Related papers: Shortest LCD embeddings of binary, ternary and qua…
Recent efforts in coding theory have focused on building codes for insertions and deletions, called insdel codes, with optimal trade-offs between their redundancy and their error-correction capabilities, as well as efficient encoding and…
Recently, simplicial complexes are used in constructions of several infinite families of minimal and optimal linear codes by Hyun {\em et al.} Building upon their research, in this paper more linear codes over the ring $\mathbb{Z}_4$ are…
Constructions of optimal locally repairable codes (LRCs) in the case of $(r+1) \nmid n$ and over small finite fields were stated as open problems for LRCs in [I. Tamo \emph{et al.}, "Optimal locally repairable codes and connections to…
Locally repairable codes (LRCs) have recently been widely used in distributed storage systems and the LRCs with $(r,\delta)$-locality ($(r,\delta)$-LRCs) attracted a lot of interest for tolerating multiple erasures. Ge et al. constructed…
Linear complementary dual (LCD) codes over finite fields are linear codes satisfying $C\cap C^{\perp}=\{0\}$. We generalize the LCD codes over finite fields to $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes over the ring…
One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in…
We prove that every $1$-error-correcting code over a finite field can be embedded in a $1$-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting $1$-perfect code and can be…
We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove: (1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design…
In a digital communication system, information is sent from one place to another over a noisy communication channel. It may be possible to detect and correct errors that occur during the transmission if one encodes the original information…
We construct constant-sized ensembles of linear error-correcting codes over any fixed alphabet that can correct a given fraction of adversarial erasures at rates approaching the Singleton bound arbitrarily closely. We provide several…
In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However,…
Locally Decodable Codes (LDCs) are error correcting codes which permit the recovery of any single message symbol with a low number of queries to the codeword (the locality). Traditional LDC tradeoffs between the rate, locality, and error…
Certain simplicial complexes are used to construct a subset $D$ of $\mathbb{F}_{2^n}^m$ and $D$, in turn, defines the linear code $C_{D}$ over $\mathbb{F}_{2^n}$ that consists of $(v\cdot d)_{d\in D}$ for $v\in \mathbb{F}_{2^n}^m$. Here we…
In this paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, called interior point decoding, is designed for linear vector channels. The linear vector…
In this article, we construct linear codes over the commutative non-unital ring $I$ of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are…
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and…
A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\perp}$ satisfies $C \cap C^{\perp}$= $\left\{ \mathbf{0}\right\} $. Let $LCD{[}n,k{]}$ denote the maximum of possible values of…
LCD codes are linear codes that intersect with their dual trivially. Quasi cyclic codes that are LCD are characterized and studied by using their concatenated structure. Some asymptotic results are derived. Hermitian LCD codes are…
Codes considered as structures within unit schemes greatly extends the availability of linear block and convolutional codes and allows the construction of these codes to required length, rate, distance and type. Properties of a code emanate…
In this paper, we introduce a general construction of linear codes with small dimension hull from any non LCD codes. Furthermore, we show that for any linear code $\Co$ over $\F_q$ ($q > 3$) with $dim(Hull(\Co))=h$ there exist an equivalent…