Binary Block Coding

Iii this paper we investigate a certain problem in combinatorial analysis which arises in the theory of error correcting coding. A development of coding theory is to be found in the papers of Hamming 1 and Shannon 2 ; this section is intended primarily as a presentation of the terminology used in subsequent sections. We take (0, 1) as the range of binary variables. By an n-word we mean a sequence of n symbols, each of which is 0 or 1. We call the individual symbols of an n-word the letters of the n-word. We denote by Bn the set consisting of all the 2" possible distinct n-words. The set Bn may be mapped onto the vertices of an n-dimensional cube, in the usual way, by regarding an n-word as an n-dimensional Cartesian coordinate expression. The distance d(v, v) between ??-words u and v is defined to be the number of places in which the letters of n and v differ; on the n-cube, this is seen to be the smallest number of edges in paths along edges between the vertices corresponding to u and v. The weight of an n-word n 517