Performance of Weighted Nonbinary Repeat-Accumulate Codes over GF(q) with q-ary Orthogonal Modulation

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

“…Using (2) and (3) in (1), we get the expression (4) at the top of the page for the CWE (with w > 0) of WNRMA code ensembles, where for conciseness h 0 = nw and h L = h.…”

“…However, for finite block lengths, the IOWE of a nonbinary accumulator as given by Theorem 1 in [7] using the approximation for p(k) given in [7,Eq. (3)] (which is taken from [2]) is not exact. From (5) and (6) it can easily be verified that the asymptotic spectral shape function of WNRMA code ensembles satisfies the recursive relation…”

Performance analysis of weighted nonbinary repeat multiple-accumulate codes

“…Therefore, the growth rate coefficients computed in this section coincide with those in [7]. However, for finite block lengths, the IOWE of a nonbinary accumulator as given by Theorem 1 in [7] using the approximation for p(k) given in [7,Eq. (3)] (which is taken from [2]) is not exact. From (11) and (12) it can easily be verified that the asymptotic spectral shape function of WNRMA code ensembles satisfies the recursive relation…”

“…The encoder consists of a rate R rep = 1/n nonbinary repeat code, a weighter, a random symbol interleaver, and an accumulator over a finite field GF(q) of size q. WNRA codes can be decoded iteratively using the turbo principle, and in [1] simulation results were presented that showed that these codes are superior to binary RA codes on the additive white Gaussian noise (AWGN) channel when the weighter is properly chosen. In a recent work [2], Kim et al derived an approximate input-output weight enumerator (IOWE) for the nonbinary accumulator. Based on that, approximate upper bounds on the maximum-likelihood (ML) decoding threshold of WNRA codes with qary orthogonal modulation and coherent detection over the AWGN channel were computed for different values of the repetition factor n and the field size q, showing that these codes perform close to capacity under ML decoding for large values of n and q.…”

“…In a recent paper [7], the authors considered weighted nonbinary repeat multiple-accumulate (WNRMA) code ensembles obtained from the serial concatenation of a nonbinary repeat code and the cascade of L ≥ 1 accumulators, where each encoder is followed by a nonbinary weighter, as the qary generalization of binary RMA codes [3][4][5][6]8]. Building upon the approximate IOWE for nonbinary accumulators [2], it was shown numerically in [7] that the d min of WNRMA code ensembles grows linearly with the block length, and the growth rates were estimated. However, no formal proof was provided in [7].…”