On Single-Deletion-Correcting Codes

Quantum Deletion Codes derived from Classical Deletion Codes (Extended Abstract)

This manuscript is an extended abstract version of the paper entitled ``Quantum Deletion Codes derived from Classical Deletion Codes.'' The paper contributes to the fundamental theory for quantum deletion error-correcting codes. The paper proposes a code construction condition for a partition of classical deletion error-correcting codes to derive quantum deletion error-correcting codes. The construction methods in this paper give examples of quantum codes that can correct single-quantum deletion errors and have a code rate arbitrarily close to 1, while the previously known quantum deletion cod…

Efficient Linear and Affine Codes for Correcting Insertions/Deletions

This paper studies \emph{linear} and \emph{affine} error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to have information rate at most $1/2$ (achieved by the trivial 2-fold repetition code). Previously, it was (erroneously) reported that more generally no non-trivial linear codes correcting $k$ deletions exist, i.e., that the $(k+1)$-fold repetition codes and its rate of $1/(k+1)$ are basically optimal for any $k$. We disprove this and show…

Non-asymptotic Upper Bounds for Deletion Correcting Codes

Explicit non-asymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for $q$-ary alphabet and string length $n$ is shown to be of size at most $\frac{q^n-q}{(q-1)(n-1)}$. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the …

Deletion-correcting codes and dominant vectors

In this paper we describe all pairs of binary vectors $({\bf u}, {\bf v})$ such that the set of vectors obtained by $t$ deletions in ${\bf v}$ is a subset of the set of vectors obtained by $t$ deletions in ${\bf u}$ for $t=1,2$. Such pairs play an important role for finding the value of $L_2(n,t)$, the maximum cardinality of binary $t$-deletion-correcting code of length $n$.

Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions

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, 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: …

Optimal codes in deletion and insertion metric

We improve the upper bound of Levenshtein for the cardinality of a code of length 4 capable of correcting single deletions over an alphabet of even size. We also illustrate that the new upper bound is sharp. Furthermore we will construct an optimal perfect code capable of correcting single deletions for the same parameters.

纠正多次删除错误的置换码与多重置换码

Permutation and Multi-permutation Codes Correcting Multiple Deletions

本研究改进了Ulam度量下可纠多次删除错误的置换码的Gilbert–Varshamov界，提出基于新映射的低冗余置换码构造方法并拓展至多置换场景，得到当前最优的Ulam度量多重置换码。

通过子串计数的线性哈希实现多项式更大的删除码

Polynomially larger deletion codes by linear hashing of substring counts

本文提出基于子串计数线性哈希的删除码构造，将纠正两次删除的冗余度上界从4log2n改进为3log2n，并推广到t次删除，通过标签类提取和二染色处理混淆词。

Efficient Systematic Deletions/Insertions of $0$'s Error Control Codes and the $L_{1}$ Metric (Extended version)

This paper gives some theory and efficient design of binary block systematic codes capable of controlling the deletions of the symbol ``$0$'' (referred to as $0$-deletions) and/or the insertions of the symbol ``$0$'' (referred to as $0$-insertions). The problem of controlling $0$-deletions and/or $0$-insertions (referred to as $0$-errors) is known to be equivalent to the efficient design of $L_{1}$ metric asymmetric error control codes over the natural alphabet, $\mathbb{N}$. So, $t$ $0$-insertion correcting codes can actually correct $t$ $0$-errors, detect $(t+1)$ $0$-errors and, simultaneous…

Weight Enumerators and Cardinalities for Number-Theoretic Codes

The number-theoretic codes are a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts' codes, correcting single insertion or deletion. Moreover…

Perfect Multi Deletion Codes Achieve the Asymptotic Optimality of Code Size

This paper studies on the cardinality of perfect multi deletion binary codes. The lower bound for any perfect deletion code with the fixed code length and the number of deletions, and the asymptotic achievable of Levenshtein's upper bound are shown.

Perfect Codes for Generalized Deletions from Minuscule Elements of Weyl Groups

This paper discusses a connection between insertion/deletion (ID) codes and minuscule elements of Weyl groups.

插入-删除码的改进Johnson型界

Improved Johnson-type Bounds for Insertion-Deletion Codes

通过将每个局部列表编码为二进制常重码，改进了Hayashi和Yasunaga关于插入-删除码的Johnson型界，并利用McEliece-Rodemich-Rumsey-Welch界得到渐近速率界。

Rate-Distance Trade-offs for List-Decodable Insertion-Deletion Codes

This paper presents general bounds on the highest achievable rate for list-decodable insertion-deletion codes. In particular, we give novel outer and inner bounds for the highest achievable communication rate of any insertion-deletion code that can be list-decoded from any $γ$ fraction of insertions and any $δ$ fraction of deletions. Our bounds simultaneously generalize the known bounds for the previously studied special cases of insertion-only, deletion-only, and zero-rate and correct other bounds that had been reported for the general case.

关于Varshamov-Tenengolts码与Helberg码的Naisargik映像研究

On Naisargik Images of Varshamov-Tenengolts and Helberg Codes

本文针对四元VT码与Helberg码，提出Naisargik（自然）映射与逆映射，证明其可提升两类码的删除纠错能力，拓展了$\boldsymbol{\text{Z}}_4$上码的Gray映像相关研究思路。

Covering Codes using Insertions or Deletions

A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most prior work on covering codes has focused on the Hamming metric, we consider the problem of designing covering codes defined in terms of either insertions or deletions. First, we provide new sphere-covering lower bounds on the minimum possible size of such codes. Then, we provide new existential upper bounds on the size of optimal covering codes for a single in…

An improved bound on the fraction of correctable deletions

We consider codes over fixed alphabets against worst-case symbol deletions. For any fixed $k \ge 2$, we construct a family of codes over alphabet of size $k$ with positive rate, which allow efficient recovery from a worst-case deletion fraction approaching $1-\frac{2}{k+\sqrt k}$. In particular, for binary codes, we are able to recover a fraction of deletions approaching $1/(\sqrt 2 +1)=\sqrt 2-1 \approx 0.414$. Previously, even non-constructively the largest deletion fraction known to be correctable with positive rate was $1-Θ(1/\sqrt{k})$, and around $0.17$ for the binary case. Our result pi…

所有列表可解码删除码的下界

Lower Bounds for all List-Decodable Deletion Codes

本文证明了所有列表可解码删除码的最优大小下界，改进已知结果，并给出首个非平凡下界，同时证明公共子序列和超序列数量的上界。

所有距离下对数因子更大的删除码

Logarithmically larger deletion codes of all distances

本文改进了删除码的渐近界，证明存在规模为 $\Omega_k(2^n \log n/n^{2k})$ 的 $k$-删除码，受 Jiang 和 Vardy 方法启发，并给出最长公共子序列和最短公共超序列数量的相关结果。