Representing group codes as permutation codes

Given an abstract group G, an N dimensional orthogonal matrix representation G of G, and an “initial vector” x ∈ R , Slepian defined the group code generated by the representation G to be the set of vectors Gx. If G is a group of permutation matrices, the set Gx is called a “permutation code”. For permutation codes a ’stack algorithm’ decoder exists that, in the presence of low noise, produces the maximum-likelihood estimate of the transmitted vector by using far fewer computations than the standard decoder. In this paper a new concept of equivalence of codes of different dimensions is presented which is weaker than the usual definition of equivalent codes. We show that every group code is (weakly) equivalent to a permutation code and we discuss the minimal degree of this permutation code. ∗J. Karlof is with the Mathematical Sciences Department • University of North Carolina • Wilmington, NC 28403 †E. Biglieri and E. Viterbo are with Dipartimento di Elettronica • Politecnico di Torino • I-10129 Torino (Italy) Their contribution to this research was sponsored by CNR under “Progetto Finalizzato Trasporti.”