A Study of Deep Holes in the First-Order Reed-Muller Codes

Abstract

Abstract. In analogy with the theory of integral lattices, the concept of the deep hole is defined for binary linear codes. In this paper we study combinatorial configurations that come from the deep hole cosets in the first-order Reed-Muller codes of lengths 2 m (m = 3, 4, 5, 6). As consequences we obtain the Hamming association subschemes of various orders.