On the minimum distance properties of weighted nonbinary repeat multiple-accumulate codes

We consider weighted nonbinary repeat multiple-accumulate (WNRMA) code ensembles obtained from the serial concatenation of a nonbinary rate-1/n repetition code and the cascade of L≥1 accumulators, where each encoder is followed by a nonbinary random weighter. We derive the exact weight enumerator of nonbinary accumulators and subsequently give the weight enumerators for WNRMA code ensembles. We formally prove that the symbol-wise minimum distance of WNRMA code ensembles asymptotically grows linearly with the block length when L≥3 and n≥2, and L=2 and n≥3, for all powers of primes q ≥3 considered, where q is the field size. Thus, WNRMA code ensembles are asymptotically good for these parameters.