Bounds on the List-Decoding Radius of Reed--Solomon Codes

Ruckenstein, G.; Roth, Ron M.

doi:10.1137/s0895480101395506

Cited by 3 publications

(5 citation statements)

References 10 publications

(9 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we clarify that the above results are consistent with the work in [13,22], in which the list-l…”

Section: Remarkssupporting

confidence: 89%

“…Finally, we clarify that the above results are consistent with the work in [13,22], in which the list-l Guruswami-Sudan bound is shown to be tight only when the algorithm degenerates to the classical (list-1) hard-decision decoding.…”

Section: Perform Error Correction and Evaluate Ysupporting

confidence: 85%

“…⌋ as in the case of polynomial interpolation, but a more sophisticated function of deg 1,w (Q(x, y)), as will be characterized in (22) in an optimal setup. interpolation points are…”

Section: B Rational Interpolationmentioning

confidence: 99%

“…where deg 1,w (Q(x, y)) satisfies (22) and m satisfies (30). By assumption the above inequality holds when e = t, i.e., (2t − 2t 0 )P y tm − t + t 0 < 1.…”

mentioning

confidence: 99%

“…2n(t−t 0 ) = 7. Let the degree of y be chosen as P y = tm 2t−2t 0 = 16, following(23), and the degree of Q(x, y), deg 1,w (Q(x, y)), as in(22). As a result, the degrees of freedom is (tm − P y (t − t 0 ))(P y + 1) = 425 whereas the number of linear constraints is nm(m + 1)/2 = 420.…”

mentioning

confidence: 99%

See 4 more Smart Citations

New List Decoding Algorithms for Reed–Solomon and BCH Codes

Wu¹

2008

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

In this paper we devise a rational curve fitting algorithm and apply it to the list decoding of Reed-Solomon and BCH codes. The resulting list decoding algorithms exhibit the following significant properties.• The algorithm achieves the limit of list error correction capability (LECC) n(1 − √ 1 − D)for a (generalized) (n, k, d = n − k + 1) Reed-Solomon code, which matches the Johnson bound, where D △ = d n denotes the normalized minimum distance. The algorithmic complexity is O n 6 (1 − √ 1 − D) 8 . In comparison with the Guruswami-Sudan algorithm, which exhibits the same LECC, the proposed requires a multiplicity (which dictates the algorithmic complexity) significantly smaller than that of the Guruswami-Sudan algorithm in achieving a given LECC, except for codes with code-rate below 0.15. In particular, for medium-to-high rate codes, the proposed algorithm reduces the multiplicity by orders of magnitude. Moreover, for any ǫ > 0, the intermediate LECC t = ⌊ǫ · d 2 + (1 − ǫ) · (n − n(n − d))⌋ can be achieved by the proposed algorithm with multiplicity m = ⌊ 1 ǫ ⌋. Its list size is shown to be upper bounded by a constant with respect to a fixed normalized minimum distance D, rendering the algorithmic complexity quadratic in nature, O(n 2 ).• By utilizing the unique properties of the Berlekamp algorithm, the algorithm achieves the LECC limit n 2 (1 − √ 1 − 2D) for a narrow-sense (n, k, d) binary BCH code, which matches the Johnson bound for binary codes. The algorithmic complexity is O n 6 (1 − √ 1 − 2D) 8 . Moreover, for any ǫ > 0, the intermediate LECC t = ⌊ǫ · d 2 + (1 − ǫ) · n− √ n(n−2d) 2 ⌋ can be achieved by the proposed algorithm with multiplicity m = ⌊ 1 2ǫ ⌋. Its list size is shown to be upper bounded by a constant, rendering the algorithmic complexity quadratic in nature, O(n 2 ).

show abstract

“…Finally, we clarify that the above results are consistent with the work in [13,22], in which the list-l…”

Section: Remarkssupporting

confidence: 89%

Section: Perform Error Correction and Evaluate Ysupporting

confidence: 85%

“…⌋ as in the case of polynomial interpolation, but a more sophisticated function of deg 1,w (Q(x, y)), as will be characterized in (22) in an optimal setup. interpolation points are…”

Section: B Rational Interpolationmentioning

confidence: 99%

“…where deg 1,w (Q(x, y)) satisfies (22) and m satisfies (30). By assumption the above inequality holds when e = t, i.e., (2t − 2t 0 )P y tm − t + t 0 < 1.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations