On L1 metric asymmetric/unidirectional error control codes, constrained weight codes and σ-codes
Luca G. Tallini, Bella Bose · 2013
The general theory on partially asymmetric (t, t+)-EC/(d-, d+)-ED m-ary codes for the L1distance is developed. In this metric, such codes are capable of correcting t-or less negative errors, detecting d or less negative errors, correcting t+or less positive errors, and simultaneously detecting d+or less positive errors. Based on the elementary symmetric function, a wide class of these codes with efficient decoding algorithms are given. Let S(m, n, w, D) be the set of all the m-ary words of length n with real sum of their components being equal to w mod D. Any subset of S(m, n, w, D) is called m-ary constrained weight (CW) code of length n and is known to be a (D - 1)-UED code. Given a field, K, of prime characteristic p, some m-ary CW codes of length n ≤ |K| - 1 are defined. Such codes are (t-, t+)-EC/(d-, d+)-ED and have a redundancy of ρ(c) = n - logm|C| ≤ ρ {S(m, n, w, d + 1)) + t logm|K|, with t = min{t_ + d+, d_ + t+}, d = max{t_ + d+, d_ + t+} and w ∈ IN. In particular, for t ≤ p-1, a class of essentially linear and systematic (hence, easy to encode) m-ary (t_, t+)-EC/(d_, d+)-ED CW σ-codes with length nm|K| + ⌈d/(m-1)⌉ check digits are given. Also, some new hybrid partially asymmetric/unidirectional/symmetric error control codes are given and shown to be equivalent to the partially asymmetric (t_, t+)-EC/(d_, d+)-ED m-ary codes.