Group Codes and Walsh Functions

The set of Walsh functions, wal(j,?), is the character group of the dyadic group. For O?j?2k it is shown that they may also be derived from the character table of the abstract Abelian group Ck generated by k elements of order two. The method uses Slepians modular representation table3] to compute the 2k irreducible representations (each of degree one) of Ck. The character table, K, is a 2kx2k square array of +1's and -l's and, considered as a matrix, the orthogonality relationships for characters show that K has the Hadamard property, K]K]T = 2K I]. In fact, for the proper ordering of the group elements in the construction of the modular representation table it is the Hadamard matrix, the entries of whose ith row take on the values of the Walsh function wal (i,?) in each of ?/2k subintervals. In a similar way other permutations of the modular representation table define different functions taking on the values +l, -l, also orthogonal and in a one to one relationship to the Walsh functions. Since an n place binary group code with k information places is isomorphic to Ck,3] each code can thus be used to generate real functions orthogonal over a given interval or period ?. In the special case of cyclic codes where the elements of the code interpreted as polynomials form an ideal in a polynomial ring of characteristic two, the group operation used in deriving the character table is of course, addition.