Explicit Constructions of Two-Dimensional Reed-Solomon Codes in High Insertion and Deletion Noise Regime

Duc, Tai Do; Liu, Shu; Tjuawinata, Ivan; Xing, Chaoping

doi:10.1109/tit.2021.3065618

Cited by 16 publications

(15 citation statements)

References 29 publications

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“…In this paper, we showed that if C is an [n, k] linear code over F q with n > k ≥ 2, then the minimum insdel distance of C is at most 2n − 2k (see Theorem A). This result significantly improves the previously known results in [4] and [5] as we mentioned in the Introduction section. We gave a sufficient condition under which a two-dimensional Reed-Solomon code of length n over F q has minimum insdel distance 2n − 4 (see Theorem B); as a corollary, we showed that the conditions listed in Theorem B are easy to achieve (see Corollary C).…”

Section: Discussionsupporting

confidence: 90%

“…Consequently, we have explicitly constructed an infinite family of optimal two-dimensional Reed-Somolom codes meeting the bound in Theorem A. Comparing with [4], our methods are more direct and easy to understand.…”

Section: Discussionmentioning

confidence: 99%

“…In 2007, McAven et al [11] showed that Reed-Solomon codes of length n ≥ 3 and dimension 2 over prime fields can never meet the Singleton bound. Recently, Do Duc et al [4] improved the result by showing that when the field size is sufficiently large compared to the code length, the Singleton bound cannot be achieved by Reed-Solomon codes; more explicitly, it was shown that the minimum insdel distance of a k-dimensional Reed-Solomon code is at most 2n − 2k if the code length n satisfies n > k > 1 and q > n 2 ([4, Theorem 1]); optimal codes that meet the new bound were also constructed explicitly [4, Theorems 2 and 3]. Very recently, Liu et al in [8] established a set of sufficient conditions for two-dimensional insdel Reed-Solomon codes to have optimal asymptotic error-correcting capabilities.…”

Section: Introductionmentioning

confidence: 98%

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Improved Singleton bound on insertion-deletion codes and optimal constructions

Chen¹,

Zhang²

2021

Preprint

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Insertion-deletion codes (insdel codes for short) play an important role in synchronization error correction. The higher the minimum insdel distance, the more insdel errors the code can correct. Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any [n, k] linear code over Fq satisfies d ≤ 2n − 2k + 2. There have been some constructions of insdel codes through Reed-Solomon codes with high capabilities, but none has come close to this bound. Recently, Do Duc et al. showed that the minimum insdel distance of any [n, k] Reed-Solomon code is no more than 2n − 2k if q is large enough compared to the code length n; optimal codes that meet the new bound were also constructed explicitly. The contribution of this paper is twofold. We first show that the minimum insdel distance of any [n, k] This result improves and generalizes the previously known results in the literature. We then give a sufficient condition under which the minimum insdel distance of a two-dimensional Reed-Solomon code of length n over Fq is exactly equal to 2n − 4. As a consequence, we show that the sufficient condition is not hard to achieve; we explicitly construct an infinite family of optimal two-dimensional Reed-Somolom codes meeting the bound. MSC: 94B50.

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Section: Discussionsupporting

confidence: 90%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Improved Singleton bound on insertion-deletion codes and optimal constructions

Chen¹,

Zhang²

2021

Preprint

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“…In another line of work, Liu and Tjuawinata [LT21] and Duc, Liu, Tjuawinata, and Xing [DLTX19] constructed 2-dimensional RS-codes. Specifically, in [DLTX19, Theorem 2] the authors show, non explicitly, that there exist [n, 2] q RS-codes that can correct from n − 3 insdel errors, for q = exp(n 2 ).…”

Section: Previous Resultsmentioning

confidence: 99%

“…Specifically, in [DLTX19, Theorem 2] the authors show, non explicitly, that there exist [n, 2] q RS-codes that can correct from n − 3 insdel errors, for q = exp(n 2 ). In [DLTX19,LT21], the authors presented constructions of [n, 2] RS-codes that for every ε > 0 can correct from (1 − ε) • n insdel errors, for codes of length n = poly(1/ε) over fields of size Ω(exp((log n) 1/ε )) and Ω(exp(n 1/ε )), respectively. Lastly, in [CZ21], the authors present construction of two-dimensional RScodes where the field size is exponential in n that can correct from n − 3 insdel errors.…”

Section: Previous Resultsmentioning

confidence: 99%

Reed Solomon Codes Against Adversarial Insertions and Deletions

Con¹,

Shpilka²,

Tamo³

2021

Preprint

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In this work, we study linear error-correcting codes against adversarial insertiondeletion (insdel) errors, a topic that has recently gained a lot of attention. We focus on two different settings -codes over small alphabets and Reed-Solomon codes.Linear codes over small fields: We construct linear codes over F q , for q = poly(1/ε), that can efficiently decode from a δ fraction of insdel errors and have rate (1 − 4δ)/8 − ε. We also show that by allowing codes over F q 2 that are linear over F q , we can improve the rate to (1 − δ)/4 − ε while not sacrificing efficiency. Using this latter result, we construct fully linear codes over F 2 that can efficiently correct up to δ < 1/54 fraction of deletions and have rate R = (1 − 54 • δ)/1216. Cheng, Guruswami, Haeupler, and Li [CGHL21] constructed codes with (extremely small) rates bounded away from zero that can correct up to a δ < 1/400 fraction of insdel errors. They also posed the problem of constructing linear codes that get close to the half-Singleton bound (proved in [CGHL21]) over small fields. Thus, our results significantly improve their construction and get much closer to the bound.Reed-Solomon codes: We prove that over fields of size n O(k) there are [n, k] Reed-Solomon codes that can decode from n − 2k + 1 insdel errors and hence attain the half-Singleton bound. We also give a deterministic construction of such codes over much larger fields (of size n k O(k) ). Nevertheless, for k = O(log n/ log log n) our construction runs in polynomial time. For the special case k = 2, which received a lot of attention in the literature, we construct an [n, 2] Reed-Solomon code over a field of size O(n 4 ) that can decode from n − 3 insdel errors. Earlier construction required an exponential field size. Lastly, we prove that any such construction requires a field of size Ω(n 3 ).

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Insdel codes from subspace and rank-metric codes

Aggarwal,

Pratihar

2024

Discrete Mathematics

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Explicit Constructions of Two-Dimensional Reed-Solomon Codes in High Insertion and Deletion Noise Regime

Cited by 16 publications

References 29 publications

Improved Singleton bound on insertion-deletion codes and optimal constructions

Improved Singleton bound on insertion-deletion codes and optimal constructions

Reed Solomon Codes Against Adversarial Insertions and Deletions

Insdel codes from subspace and rank-metric codes

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