Bounds on the bit error probability of a linear cyclic code over GF(2l) and its extended code

An upper bound on the bit-error probability (BEP) of a linear cyclic code over GF(2^l) with hard-decision (HD) maximum-likelihood (ML) decoding on memoryless symmetric channels is derived. Performance results are presented for Reed-Solomon codes on GF(32), GF(64), and GF(128). Also, a union upper bound on the BEP of a linear cyclic code with either hard- or soft-decision ML decoding is developed, as well as the corresponding bounds for the extended code of a linear cyclic code. Using these bounds, which are tight at low bit error rate, the performance advantage of soft-decision (SD) ML and HD ML over bounded-distance (BD) decoding is established