Single-Deletion Single-Substitution Correcting Codes

Smagloy, Ilia; Welter, Lorenz; Wachter-Zeh, Antonia; Yaakobi, Eitan

doi:10.48550/arxiv.2005.09352

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…We refer to [28,48,44,30,12,13,1,34,33,39,3,16] for the historic development of insertion-deletion error-correcting codes. For the recent breakthroughs and constructions we refer to [18,19,20,22,9,16,10,15,38,37,40,41,42,27,45] and a nice latest survey [21]. Efficient coding attaining the near-Singleton optimal rate-distance tradeoff was achieved in [18,19].…”

Section: Introductionmentioning

confidence: 99%

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

Chen¹

2021

Preprint

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The insertion-deletion codes was motivated to correct the synchronization errors. In this paper we prove several Singleton type upper bounds on the insdel distance of linear insertion-deletion codes, based on the generalized Hamming weights and the formation of minimum Hamming weight codewords. Our bound are stronger than some previous known bounds. These upper bounds are valid for any fixed ordering of coordinate positions. We apply these upper bounds to some binary cyclic codes and binary Reed-Muller codes with any coordinate ordering, and some binary Reed-Muller codes and one algebraic-geometric code with certain special coordinate ordering.

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Section: Introductionmentioning

confidence: 99%

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

Chen¹

2021

Preprint

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“…In other words, quantum error-correcting codes that can correct two or more deletion errors have not been found to date. Furthermore, in classical coding theory, the construction of codes that can correct both substitution and deletion errors has attracted much attention [23,24]; however, no such codes have yet been found in quantum coding theory.…”

Section: Introductionmentioning

confidence: 99%

Permutation-Invariant Quantum Codes for Deletion Errors

Shibayama,

Hagiwara

2021

Preprint

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This paper presents conditions for constructing permutation-invariant quantum codes for deletion errors and provides a method for constructing them. Our codes give the first example of quantum codes that can correct two or more deletion errors. Also, our codes give the first example of quantum codes that can correct both multiple-qubit errors and multiple-deletion errors. We also discuss a generalization of the construction of our codes at the end.

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“…We refer to [56,57,58,79,77,59,19,22,1,64,63,71,51,5,4,31,33,23,61] for the historic development of insertion-deletion error-correcting codes. For the recent breakthroughs and constructions we refer to [37,38,39,43,13,33,15,28,70,69,73,74,75,55,78,13,45,34,18] and a nice latest survey [42].…”

Section: Introductionmentioning

confidence: 99%

Explicit Good Subspace-metric Codes and Subset-metric Codes

Chen¹

2021

Preprint

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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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Single-Deletion Single-Substitution Correcting Codes

Cited by 3 publications

References 12 publications

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

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

Permutation-Invariant Quantum Codes for Deletion Errors

Explicit Good Subspace-metric Codes and Subset-metric Codes

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