Related papers: Rate $(n-1)/n$ Systematic MDS Convolutional 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…
The singleton defect of an $[n,k,d]$ linear code ${\cal C}$ is defined as $s({\cal C})=n-k+1-d$. Codes with $S({\cal C})=0$ are called maximum distance separable (MDS) codes, and codes with $S(\cal C)=S(\cal C ^{\bot})=1$ are called near…
In \textit{Distributed Storage Systems} (DSSs), usually, data is stored using replicated packets on different chunk servers. Recently a new paradigm of \textit{Fractional Repetition} (FR) codes have been introduced, in which, data is…
We investigate when a maximum distance separable ($MDS$) code over $F_q$ is also completely regular ($CR$). For lengths $n=q+1$ and $n=q+2$ we provide a complete classification of the $MDS$ codes that are $CR$ or at least uniformly packed…
A $q$-ary maximum distance separable (MDS) code $C$ with length $n$, dimension $k$ over an alphabet $\mathcal{A}$ of size $q$ is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the Hamming distance between two…
Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that…
Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters…
Maximum distance separable (MDS) array codes are widely employed in modern distributed storage systems to provide high data reliability with small storage overhead. Compared with the data access latency of the entire file, the data access…
An erasure channel with a fixed alphabet size $q$, where $q \gg 1$, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming…
This paper presents a construction for high-rate MDS codes that enable bandwidth-efficient repair of a single node. Such MDS codes are also referred to as the minimum storage regenerating (MSR) codes in the distributed storage literature.…
Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most…
Maximum distance separable (MDS) codes are widely used in distributed storage systems as they provide optimal fault tolerance for a given amount of storage overhead. The seminal work of Dimakis~\emph{et al.} first established a lower bound…
In order to correct the pair-errors generated during the transmission of modern high-density data storage that the outputs of the channels consist of overlapping pairs of symbols, a new coding scheme named symbol-pair code is proposed. The…
The Doob graph $D(m,n)$, where $m>0$, is the direct product of $m$ copies of The Shrikhande graph and $n$ copies of the complete graph $K_4$ on $4$ vertices. The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as…
A $q$-ary code of length $n$, size $M$, and minimum distance $d$ is called an $(n,M,d)_q$ code. An $(n,q^{k},n-k+1)_q$ code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified.…
Abundant high-rate (n, k) minimum storage regenerating (MSR) codes have been reported in the literature. However, most of them require contacting all the surviving nodes during a node repair process, resulting in a repair degree of d=n-1.…
We study the construction of $d$-deletion-correcting binary codes by formulating the problem as a Maximum Clique Problem (MCP). In this formulation, vertices represent candidate codewords and edges connect pairs whose longest common…
This paper studies the decoding capabilities of maximum distance profile (MDP) convolutional codes over the erasure channel and compares them with the decoding capabilities of MDS block codes over the same channel. The erasure channel…
An $[n,k,d]$ linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) if $d=n-k+1$ or $d=n-k$, respectively. If a code and its dual code are both AMDS, then the code is said to be near maximum…
An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters [n,k,d] and the list size L. L-MDS codes are then defined as codes that attain this bound (under a slightly…