A Low-Complexity Method for Chase-Type Decoding of Reed-Solomon Codes

Abstract

Abstract-In this work we present a low-complexity implementation of Chase-type decoding of Reed-Solomon Codes. In such, we first use the soft-information available at the channel output to construct a test-set of 2 η vectors, equivalent in all except the η << n least reliable coordinate positions. We then give an interpolation procedure to construct a set of 2 η bivariate polynomials, with the roots of each specified by its corresponding test-vector. Here, test-vector similarity is exploited to share much of the required computation. Finally, we obtain the candidate message from the single z-linear factor of each bivariate polynomial. Although we provide an expression for the direct computation of each candidate message, the complexity of repeating this computation for each interpolation polynomial is prohibitive. We, thus, also present a reduced-complexity factorization (RCF) method to select a single polynomial that, with high probability, contains the correctly decoded message in its z-linear factor. Although suboptimal, the loss in performance of RCF decreases rapidly with increasing code length. We provide extensive simulation results showing that a significant performance increase over traditional hard-decision decoding is achievable with a comparable computational complexity (as implemented with the BerlekampMassey Algorithm).

High-speed VLSI architecture for low-complexity chase soft-decision Reed-Solomon decoding

“…Hence it is no longer a bottleneck. A selection technique has been proposed in [6] to pass the interpolation output of only one test vector to the factorization step at the cost of small performance degradation. Nevertheless, the factorization architecture still accounts for a significant proportion of the overall decoder area.…”

“…The polynomial selection scheme in [6] passes the interpolation output of only one test vector to the following steps of the LCC decoding. This selection is not explicitly shown in Fig.…”

“…Nevertheless, previous soft-decision decoding algorithms either can only achieve limited coding gain or have very high complexity. Algebraic soft-decision (ASD) decoding algorithms [2], [3], [4], [5], [6], [7] for RS codes have been developed recently. By incorporating the probability information from the channel into the algebraic interpolation process developed by Sudan and Guruswami [8], [9], these algorithms can achieve significant coding gain with a complexity that is polynomial with respect to the codeword length.…”

“…For the purpose of practical implementation, simple multiplicity assignment schemes are preferred. The multiplicity assignment in the Kotter-Vardy (KV) algorithm [2] can be implemented by constant multiplications followed by the floor function, and those in the low-complexity Chase (LCC) [6] and bit-level generalized minimum distance decoding (BGMD) [7] algorithms can be implemented by comparators. Smaller multiplicities translate to lower complexity in the interpolation and factorization steps.…”

Soft-Input, Iterative, Reed-Solomon Decoding using Redundant Parity-Check Equations

“…We denote by H q the traditional, Vandermonde, parity-check matrix of C q (n, k) over GF (2 q ) and H q,2 its binary image [7]. We note that the former has density lower bounded by (k + 1)/n ≈ k/n, while the latter by k/nq.…”