A Correspondence Between Equivalence Classes of Switching Functions and Group Codes

The correspondence defined below was used to convert Slepian´s tabulation of the number of equivalence classes of (m, r) group codes^[1] into hitherto unpublished data of relevance to switching theory.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≫ 2ⁿ-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.