Deep Holes of Projective Reed-Solomon Codes

Zhang, Jun; Wan, Daqing; Kaipa, Krishna

doi:10.1109/tit.2019.2940962

Cited by 17 publications

(14 citation statements)

References 32 publications

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“…There are many papers connected with distinct aspects of the weight distribution of the code cosets, see e.g. [1, 9-14, 16, 18-20], [21,Section 6.3], [22,26], [31,Section 7], [32,34,35], [36,Section 1.4.6], [38], [39,Sections 5.5,6.6,6.9], [42,44,45], and the references therein.…”

Section: Introduction the Main Resultsmentioning

confidence: 99%

“…There are useful relations between the deep holes and the bounds on the size of the lists in the list decoding of GRS codes, see e.g. [10,Sections 4.3,4.8], [30,33,34], [41,Chapter 9], [44,45] and the references therein.…”

Section: Introduction the Main Resultsmentioning

confidence: 99%

“…Theorem 4.2 is used in Corollary 4.3 and Theorems 6.3 and 7.6 which are the basic ones in Sections 6 and 7, respectively. The results of Theorem 4.2 matter because the weight d−1 cosets consist of the farthest-off vectors (deep holes) of an MDS code.The deep holes are important for investigations of decoding[30,33,34,41,44,45] and multiple coverings[4,5,7,17], see Introduction and Section 7 for details.Corollary 4.3 accumulates the results of Theorems 4.1, 4.Corollary 4.3. (i): For an [n, k, d] q MDS code of distance d ≥ 2, all cosets of weight 1 have the same weight distribution of the form (4.4), if d = 2, and (4.3), if d ≥ 3.…”

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confidence: 99%

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On the weight distribution of the cosets of MDS codes

Davydov¹,

Marcugini²,

Pambianco³

2023

AMC

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<p style='text-indent:20px;'>The weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula> using the known numbers of vectors of weights <inline-formula><tex-math id="M2">\begin{document}$ \le d-2 $\end{document}</tex-math></inline-formula> in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights <inline-formula><tex-math id="M3">\begin{document}$ W $\end{document}</tex-math></inline-formula>. (The weight <inline-formula><tex-math id="M4">\begin{document}$ W $\end{document}</tex-math></inline-formula> of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered <inline-formula><tex-math id="M5">\begin{document}$ W $\end{document}</tex-math></inline-formula> or regions of <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>, special relations more simple than the general ones are obtained. For the MDS code cosets of weight <inline-formula><tex-math id="M7">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> and weight <inline-formula><tex-math id="M8">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight <inline-formula><tex-math id="M9">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> (as well as <inline-formula><tex-math id="M10">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula>) have the same weight distribution. The cosets of weight <inline-formula><tex-math id="M11">\begin{document}$ W = 2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ W = d-2 $\end{document}</tex-math></inline-formula> may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane <inline-formula><tex-math id="M13">\begin{document}$ \mathrm{PG}(2,q) $\end{document}</tex-math></inline-formula> are also considered. For MDS codes of covering radius <inline-formula><tex-math id="M14">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> we obtain the number of the weight <inline-formula><tex-math id="M15">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius <inline-formula><tex-math id="M16">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space <inline-formula><tex-math id="M17">\begin{document}$ \mathrm{PG}(N,q) $\end{document}</tex-math></inline-formula>.</p>

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Section: Introduction the Main Resultsmentioning

confidence: 99%

Section: Introduction the Main Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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On the weight distribution of the cosets of MDS codes

Davydov¹,

Marcugini²,

Pambianco³

2023

AMC

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“…, then u is also called a deep hole of generalized projective Reed-Solomon code GPRS q (D, k). In 2016, Zhang and Wan [10] studied the deep holes of projective Reed-Solomon code GPRS(F q , k). In fact, under the assumption that the only deep holes of GRS q (F q , k) are those received codewords whose Lagrange interpolation polynomials are of degree k, they proved the following results by solving a subset sum problem.…”

Section: We Writementioning

confidence: 99%

Some results on deep holes of generalized projective Reed-Solomon codes

Xiao-li¹,

2019

AIMS Mathematics

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“…The covering radius and deep holes of a linear code embedded with Hamming metric were studied extensively [1,2,3,4,5,10,12,14,16,22,23,24,25,26,27], in which MDS codes such as generalized Reed-Solomon codes, standard Reed-Solomon codes and projective Reed-Solomon codes were explored deeply. Gabidulin codes were introduced by Gabidulin in [7] and independently by Delsarte in [6].…”

Section: Introductionmentioning

confidence: 99%

On deep holes of Gabidulin codes

Fang

Wang

Wan

2018

Finite Fields and Their Applications

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In this paper, we study deep holes of Gabidulin codes in both rank and Hamming metrics. Specifically, first, we give a tight lower bound for the distance of any word to a Gabidulin code and a sufficient and necessary condition for achieving this lower bound as well. Then, a class of deep holes of a Gabidulin code are discovered. Furthermore, we obtain some other deep holes for certain Gabidulin codes.

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Deep Holes of Projective Reed-Solomon Codes

Cited by 17 publications

References 32 publications

On the weight distribution of the cosets of MDS codes

On the weight distribution of the cosets of MDS codes

Some results on deep holes of generalized projective Reed-Solomon codes

On deep holes of Gabidulin codes

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