Related papers: Linear complete symmetric rank-distance codes
In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the…
This preprint is of a chapter to appear in {\it Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and Applied Mathematics}, K.-U. Schmidt and A. Winterhof (eds.).…
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…
Maximum rank distance codes denoted MRD-codes are the equivalent in rank metric of MDS-codes. Given any integer $q$ power of a prime and any integer $n$ there is a family of MRD-codes of length $n$ over $\FF{q^n}$ having polynomial-time…
The projective space of order $n$ over a finite field $\F_q$ is a set of all subspaces of the vector space $\F_q^{n}$. In this work, we consider error-correcting codes in the projective space, focusing mainly on constant dimension codes. We…
Linearized Reed-Solomon (LRS) codes form an important family of maximum sum-rank distance (MSRD) codes that generalize both Reed--Solomon codes and Gabidulin codes. In this paper we study the equivalence problem for LRS codes and determine…
Constructions of distance-optimal codes and quasi-perfect codes are challenging problems and have attracted many attentions. In this paper, we give the following three results. 1) If $\lambda|q^{sm}-1$ and $\lambda…
We present the theory of linear rank-metric codes from the point of view of their fundamental parameters. These are: the minimum rank distance, the rank distribution, the maximum rank, the covering radius, and the field size. The focus of…
We provide a geometric characterization of $k$-dimensional $\mathbb{F}_{q^m}$-linear sum-rank metric codes as tuples of $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We then use this characterization to study one-weight codes in the…
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…
Sum-rank metric codes, as a generalization of Hamming codes and rank metric codes, have important applications in fields such as multi-shot linear network coding, space-time coding and distributed storage systems. The purpose of this study…
We revisit and extend the connections between $\mathbb{F}_{q^m}$-linear rank-metric codes and evasive $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We give a unifying framework in which we prove in an elementary way how the parameters…
We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise…
By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new…
In this note, we provide a description of the elements of minimum rank of a generalized Gabidulin code in terms of Grassmann coordinates. As a consequence, a characterization of linearized polynomials of rank at most $n-k$ is obtained, as…
In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on $\mathbb{S}_{n}(\mathbb{F}_{q})$, the space of $n\times n$ symmetric matrices over $\mathbb{F}_{q}$. Using this…
The rank of a scattered $\mathbb{F}_q$-linear set of $\mathrm{PG}(r-1,q^n)$, $rn$ even, is at most $rn/2$ as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of $r$,…
We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have…
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…
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…