On balanced codes over the m-th roots of unity, m=4

Abstract

This paper completely solves the coding problem 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 /spl Phi/4 = {-1,+1}. In particular, the paper proposes efficient constructions for balanced codes over /spl Phi//sub 4/ with k information digits and r=2log/sub 4/k -log/sub 4/s(k) + /spl Theta/(1), check digits, where s(k) is any function such that 1/spl les/s(k)/spl les/k. Such balanced codes can be implemented with T=O(klogk:+ ks(k)) digit operations, by storing A=O(k+s/sup 3/(k)) memory digits. The paper also shows that the minimum redundancy for a balanced code over /spl Phi//sub 4/ of length n is r/sub min/ /spl ap/ Iog/sub 4/n+0.326.