Construction of BCH codes over certain rings

Antonio Aparecido de Andrade, Tariq Shah

DOI: 10.14209/sbrt.2013.147

Evento: XXXI Simpósio Brasileiro de Telecomunicações (SBrT2013)

Keywords: Units of a local ring BCH code McCoy rank direct product of local rings.

Abstract

In this work, we present a construction technique for the generator polynomials of a sequence of BCH codes having symbols from a direct product of residue integer rings Zpmj , for 1 ≤ j ≤ t. We construct a chain of unitary commutative rings A0 ⊂ A1 ⊂ · · · ⊂ At−1 ⊂ At, for a positive integer t, where each Ai is constructed by a direct product of suitable Galois rings. Corresponding the this chain there is another chain of rings K0 ⊂ K1 ⊂ · · · ⊂ Kt−1 ⊂ Kt constituted through the direct product of their residue fields. Moreover, we obtain the cyclic subgroups Gni containing the roots of the polynomials xni − 1 in the commutative rings Ai, for 0 ≤ i ≤ t. The core of the construction of these BCH codes is the factorization of the polynomials xni − 1 over the cyclic subgroups Gni, for i = 1, 2, · · · , t.

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