Related papers: Multiset Deletion-Correcting Codes: Bounds and Con…
We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension $\mathrm{d}_\mathrm{L}$ in PI states using physical qudits of dimension $\mathrm{d}_\mathrm{P}$. We extend the Knill--Laflamme…
We give a complete determination of the exact optimal worst-case repair bandwidth and repair I/O for linear exact repair of $(n,n-2,2)$ MDS array codes over every finite field $\mathbb{F}_q$ and for every admissible code length $3\le n\le…
In this paper, we revisit the problem of finding the longest systematic-length $k$ for a linear minimum storage regenerating (MSR) code with optimal repair of only systematic part, for a given per-node storage capacity $l$ and an arbitrary…
We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\cal A}_q(9, 4; 3) \ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant…
Locally repairable codes enables fast repair of node failure in a distributed storage system. The code symbols in a codeword are stored in different storage nodes, such that a disk failure can be recovered by accessing a small fraction of…
In their 2007 book, Tsfasman and Vl\v{a}du\c{t} invite the reader to reinterpret existing coding theory results through the lens of projective systems. Redefining linear codes as projective systems provides a geometric vantage point. In…
In this work, we consider $q$-ary signature codes of length $k$ and size $n$ for a noisy adder multiple access channel. A signature code in this model has the property that any subset of codewords can be uniquely reconstructed based on any…
We study perfect error-correcting codes in $\mathbb{Z}^n$ for the symmetric limited-magnitude error channel, where at most $e$ coordinates of an integer vector may be altered by a value whose magnitude is at most $s$. Geometrically, such…
Recent interest on permutation rank modulation shows the Kendall tau metric as an important distance metric. This note documents our first efforts to obtain upper bounds on optimal code sizes (for said metric) ala Delsarte's approach. For…
Non-overlapping codes are a set of codewords such that the prefix of each codeword is not a suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword…
We examine the issue of separation and code design for networks that operate over finite fields. We demonstrate that source-channel (or source-network) separation holds for several canonical network examples like the noisy multiple access…
Transmit a codeword $x$, that belongs to an $(\ell-1)$-deletion-correcting code of length $n$, over a $t$-deletion channel for some $1\le \ell\le t<n$. Levenshtein, in 2001, proposed the problem of determining $N(n,\ell,t)+1$, the minimum…
In coding theory, handling errors that occur when symbols are inserted or deleted from a transmitted message is a long-standing challenge. Optimising redundancy for insertion and deletion channels remains a key open problem with significant…
We introduce a new analysis technique to derive a single-letter upper bound on the mismatch capacity of a stationary, single-user, memoryless channel with a decoding metric $q$. Our bound is obtained by considering a multicast transmission…
A code is said to be a $r$-local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most $r$ other coordinates. When some of the $r$ coordinates are also erased, the $r$-local LRC can not accomplish the…
A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes…
In this paper, we investigate binary reconstruction codes capable of correcting one deletion and one substitution. We define the \emph{single-deletion single-substitution ball} function $ \mathcal{B} $ as a mapping from a sequence to the…
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. In this paper, we address…
We propose a new interpolation-based error decoding algorithm for $(n,k)$ Reed-Solomon (RS) codes over a finite field of size $q$, where $n=q-1$ is the length and $k$ is the dimension. In particular, we employ the fast Fourier transform…
For nonnegative integers $q,n,d$, let $A_q(n,d)$ denote the maximum cardinality of a code of length $n$ over an alphabet $[q]$ with $q$ letters and with minimum distance at least $d$. We consider the following upper bound on $A_q(n,d)$. For…