Maximal Group Codes with Specified Minimum Distance

Abstract

All n-digit maximal block codes with a specified minimum distance d such that 2d ≥ n can be constructed from the Hadamard matrices. These codes meet the Plotkin bound. In this paper we construct all maximal group codes in the region 2d ≥ n, where d is a specified minimum distance and n is the number of digits per code word. Unlike the case of block codes, the Plotkin upper limit, in general, fails to determine the number of code words B(n, d) in a maximal group code in the region 2d ≥ n. We show that the value of B(n, d) largely depends on the binary structure of the number d. An algorithm is developed that determines B(n, d), the maximum number of code words for given d and n ≤ 2d. The maximal code is, then, given by its modular representation, explicitly in terms of certain binary coefficients and constants related to n and d. As a side result, we obtain a new upper bound on the number of code words in the region 2d < n which is, in general, stronger than Plotkin's extended bound.