On m-ary balanced codes with parallel decoding

An m-ary block code, m = 2, 3, 4, ..., of length n ∈ IN is called balanced if, and only if, every codeword is balanced; that is, the real sum of the codeword components, or weight, is equal to ⌊(m - 1)n/2⌋. This paper presents a tight generalization of Knuth´s complementation method with parallel (hence, fast) decoding scheme. Let (_wⁿ)_m indicate the number of m-ary words of length n and weight w ∈ {0, 1, ..., (m-1)n}. A simple implementation of the scheme uses (m - 1)k + m mod 2 balancing functions to make a k ∈ IN digit information word to be balanced. So, r ∈ IN check digits can be used to balance k ≤ [(⌊_(m-1)^r r/2⌋)_m -m mod 2]/(m - 1) information digits. A refined implementation of the parallel decoding scheme uses r check digits to balance k ≤ (m^r -1)/(m-1) information digits.