[Paper Review] On the Analysis of Weighted Nonbinary Repeat Multiple-Accumulate Codes

Eirik Rosnes, Alexandre Graell i Amat|arXiv (Cornell University)|Jan 31, 2011

Cooperative Communication and Network Coding11 references3 citations

TL;DR

This paper provides a formal analysis of weighted nonbinary repeat multiple-accumulation (WNRMA) codes, deriving exact input-output weight enumerators for nonbinary accumulators and proving that the symbol-wise minimum distance of WNRMA code ensembles grows linearly with block length for $ L \geq 3, n \geq 2 $, and $ L = 2, n \geq 3 $, when $ q \geq 3 $ is a prime power. The results establish that WNRMA codes are asymptotically good and exhibit near-capacity performance under maximum-likelihood decoding.

ABSTRACT

In this paper, we consider weighted nonbinary repeat multiple-accumulate (WNRMA) code ensembles obtained from the serial concatenation of a nonbinary rate-1/n repeat code and the cascade of L>= 1 accumulators, where each encoder is followed by a nonbinary random weighter. The WNRMA codes are assumed to be iteratively decoded using the turbo principle with maximum a posteriori constituent decoders. 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. We also give iterative decoding thresholds, computed by an extrinsic information transfer chart analysis, on the q-ary symmetric channel to show the convergence properties. Finally, we consider the binary image of WNRMA code ensembles and compare the asymptotic minimum distance growth rates with those of binary repeat multiple-accumulate code ensembles.

Motivation & Objective

• To formally prove that the minimum distance of weighted nonbinary repeat multiple-accumulate (WNRMA) code ensembles grows linearly with block length for specific parameter ranges.
• To derive exact input-output weight enumerators (IOWE) for nonbinary accumulators, enabling precise analysis of WNRMA code ensembles.
• To analyze the asymptotic behavior of WNRMA codes on the $ q $-ary symmetric channel using extrinsic information transfer (EXIT) charts for iterative decoding convergence.
• To compare the binary image of WNRMA codes with binary RMA codes, assessing minimum distance growth rates and performance under maximum-likelihood decoding.

Proposed method

• Derives an exact closed-form expression for the input-output weight enumerator (IOWE) of a nonbinary accumulator over $ \mathrm{GF}(q) $.
• Uses the exact IOWE to compute the average weight enumerator (WE) of WNRMA code ensembles through algebraic composition of constituent components.
• Applies asymptotic analysis techniques to the average WE to determine the growth rate of the symbol-wise minimum distance with block length.
• Employs extrinsic information transfer (EXIT) chart analysis to compute iterative decoding thresholds on the $ q $-ary symmetric channel (QSC).
• Derives the average binary weight enumerator of the binary image of WNRMA codes and analyzes its asymptotic minimum distance growth.
• Uses l’Hôpital’s rule and Taylor expansion to evaluate limits in the asymptotic analysis, particularly near $ u = 0 $, to determine the sign of the growth rate coefficient.

Experimental results

Research questions

• RQ1 Does the symbol-wise minimum distance of WNRMA code ensembles grow linearly with block length for $ L \geq 3, n \geq 2 $, and $ L = 2, n \geq 3 $, when $ q \geq 3 $ is a prime power?
• RQ2 What is the exact input-output weight enumerator of a nonbinary accumulator over $ \mathrm{GF}(q) $, and how does it enable precise analysis of WNRMA code ensembles?
• RQ3 How do the iterative decoding thresholds of WNRMA codes on the $ q $-ary symmetric channel compare across different values of $ n $ and $ q $?
• RQ4 How does the minimum distance growth rate of the binary image of WNRMA codes compare with that of binary RMA codes for the same parameters?
• RQ5 What is the asymptotic behavior of the minimum distance growth coefficient as $ q \to \infty $, and how does it compare to the Gilbert-Varshamov bound?

Key findings

• The symbol-wise minimum distance of WNRMA code ensembles grows linearly with block length when $ L \geq 3, n \geq 2 $, and $ L = 2, n \geq 3 $, for all prime power $ q \geq 3 $, proving that these codes are asymptotically good.
• The exact input-output weight enumerator of a nonbinary accumulator is derived, enabling precise computation of the average weight enumerator of WNRMA code ensembles.
• Iterative decoding thresholds on the $ q $-ary symmetric channel are computed via EXIT chart analysis, showing good convergence properties for practical values of $ q $.
• The minimum distance growth rate coefficient of WNRMA codes is very close to the Gilbert-Varshamov bound for moderate $ q $, but decreases with increasing $ q $, leading to a widening gap to the bound at large $ q $.
• The binary image of WNRMA codes exhibits improved minimum distance growth rates with increasing $ q $, and performs very close to capacity under maximum-likelihood decoding on the AWGN channel.
• For the binary image, the asymptotic minimum distance growth rate improves with $ q $, and the growth rate coefficient remains positive and significant for the considered parameter ranges.