Related papers: Algebraic structures of MRD Codes
In this work we investigate unions of lifted MRD codes of a fixed dimension and minimum distance and derive an explicit formula for the cardinality of such codes. This will then imply a lower bound on the cardinality of constant dimension…
We provide an algorithm to construct unitary matrices over finite fields. We present various constructions of Hermitian self-dual code by means of unitary matrices, where some of them generalize the quadratic double circulant constructions.…
A basic problem for the constant dimension subspace coding is to determine the maximal possible size A_q (n, d, k) of a set of k-dimensional subspaces in Fnq such that the subspace distance satisfies d(U, V )> or =d for any two different…
Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in…
We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods…
We present an explicit construction of a sequence of rate $1/2$ Wozencraft ensemble codes (over any fixed finite field $\mathbb{F}_q$) that achieve minimum distance $\Omega(\sqrt{k})$ where $k$ is the message length. The coefficients of the…
Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly…
We determine the automorphism group of Gabidulin codes of full length and characterise when these codes are equivalent to self-dual codes.
For each odd prime power $q$, let $4 \leq n\leq q^{2}+1$. Hermitian self-orthogonal $[n,2,n-1]$ codes over $GF(q^{2})$ with dual distance three are constructed by using finite field theory. Hence, $[[n,n-4,3]]_{q}$ quantum MDS codes for $4…
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…
The study of MDS self-dual codes has attracted lots of attention in recent years. There are many papers on determining existence of $q-$ary MDS self-dual codes for various lengths. There are not existence of $q-$ary MDS self-dual codes of…
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…
Maximum distance separable (in short, MDS), near MDS (in short, NMDS), and self-orthogonal codes play a pivotal role in algebraic coding theory, particularly in applications such as quantum communications and secret sharing scheme.…
Sum-rank-metric codes have wide applications in universal error correction, multishot network coding, space-time coding and the construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric…
In this note, we give a construction of codes on algebraic function field $F/ \mathbb{F}_{q}$ using places of $F$ (not necessarily of degree one) and trace functions from various extensions of $\mathbb{F}_{q}$. This is a generalization of…
Self-dual maximum distance separable codes (self-dual MDS codes) and self-dual near MDS codes are very important in coding theory and practice. Thus, it is interesting to construct self-dual MDS or self-dual near MDS codes. In this paper,…
Given $\mathbb{F}_q$ the finite field with $q$ elements and an integer $n\geq 2$, a flag is a sequence of nested subspaces of $\mathbb{F}_q^n$ and a flag code is a nonempty set of flags. In this context, the distance between flags is the…
In this paper, we consider the Reed-Muller (RM) codes. For the first order RM code, we prove that it is unique in the sense that any linear code with the same length, dimension and minimum distance must be the first order RM code; For the…
Let $n_q(M,d)$ be the minimum length of a $q$-ary code of size $M$ and minimum distance $d$. Bounding $n_q(M,d)$ is a fundamental problem that lies at the heart of coding theory. This work considers a generalization $n^\bx_q(M,d)$ of…
This paper is concerned with some Algebraic Geometry codes on Jacobians of genus 2 curves. We derive a lower bound for the minimum distance of these codes from an upper "Weil type" bound for the number of rational points on irreducible…