Computing Error Distance of Reed-Solomon Codes

Zhu, Guizhen; Wan, Daqing

doi:10.1007/978-3-642-29952-0_24

Cited by 16 publications

(12 citation statements)

References 14 publications

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“…Further evidence in favour of the conjecture has been provided by Li-Wan [13], Liao [11], Zhu-Wan [20], using polynomial congruences and estimates of character sums, Cafure-Matera-Privitelli [1] using algebraic geometry, and Li-Zhu [14] by explicitly counting solutions to certain polynomial equations. To list the one by Li-Zhu, Theorem 4 (Li-Zhu).…”

Section: For D = F Qmentioning

confidence: 99%

Deep holes in Reed–Solomon codes based on Dickson polynomials

Keti

Wan

2016

Finite Fields and Their Applications

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For an [n, k] Reed-Solomon code C, it can be shown that any received word r lies a distance at most n − k from C, denoted d(r, C) ≤ n − k. Any word r meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of C is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear with a special uniformity. To find families of received words that are not deep holes, we reduce to a subset sum problem (or equivalently, a Dickson polynomial-variation of Waring's problem) and find solution conditions by applying an argument using estimates on character sums indexed over the evaluation set.

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Section: For D = F Qmentioning

confidence: 99%

Deep holes in Reed–Solomon codes based on Dickson polynomials

Keti

Wan

2016

Finite Fields and Their Applications

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“…[1] improved the result in [17] a little bit by using tools of algebraic geometry. Liao [19] gave a tighter estimation of error distance, which was improved by Zhu and Wan [27].…”

Section: Proposition 23 ([16]mentioning

confidence: 99%

Deep holes of generalized Reed-Solomon codes

Zhang¹,

Fu²,

Liao³

2013

Sci. Sin.-Math.

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In this paper, deep holes of Reed-Solomon (RS) codes are studied. A new class of deep holes for generalized affine RS codes is given if the evaluation set satisfies certain combinatorial structure. Three classes of deep holes for projective Reed-Solomon (PRS) codes are constructed explicitly. In particular, deep holes of PRS codes with redundancy three are completely obtained when the characteristic of the finite field is odd. Most (asymptotically of ratio 1) of the deep holes of PRS codes with redundancy four are also obtained.

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“…Using Weil's character sum estimate and Li-Wan's new sieve [11] for distinct coordinates counting, Zhu and Wan [21] showed the following result: …”

Section: Conjecture 2 ([20])mentioning

confidence: 99%

On Determining Deep Holes of Generalized Reed-Solomon Codes

Cheng

Zhuang

2013

Algorithms and Computation

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Abstract. For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized ReedSolomon codes is proved to be co-NP-complete [9] [5]. For the extended Reed-Solomon codes RSq(Fq, k), a conjecture was made to classify deep holes in [5]. Since then a lot of effort has been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for generalized Reed-Solomon codes RSp(D, k), where p is a prime, |D| > k p−1 2. Our techniques are built on the idea of deep hole trees, and several results concerning the Erdös-Heilbronn conjecture.

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Computing Error Distance of Reed-Solomon Codes

Cited by 16 publications

References 14 publications

Deep holes in Reed–Solomon codes based on Dickson polynomials

Deep holes in Reed–Solomon codes based on Dickson polynomials

Deep holes of generalized Reed-Solomon codes

On Determining Deep Holes of Generalized Reed-Solomon Codes

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