A Correspondence Between Equivalence Classes of Switching Functions and Group Codes

Abstract

The correspondence defined below was used to convert Slepian's tabulation of the number of equivalence classes of (m, r) group codes <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</sup> into hitherto unpublished data of relevance to switching theory. Specifically, Table II lists the number of equivalence classes of switching functions of weight m≪ 20 or m≫ 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -20 in nine or fewer arguments under the group of linear transformations on its argument variables. The correspondence is established by means of an m×n binary matrix, all of whose rows are distinct. The rows of this matrix define m points at which a switching function of n arguments takes on unit value. If the rank of this matrix (over the 2-element field) is r, then its columns generate an r-dimensional subspace of binary m-tuples which, by definition, is the message set of an (m, r) group code.