相关论文: Upper bounds on the rate of linear $q$-ary $k$-has…

We develop an algorithm for computing the weight distribution of a linear $[n,k]$ code over a finite field $\mathbb{F}_q$. We represent the codes by their characteristic vector with respect to a given generator matrix and a special type of…

Let $k\leq n$ be two positive integers and $q$ a prime power. The basic question in minimal linear codes is to determine if there exists an $[n,k]_q$ minimal linear code. The first objective of this paper is to present a new sufficient and…

New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths.…

We define a metric on $\mathbb{F}_q^n$ using the linear complexity of finite sequences. We will then develop a coding theory for this metric. We will give a Singleton-like bound and we will give constructions of subspaces of…

Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely…

Let $q=p^\alpha$ be a fixed prime power, $k\geq 2$ be an integer. We give a new upper bound for the size of $k$-wise $q$-modular $L$-avoiding $L$-intersecting set systems, where $L$ is any proper subset of $\{0, \ldots , q-1\}$. Our proof…

The class of k-bit delay decodable codes, source codes allowing decoding delay of at most k bits for k >= 0, can attain a shorter average codeword length than Huffman codes. This paper discusses the general properties of the class of k-bit…

A code is called a $q$-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index $i$ and a received word $w$ close to an encoding of a message $x$, outputs $x_i$ by querying only at most $q$…

A realistic Quantum Key Distribution (QKD) protocol necessarily deals with finite resources, such as the number of signals exchanged by the two parties. We derive a bound on the secret key rate which is expressed as an optimization problem…

Given a finite relational language $\mathcal{L}$, a hereditary $\mathcal{L}$-property is a class of finite $\mathcal{L}$-structures closed under isomorphism and substructure. The speed of $\mathcal{H}$ is the function which sends an integer…

For every natural number $n\geq 2$ and every finite sequence $L$ of natural numbers, we consider the set $UD_n(L)$ of all uniquely decodable codes over an $n$-letter alphabet with the sequence $L$ as the sequence of code word lengths, as…

A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over…

High-rate quantum error correcting (QEC) codes encode many logical qubits in a given number of physical qubits, making them promising candidates for quantum computation. Implementing high-rate codes at a scale that both frustrates classical…

We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold…

After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we…

Separable codes were introduced to provide protection against illegal redistribution of copyrighted multimedia material. Let $\mathcal{C}$ be a code of length $n$ over an alphabet of $q$ letters. The descendant code ${\sf…

Lifted codes are a class of evaluation codes attracting more attention due to good locality and intermediate availability. In this work we introduce and study quadratic-curve-lifted Reed-Solomon (QC-LRS) codes, where the codeword symbols…

Let $\mathscr{S}_n(q)$ denote the set of symmetric bilinear forms over an $n$-dimensional $\mathbb{F}_q$-vector space. A subset $\mathcal{C}$ of $\mathscr{S}_n(q)$ is called a $d$-code if the rank of $A-B$ is larger than or equal to $d$ for…

Traditional error-correcting codes (ECCs) assume a fixed message length, but many scenarios involve ongoing or indefinite transmissions where the message length is not known in advance. For example, when streaming a video, the user should…

We prove an upper bound on the Shannon capacity of a graph via a linear programming variation. We show that our bound can outperform both the Lov\'asz theta number and the Haemers minimum rank bound. As a by-product, we also obtain a new…