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タイトル： Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions
著者名： Ryan Gabrys, Venkatesan Guruswami, Joao Ribeiro, Ke Wu
アクセス： https://arxiv.org/pdf/2112.09971.pdf

Abstract： We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, who proved that binary Varshamov-Tenengolts codes correct one arbitrary edit error, i.e., one deletion or one substitution, with
nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations…

タイトル： Constructions of asymptotically optimal codebooks with respect to Welch bound and Levenshtein bound
著者名： G Wang, DM Xu, FW Fu
ジャーナル： Advances in Mathematics of Communications, 2021
アクセス： https://www.aimsciences.org/article/doi/10.3934/amc.2021065

Abstract： Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in code division multiple access communication systems. In this paper, several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound. Specially, a class of optimal codebooks with respect to the Levenshtein bound is obtained. These classes of codebooks are constructed by selecting certain rows deterministically from circulant matrices, Fourier matrices and Hadamard matrices, respectively. The construction methods and parameters of some codebooks provided in this paper are new.

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タイトル： List-decodable Codes for Single-deletion Single-substitution with List-size Two
著者名： Wentu Song, Kui Cai, and Tuan Thanh Nguyen
アクセス： https://arxiv.org/abs/2201.02013v1

Abstract： In this paper, we present an explicit construction of list-decodable codes for single-deletion and single-substitution with list size two and redundancy 3 log n+4, where n is the block length of the code. Our construction has lower redundancy than the best known explicit construction by Gabrys et al. (arXiv 2021), whose redundancy is 4 log n + O(1)

タイトル： A new construction of linear codes with one-dimensional hull
著者名： Lin Sok
ジャーナル： Designs, Codes and Cryptography
アクセス： https://link.springer.com/article/10.1007/s10623-021-00991-4

Abstract： The hull of a linear code C is the intersection of C with its dual C⊥. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The hull of linear codes has recently found its application to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, we provide a new method to construct linear codes with one-dimensional hull. This construction method improves the code lengths and dimensions of the recent results given by the author. As a consequence, we derive several new classes of optimal linear codes with one-dimensional hull. Some new EAQECCs are presented.

タイトル： Design and Performance of Low-Density Parity-Check Codes for
Noisy Channels with Synchronization Errors
著者名： Ryo SHIBATA, Hiroyuki YASHIMA
ジャーナル： IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
DOI： https://doi.org/10.1587/transfun.2021EAL2013

Abstract：  In this letter, we study low-density parity-check (LDPC)
codes for noisy channels with insertion and deletion (ID) errors. We
first propose a design method of irregular LDPC codes for such channels,
which can be used to simultaneously obtain degree distributions for
different noise levels. We then show the asymptotic/finite-length
decoding performances of designed codes and compare them with the
symmetric information rates of cascaded ID-noisy channels. Moreover, we
examine the relationship between decoding performance and a code
structure of irregular LDPC codes.

タイトル： Coding for Sequence Reconstruction for Single Edits
著者名： Kui Cai, Han Mao Kiah, Tuan Thanh Nguyen, Eitan Yaakobi
ジャーナル： IEEE Transactions on Information Theory
DOI：https://10.1109/TIT.2021.3122798

Abstract： The sequence reconstruction problem, introduced by
Levenshtein in 2001, considers a communication scenario where the sender
transmits a codeword from some codebook and the receiver obtains
multiple noisy reads of the codeword. The common setup assumes the
codebook to be the entire space and the problem is to determine the
minimum number of distinct reads that is required to reconstruct the
transmitted codeword. Motivated by modern storage devices, we study a
variant of the problem where the number of noisy reads N is fixed.
Specifically, we design reconstruction codes that reconstruct a codeword
from N distinct noisy reads. We focus on channels that introduce a
single edit error (i.e. a single substitution, insertion, or deletion)
and their variants, and design reconstruction codes for all values of N . In particular, for the case of a single edit, we show that as the
number of noisy reads increases, the number of redundant symbols
required can be gracefully reduced from logqn+O(1) to logqlogqn+O(1) ,
and then to O(1) , where n denotes the length of a codeword. We also
show that these reconstruction codes are asymptotically optimal. Finally,
via computer simulations, we demonstrate that in certain cases,
reconstruction codes can achieve similar performance as classical error-
correcting codes with less redundant symbols.

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タイトル：Asymptotic Behavior and Typicality Properties of Runlength-Limited Sequences
著者名 : Mladen Kovačević; Dejan Vukobratovi
ジャーナルIEEE Transactions on Information Theory
DOI : https://doi.org/10.1109/TIT.2021.3134871

Abstract : This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained by using the methods of multivariate analytic combinatorics, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion—q-ary sequences with fixed Manhattan weight—is briefly discussed, and an application in coding for molecular timing channels is illustrated.

タイトル：A Quaternary Code Correcting a Burst of at Most Two Deletion or Insertion Errors in DNA Storage
著者名 : Thi-Huong Khuat; Sunghwan Kim
ジャーナル：Entropy 
号数 : Volume 23 Issue 12
DOI : https://doi.org/10.3390/e23121592

Abstract : Due to the properties of DNA data storage, the errors that occur in DNA strands make error correction an important and challenging task. In this paper, a new code design of quaternary code suitable for DNA storage is proposed to correct at most two consecutive deletion or insertion errors. The decoding algorithms of the proposed codes are also presented when one and two deletion or insertion errors occur, and it is proved that the proposed code can correct at most two consecutive errors. Moreover, the lower and upper bounds on the cardinality of the proposed quaternary codes are also evaluated, then the redundancy of the proposed code is provided as roughly 2log48n.

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