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Register a new userThe interplay of different metrics for the construction of constant dimension codes
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Added by
Sascha Kurz
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Sascha Kurz
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A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies $d_S(U,W):=2k-2dim(Ucap W)ge d$ for all pairs of different codewords $U$, $W$. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for $A_q(n,d;k)$ are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show up the potential for further improvements. As examples we give improved constructions for the cases $A_q(10,4;5)$, $A_q(11,4;4)$, $A_q(12,6;6)$, and $A_q(15,4;4)$. We also derive general upper bounds for subcodes arising in those constructions.
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Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$-ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special case). This discovery inspired some recent studies on length of an optimal locally recoverable code. It was shown in cite{LXY} that a $q$-ary optimal locally recoverable code is unbounded for $d=3,4$. Soon after, it was proved that a $q$-ary optimal locally recoverable code with distance $d$ and locality $r$ can have length $Omega_{d,r}(q^{1 + 1/lfloor(d-3)/2rfloor})$. Recently, an explicit construction of $q$-ary optimal locally recoverable codes for distance $d=5,6$ was given in cite{J18} and cite{BCGLP}. In this paper, we further investigate construction optimal locally recoverable codes along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and cite{J18}, we equip parity-check matrices with the Vandermond structure. It is turns out that a parity-check matrix with the Vandermond structure produces an optimal locally recoverable code must obey certain disjoint property for subsets of $mathbb{F}_q$. To our surprise, this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in cite{GXY} for $dgeq 7$. In addition, for $d=6$, we are able to remove the constraint required in cite{J18} that $q$ is even.
One of the main problems in random network coding is to compute good lower and upper bounds on the achievable cardinality of the so-called subspace codes in the projective space $mathcal{P}_q(n)$ for a given minimum distance. The determination of the exact maximum cardinality is a very tough discrete optimization problem involving a huge number of symmetries. Besides some explicit constructions for textit{good} subspace codes several of the most success full constructions involve the solution of discrete optimization subproblems itself, which mostly have not been not been solved systematically. Here we consider the multilevel a.k.a. Echelon--Ferrers construction and given lower and upper bounds for the achievable cardinalities. From a more general point of view, we solve maximum clique problems in weighted graphs, where the weights can be polynomials in the field size $q$.
Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[2^m+1, 2u-1, 2^m-2u+3]$ MDS codes for $1 leq u leq 2^{m-1}$, which are cyclic, reversible and BCH codes over $mathrm{GF}(2^m)$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, infinite families of $3$-designs from these quaternary codes are presented.
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
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.
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