Uniqueness of Four Covering Arrays Equivalent to Error-Correcting Codes

Abstract

A covering array CA(N;t,k,v) is an N × k array over v symbols where every N × t subarray contains as a row each t-tuple over v symbols at least once. Two covering arrays are isomorphic of one can be obtained from the other by permutations of rows, columns, and symbols in the columns. Isomorphic covering arrays form equivalence classes in the set of all CA(N;t,k,v). The problem of classifying covering arrays consists in generating one element of each isomorphism class; if there is only one isomorphism class, then CA(N;t,k,v) is unique. This work introduces two parallel versions of a previously reported algorithm to classify covering arrays. By using these algorithms we determine the uniqueness of the covering arrays CA(32;4,13,2), CA(64;5,14,2), CA(128;6,15,2), and CA(256;7,16,2). We also find that these four covering arrays are equivalent respectively to the unique error-correcting codes (13,32,6), (14,64,6), (15,128,6), and (16,256,6), where (n,M,d) denotes a code with word length n, M code words, and minimum distance d.