Coordinate-ordering-free Upper Bounds for Linear Insertion-Deletion Codes

Chen, Hao

doi:10.48550/arxiv.2106.10782

Cited by 3 publications

(10 citation statements)

References 39 publications

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“…The half-Singleton bounds in Theorem 2.1 for linear subspace-metric and subset-metric codes are similar to the half-Singleton bound for linear insertiondeletion codes, see [15,12].…”

Section: Subspace-metric and Subset-metric Codesmentioning

confidence: 67%

“…for the insdel distance of a linear [n, k] q insertion and deletion code, see [12]. This half-Singleton bound for linear insertion-deletion codes can be generalized to the strong half-Singleton bound based on the generalized Hamming weights…”

Section: Introductionmentioning

confidence: 99%

“…we refer to [12]. The existence and explicit algorithmic construction of binary linear code sequence satisfying δ = lim d insdel 2nt > 0 but with rate R < 1 2 was given in [15].…”

Section: Introductionmentioning

confidence: 99%

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Explicit Good Subspace-metric Codes and Subset-metric Codes

Chen¹

2021

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In this paper motivated from subspace coding we introduce subspacemetric and subset-metric codes. These are coordinate-position independent pseudometrics and suitable for the folded codes introduced by Guruswami and Redra. The half-Singleton upper bounds for linear subspace-metric and subset-metric codes are proved. Subspace distances and subset distances of codes are natural lower bounds for insdel distances of codes, and then can be used to lower bound the insertion-deletion error-correcting capabilities of codes. The problem to construct efficient insertion-deletion error-correcting codes is notorious difficult and has attracted a long-time continuous efforts. The recent breakthrough is the algorithmic construction of near-Singleton optimal rate-distance tradeoff insertion-deletion code families by B. Haeupler and A. Shahrasbi in 2017 from their synchronization string technique. However most nice codes in these recent results are not explicit though many of them can be constructed by highly efficient algorithms. Our subspace-metric and subset-metric codes can be used to construct systemic explicit well-structured insertion-deletion codes. We present some near-optimal subspace-metric and subset-metric codes from known constant dimension subspace codes. By analysing the subset distances of folded codes from evaluation codes of linear mappings, we prove that they have high subset distances and then are explicit good insertion-deletion codes.

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“…The half-Singleton bounds in Theorem 2.1 for linear subspace-metric and subset-metric codes are similar to the half-Singleton bound for linear insertiondeletion codes, see [15,12].…”

Section: Subspace-metric and Subset-metric Codesmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

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Explicit Good Subspace-metric Codes and Subset-metric Codes

Chen¹

2021

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“…Observe that the codes that we construct in Theorem 1.10 are explicit, can decode from n − 3 errors, and are defined over a field of size O(n 4 ). In a very recent work by Chen [Che21] gave some new Singleton-type upper bounds on the insdel distance of linear codes and applied them to certain families of Reed-Muller and algebraic geometry codes. Chen also speculated that RS-codes have "moderate good insertion-deletion error-correcting capabilities" and conjectures that some RS-codes attain the half-Singleton bound.…”

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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“…Their half-Singleton bound for an [n, k] linear code C over F q can be reformulated as d I (C) ≤ max{2(n − 2k + 2), 2}. For a simpler proof we refer to [5]. A new coordinate-ordering free upper bound was also given in [5].…”

Section: Introductionmentioning

confidence: 99%

Strict Half-Singleton Bound, Strict Direct Upper Bound for Linear Insertion-Deletion Codes and Optimal Codes

Ji¹,

Zheng²,

Chen³

et al. 2022

Preprint

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Insertion-deletion codes (insdel codes for short) are used for correcting synchronization errors in communications, and in other many interesting fields such as DNA storage, date analysis, racetrack memory error correction and language processing, and have recently gained a lot of attention. To determine the insdel distances of linear codes is a very challenging problem. The half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li is a basic upper bound on the insertion-deletion error-correcting capabilities of linear codes. On the other hand the natural direct upper bound d I (C) ≤ 2d H (C) is valid for any insdel code. In this paper, for a linear insdel code C we propose a strict half-Singleton upper bound d I (C) ≤ 2(n − 2k + 1) if C does not contain the codeword with all 1s, and a stronger direct upper bound d I (C) ≤ 2(d H (C)−t) under a weak condition, where t ≥ 1 is a positive integer determined by the generator matrix. We also give optimal linear insdel codes attaining our strict half-Singleton bound and direct upper bound, and show that the code length of optimal binary linear insdel codes with respect to the (strict) half-Singleton bound is about twice the dimension. Interestingly explicit optimal linear insdel codes attaining the (strict) half-Singleton bound, with the code length being independent of the finite field size, are given.

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Coordinate-ordering-free Upper Bounds for Linear Insertion-Deletion Codes

Cited by 3 publications

References 39 publications

Explicit Good Subspace-metric Codes and Subset-metric Codes

Explicit Good Subspace-metric Codes and Subset-metric Codes

Reed Solomon Codes Against Adversarial Insertions and Deletions

Strict Half-Singleton Bound, Strict Direct Upper Bound for Linear Insertion-Deletion Codes and Optimal Codes

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