Group Codes for the Gaussian Channel

A class of codes for use on the Gaussian channel, called group codes, is defined and investigated. Roughly speaking, all words in a group code are on an equal footing: each has the same error probability and the same disposition of neighbors. A decomposition theorem shows every group code to be equivalent to a direct sum of certain basic group codes generated by real-irreducible representations of a finite group associated with the code. Some theorems on distances between words in group codes are demonstrated. The difficult problem of finding group codes with large nearest neighbor distance is discussed in detail.