On efficient balanced codes over the mth roots of unity

Abstract

Let /spl Phi//sub m//spl sube/ /spl Copf/ be the set of all mth roots of unity, m/spl isin/ IN. A balanced code over /spl Phi//sub m/ is a block code over the alphabet /spl Phi//sub m/ such that each code word is balanced; that is, the complex sum of its components (or weight) is equal to 0. Let B/sub m/(n) be the set of all balanced words of length n over /spl Phi//sub m/. In this correspondence, it is shown that when m is a prime number, the set B/sub m/(n) is not empty if, and only if, m divides n. In this case, the minimum redundancy for a balanced code over /spl Phi//sub m/ of length n is. On the other hand, it is shown that when m=4, the set B/sub 4/(n) is not empty if, and only if, n is even, and in this case, the minimum redundancy for a balanced code over /spl Phi//sub 4/ of length n is. Further, this correspondence completely solves the problem of designing efficient coding methods for balanced codes over /spl Phi//sub m/, when m=4. In fact, it reduces the problem of designing efficient coding schemes for balanced codes over /spl Phi//sub 4/ to the design of efficient balanced codes over the usual bipolar alphabet /spl Phi//sub 2/={-1,+1}.