Given an abstract group /spl Gscr/, an N-dimensional orthogonal matrix representation G of /spl Gscr/, and an "initial vector" x/spl isin/R/sup N/, 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 correspondence, 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.

This correspondence contains a continuation of the table contained in Karlof (1989) listing the minimum distance of optimal permutation codes generated by primitive permutation groups of degree less than or equal to 20. We use a nonlinear programming algorithm developed using geometric programming techniques.

A decoding algorithm for permutation codes that is equivalent to maximum-likelihood decoding, but less complex than the correlation decoder, is presented. A general construction for iteratively maximum-likelihood decodable (IMLD) codes is proved and used to construct IMLD codes for every dimension n. D. Slepian (1965) defined permutation modulation codes and presented an efficient algorithm for their decoding. Slepian's decoding scheme is one of the principal components of the permutation code decoding algorithm presented.>

The initial vector problem is considered for group codes for the Gaussian channel and, in particular, for codes generated by permutation groups. Variant I and variant II permutation codes are defined, and optimal initial vectors for these codes are constructed. A numerical algorithm for constructing optimal group codes is presented together with examples of this algorithm applied to several permutation groups.>

TheM-receiver Gaussian broadcast channel is a communication system in which a single codeword is transmitted overMdistinct Gaussian channels and is received byMreceivers. The receivers have no contact with each other and the channels have different signal-to-noise ratios. The purpose of this correspondence is to define the concept of group code for theM-receiver Gaussian broadcast channel and study permutation codes as a special case. Such a code is generated by an initial vectorx, a groupGof orthogonaln-by-nmatrices, and a sequence of subgroups ofG.

The two-receiver Gaussian broadcast channel is a model of a communication system where a single codeword is transmitted over two distinct Gaussian channels that have different signal-to, noise ratios. The receiver with the better signal-to-noise ratio decodes all of the information carried by the codeword, while the other receiver decodes only some of the information. The concept of group codes for the Gaussian broadcast channel is developed and applied to permutation codes for the Gaussian broadcast channel. A group code for the Gaussian broadcast channel is a group code for the single-user Gaussian channel in which a subgroup of the generating matrix group is used to partition the codewords into disjoint subsets called clouds. Distinct clouds represent distinct messages for the noisier channel while the individual codewords represent distinct messages for the other channel. The clouds share some of the distance properties of group codes. Necessary and sufficient conditions are given for finding good group codes for the Gaussian broadcast channel (in terms of minimum distance).

The question of the existence of nonplanar[M,n]group codes for the Gaussian channel has been settled except in the case ofnodd andModd and composite. For this unsettled case, it is shown that the existence of a nonplanar[M,n]group code is implied by the existence of a groupGsatisfying i)|G|is even, ii)Ghas a faithful complex irreducible representationTof the first kind, and iii)Trestricted to the two-Sylow subgroup ofGcontains the identity representation. A partial converse of this existence result is also given. Finally, it is shown that for each oddnnot of the form2^{m} - 1, there exists a nonplanar[M,n ]group code withModd and composite.