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Register a new userA remark on ${mathbb F}_{q^n}$-Linear MRD codes
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Added by
Luca Giuzzi DPhil
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 well as parametric equations for MRD-codes of distance $d=n-k+1$.
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We determine the proportion of $[3times 3;3]$-MRD codes over ${mathbb F}_q$ within the space of all $3$-dimensional $3times3$-rank-metric codes over the same field. This shows that for these parameters MRD codes are sparse in the sense that the proportion tends to $0$ as $qrightarrowinfty$. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=F_{q}+vF_{q}+v^{2}F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $F_q$ and extend these to codes over $R$.
In this paper, we give conditions for the existence of Hermitian self-dual $Theta-$cyclic and $Theta-$negacyclic codes over the finite chain ring $mathbb{F}_q+umathbb{F}_q$. By defining a Gray map from $R=mathbb{F}_q+umathbb{F}_q$ to $mathbb{F}_{q}^{2}$, we prove that the Gray images of skew cyclic codes of odd length $n$ over $R$ with even characteristic are equivalent to skew quasi-twisted codes of length $2n$ over $mathbb{F}_q$ of index $2$. We also extend an algorithm of Boucher and Ulmer cite{BF3} to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over $mathbb{F}_q+umathbb{F}_q$.
In this paper, the investigation on the algebraic structure of the ring $frac{mathbb{F}_q[v]}{langle,v^q-v,rangle}$ and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $frac{mathbb{F}_q[v]}{langle,v^q-v,rangle}$ are given.
It is well known that linear rank-metric codes give rise to $q$-polymatroids. Analogously to classical matroid theory one may ask whether a given $q$-polymatroid is representable by a rank-metric code. We provide a partial answer by presenting examples of $q$-matroids that are not representable by ${mathbb F}_{q^m}$-linear rank-metric codes. We then go on and introduce deletion and contraction for $q$-polymatroids and show that they are mutually dual and that they correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated $q$-polymatroid.
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