”

“…This paper seeks to address the main open problem given in [1]. The open problem was trying to establish a tight bound for the rank of an adjacency matrix of a directed strongly regular graph over a finite field, which is equivalent to finding the dimension of the codes.…”

“…The open problem was trying to establish a tight bound for the rank of an adjacency matrix of a directed strongly regular graph over a finite field, which is equivalent to finding the dimension of the codes. The statement was conjectured in [1] as follows: Conjecture 1. [1] If p j divides det(A) = 0 exactly, then the rank of A over F p is v − j.…”

“…The statement was conjectured in [1] as follows: Conjecture 1. [1] If p j divides det(A) = 0 exactly, then the rank of A over F p is v − j.…”