Let Zm={0,1,...,(m-1)} be the m-ary alphabet, m in IN. This paper gives some new theory and efficient designs of Zm linear error control codes based on the elementary symmetric functions of m-ary words. Here, a Zm linear code is a sub-module of the module (Zm^{n}, + mod m, Zm, * mod m), n in IN, and the errors are measured in the L1 or Lee metric. In particular, starting from a very general class of Goppa-like Zm linear codes, given a field, K, of characteristic p=char(K) in IN, we consider a generalization of the BCH codes to the m-ary alphabet for m=p^l, l in IN. For this BCH-like codes we are able to prove a BCH-like bound with respect to the L1 and Lee distances. Hence, focusing on cyclic codes, we present a new class of (d-1) asymmetric error correcting Zm linear cyclic codes, Cd, of length n, with d,m=p^l,p,l,n in IN, d <= m, p=char(K)=2,3,5,... prime, n <= |K|, whose redundancy is only rho(Cd)=n-log_m|Cd|<=(d-1)log_{m}|K|.

On Zm-linear codes and elementary symmetric functions (ad invito)

Tallini, Luca G.;
2015-01-01

Abstract

Let Zm={0,1,...,(m-1)} be the m-ary alphabet, m in IN. This paper gives some new theory and efficient designs of Zm linear error control codes based on the elementary symmetric functions of m-ary words. Here, a Zm linear code is a sub-module of the module (Zm^{n}, + mod m, Zm, * mod m), n in IN, and the errors are measured in the L1 or Lee metric. In particular, starting from a very general class of Goppa-like Zm linear codes, given a field, K, of characteristic p=char(K) in IN, we consider a generalization of the BCH codes to the m-ary alphabet for m=p^l, l in IN. For this BCH-like codes we are able to prove a BCH-like bound with respect to the L1 and Lee distances. Hence, focusing on cyclic codes, we present a new class of (d-1) asymmetric error correcting Zm linear cyclic codes, Cd, of length n, with d,m=p^l,p,l,n in IN, d <= m, p=char(K)=2,3,5,... prime, n <= |K|, whose redundancy is only rho(Cd)=n-log_m|Cd|<=(d-1)log_{m}|K|.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11575/89311
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