Efficient balanced codes over the mth roots of unity

Let Phi(m) subset of C be the set of all mth roots of unity, m is an element of IN. A balanced code over Phi(m) is a block code over the alphabet Phi(m) such that each code word is balanced; that is, the complex sum of its components (or weight) is equal to 0. Let B-m (n) be the set of all balanced words of length n over Phi(m). In this correspondence, it is shown that when m is a prime number, the set B-m (n) is not empty if, and only if, m divides n. In this case, the minimum redundancy for a balanced code over Phi(m) of length n is rho(B-m(n)) = n [log(m) vertical bar B-m(n)vertical bar] approximate to[(m-1)/2]log(m)(2 pi n) - m/2. On the other hand, it is shown that when m = 4, the set B-4 (n) is not empty if, and only if, n is even, and in this case, the minimum redundancy for a balanced code over Phi(4) of length n is rho(B-4(n)) = n - [log(4) vertical bar B-4(n)vertical bar] approximate to log(4) n + 0.326. Further, this correspondence completely solves the problem of designing efficient coding methods for balanced codes over Phi(m), when m = 4. In fact, it reduces the problem of designing efficient coding schemes for balanced codes over Phi(4) to the design of efficient balanced codes over the usual bipolar alphabet Phi(2) = {-1,+1}.[...]