Some Further Theory of Group Codes

This paper is a collection of results on the theory of group error-correcting codes for use 011 binary channels. It investigates f u r t h e r certain topics introduced in an earlier paper 1 by the author. T h e reader will he assumed to be familiar with the contents of this earlier paper as well as with the general nature of the coding problem in information theory. T h e evident trend to digital transmission systems has given rise in recent years to an increased interest in coding as a possible practical means of error control. Lacking an "explicit solution" to the coding problem in any real sense, many investigators have chosen in an ad hoc manner promising special classes of parity-check codes and have examined their properties. A large and useful literature of special codes has resulted. T h e approach taken here is different. No special codes are examined; rather, we a t t e m p t to shed some additional light 011 the structure of the class of all group codes. Our original aim was to parametrize in some manner the various equivalence classes of group codes. If such a parametrization could be effected, one could then hope to express the error probability of a code in terms of the parameters, and possibly to see how to choose the parameters to obtain codes of small error probability. We have fallen far short of this goal. T h e main results to be found in this paper arc as follows. A natural dual for a group cod? is defined. For any two group codes, a product code and a sum code are defined and certain properties of these operations are investigated These operations have the important property of 1219