Related papers: Embedding in $q$-ary $1$-perfect codes and partiti…
We present error-correcting codes that achieve the information-theoretically best possible trade-off between the rate and error-correction radius. Specifically, for every $0 < R < 1$ and $\eps> 0$, we present an explicit construction of…
We study codes that are list-decodable under insertions and deletions. Specifically, we consider the setting where a codeword over some finite alphabet of size $q$ may suffer from $\delta$ fraction of adversarial deletions and $\gamma$…
Error-correction codes are central for fault-tolerant information processing. Here we develop a rigorous framework to describe various coding models based on quantum resource theory of superchannels. We find, by treating codings as…
We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of…
Codes that can correct up to $t$ symmetric errors and detect all unidirectional errors, known as $t$-EC-AUED codes, are studied in this paper. Given positive integers $q$, $a$ and $t$, let $n_q(a,t+1)$ denote the length of the shortest…
Tandem duplication is the process of inserting a copy of a segment of DNA adjacent to the original position. Motivated by applications that store data in living organisms, Jain et al. (2017) proposed the study of codes that correct tandem…
Mixed codes, which are error-correcting codes in the Cartesian product of different-sized spaces, model degrading storage systems well. While such codes have previously been studied for their algebraic properties (e.g., existence of perfect…
We consider the embedding problem in coding theory: given an independence (a code-related property) and an independent language $L$, find a maximal independent language containing $L$. We consider the case where the code-related property is…
From the set of operators for errors and its correction code, we introduce the so-called complete unitary transformation. It can be used for encoding while the inverse of it can be applied for correcting the errors of the encoded qubit. We…
We study data structures in the presence of adversarial noise. We want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object, even if the data structure has been…
In steganography, we always hope to maximize the embedding payload subject to an upper-bounded distortion. We need suitable distortion measurement to evaluate the embedding impact. However, different distortion functions exposes different…
A function $\varphi:\{0,1\}^n \to \{0,1\}^N$ is called an isometric embedding of the $n$-dimensional Hamming metric space to the $N$-dimensional edit metric space if, for all $x,y\in\{0,1\}^n$, the Hamming distance between $x$ and $y$ is…
One peculiarity with deletion-correcting codes is that perfect $t$-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius $t$ with respect to the…
A tiling of the $n$-dimensional Hamming cube gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on the $n$-dimensional Hamming cube which are determined by a weight…
We study a discrete model of repelling particles, and we show using linear programming bounds that many familiar families of error-correcting codes minimize a broad class of potential energies when compared with all other codes of the same…
Designing quantum error correcting codes that promise a high error threshold, low resource overhead and efficient decoding algorithms is crucial to achieve large-scale fault-tolerant quantum computation. The concatenated quantum Hamming…
This paper provides a new instance of quantum deletion error-correcting codes. This code can correct any single quantum deletion error, while our code is only of length 4. This paper also provides an example of an encoding quantum circuit…
Entanglement-assisted quantum error-correcting codes (EAQECCs) to desired rate, error-correcting capability and maximum shared entanglement are constructed. Thus for a required rate $R$, required error-correcting capability to correct $t$…
A (tandem) duplication of length $ k $ is an insertion of an exact copy of a substring of length $ k $ next to its original position. This and related types of impairments are of relevance in modeling communication in the presence of…
Error-correcting codes are usually envisioned to counter errors by operating unitary corrections depending on the projective measurement results of some syndrome observables. We here propose a way to use them in a more integrated way, where…