A Theorem on the Distribution of Weights in a Systematic Code†

Abstract

A systematic code of word length n is a subspace of the vector space of all possible rows of n symbols chosen from a finite field. The weight of a vector is the number of its nonzero coordinates; clearly any given code contains a certain finite number of vectors of each weight from zero to n. This set of integers is called the spectrum of the code, and very little is known about it, although it appears to be important both mathematically and as a practical means of evaluating the error-detecting properties of the code. In this paper it is shown that the spectrum of a systematic code determines uniquely the spectrum of its dual code (the orthogonal vector space). In fact the two sets of integers are related by a system of linear equations. Consequently there is a set of conditions which must be satisfied by the weights which actually occur in a systematic code. If there is enough other information about the code, it is possible to use this result to calculate its spectrum.