Related papers: New MRD codes from linear cutting blocking sets
In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed-Solomon block codes to…
Just as the Hamming weight spectrum of a linear block code sheds light on the performance of a maximum likelihood decoder, the pseudo-weight spectrum provides insight into the performance of a linear programming decoder. Using properties of…
An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J.…
We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical…
Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram…
Coset constructions of $q$-ary Reed-Muller codes can be used to store secrets on a distributed storage system in such a way that only parties with access to a large part of the system can obtain information while still allowing for local…
Two skew cyclic codes can be equivalent for the Hamming metric only if they have the same length, and only the zero code is degenerate. The situation is completely different for the rank metric, where lengths of codes correspond to the…
A q-ary linear code of dimension k is called a maximum weight spectrum (MWS) code if it has the maximum possible number (viz. (q^k-1)/(q-1)) of different non-zero weights. We construct MWS codes from quasi-minimal codes, thus obtaining of…
Let $p$ be a prime such that $p \equiv 2$ or $3$ mod $5$. Linear block codes over the non-commutative matrix ring of $2 \times 2$ matrices over the prime field $GF(p)$ endowed with the Bachoc weight are derived as isometric images of linear…
New families of maximum distance separable (MDS) codes are constructed from elliptic curves by exploiting their group structures. In contrast to classical constructions based on divisors supported at a single rational point, the proposed…
An $\mathbb{F}_q$-linear code of minimum distance $d$ is called complete if it is not contained in a larger $\mathbb{F}_q$-linear code of minimum distance $d$. In this paper, we classify $\mathbb{F}_q$-linear complete symmetric…
Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of linear codes -- the generalized covering radius. The generalized covering-radius hierarchy of a…
A cyclic codes of length $n$ over the rings $Z_{2^{m}}$ of integer of modulo $2^{m}$ is a linear code with property that if the codeword $(c_0,c_1,...,c_{n-1})\in \mathcal{C}$ then the cyclic shift $(c_1,c_2,...,c_0)\in \mathcal{C}$.…
Linearized Reed-Solomon (LRS) codes are evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary…
Algebraic methods for the design of series of maximum distance separable (MDS) linear block and convolutional codes to required specifications and types are presented. Algorithms are given to design codes to required rate and required…
By using the notion of $d$-embedding $\Gamma$ of a (canonical) subgeometry $\Sigma$ and of exterior set with respect to the $h$-secant variety $\Omega_{h}(\mathcal{A})$ of a subset $\mathcal{A}$, $ 0 \leq h \leq n-1$, in the finite…
Linear codes have diverse applications in secret sharing schemes, secure two-party computation, association schemes, strongly regular graphs, authentication codes and communication. There are a large number of linear codes with few weights…
We consider the decoding of rank metric codes assuming the error matrix is symmetric. We prove two results. First, for rates $<1/2$ there exists a broad family of rank metric codes for which any symmetric error pattern, even of maximal rank…
For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to making short cycles. Also, the existing high-rate quasi-cyclic…
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then…