On efficient balanced codes over the mth roots of unity

Let Φ_m⊆ C be the set of all mth roots of unity, m∈ IN. A balanced code over Φ_m is a block code over the alphabet Φ_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 Φ_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 Φ_m of length n is. On the other hand, it is shown that when m=4, the set B₄(n) is not empty if, and only if, n is even, and in this case, the minimum redundancy for a balanced code over Φ₄ of length n is. Further, this correspondence completely solves the problem of designing efficient coding methods for balanced codes over Φ_m, when m=4. In fact, it reduces the problem of designing efficient coding schemes for balanced codes over Φ₄ to the design of efficient balanced codes over the usual bipolar alphabet Φ₂={-1,+1}.