Minimum Distance Bounds for S-regular Codes Dedicated to Jaap Seidel on the Occasion of His 80th Birthday

A code C F n is s-regular provided, for every vertex x 2 F n , if x is at distance at most s from C then the number of codewords y 2 C at distance i from x depends only on i and the distance from x to C. If denotes the covering radius of C and C is-regular, then C is said to be completely regular. Suppose C is a code with minimum distance d, strength t as an orthogonal array, and dual degree s. We prove that d 2t + 1 when C is completely regular (with the exception of binary repetition codes). The same bound holds when C is (t + 1)-regular. For unrestricted codes, we show that d s + t unless C is a binary repetition code. H(n; q) has vertex set X = F n and distance given by the Hamming metric dist(x; y) = jfh : x h 6 = y h gj : Let A denote the adjacency matrix of this graph and E j the matrix representing orthogonal projection onto the eigenspace V j belonging to eigenvalue j = n(q ? 1) ? qj (0 j n). Cf. 1, Theorem 9.2.1]. To any code C X (jCj > 1), we associate its outer distribution matrix D with rows indexed by X and columns indexed by f0; : : : ; ng and with (x; i) entry given by D xi = jfy 2 C : dist(x; y) = igj;