On L1 metric asymmetric/unidirectional error control codes, constrained weight codes and σ-codes

The general theory on partially asymmetric (t, t₊)-EC/(d_-, d₊)-ED m-ary codes for the L₁ distance 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 - log_m |C| ≤ ρ {S(m, n, w, d + 1)) + t log_m |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 n <; |K| + ⌈d/(m - 1)] - 1, k ≤ |K| - |K| - 1 information digits and n - k = tlog_m |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.