Alternant codes over arbitrary finite commutative local rings with identity are constructed in terms of parity-check matrices. The derivation is based on the factorization of x s - 1 over the unit group of an appropriate extension of the finite ring. An efficient decoding procedure which makes use of the modified Berlekamp-Massey algorithm to correct errors and erasures is presented. Furthermore, we address the construction of BCH codes over Zm under Lee metric.
codes over rings; alternant codes; BCH codes; Galois extensions of local commutative rings; algebraic decoding; modified Berlekamp-Massey algorithm; errors and erasures decoding; Lee metric
