On symmetric L1 distance error control codes and elementary symmetric functions

Based on the elementary symmetric functions, this paper gives a new wide class of Goppa like codes capable of correcting/detecting errors measured under the (symmetric) L₁ distance defined over the m-ary words, 2 ≤ m ≤ +∞. All these codes can be efficiently decoded by algebraic means with the Extended Euclidean Algorithm (EEA). In particular it is shown that if K is any field with characteristic char(K) ≠ 2, m ϵ IN U {+∞} and n, t ϵ IN then there exist m-ary codes C of length n ≤ (|K|- 1)/2 and cardinality |C| ≥ mⁿ/|K|^t which are capable of, say, correcting t errors (i. e., the minimum L₁ distance of C is d_L1 (C) ≥ 2t + 1) with t steps of EEA. Also, if K is a finite field and 2t + 1 ≤ char(K) ≠ 2 then some of these codes are (essentially) linear and, hence, easy to encode.