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On $q$-Steiner systems from rank metric codes


Arias, Francisco; de la Cruz, Javier; Rosenthal, Joachim; Willems, Wolfgang (2018). On $q$-Steiner systems from rank metric codes. Discrete Mathematics, 341(10):2729-2734.

Abstract

In this paper we prove that rank metric codes with special properties imply the existence of $q$-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a [2$d$,$d$, $d$]dually almost MRD code $C \leq \mathbb{F}^{2d}_{g^m} (2d \leq m )$ which has no code words of rank weight $a + 1$ form a $q$-Steiner system $S (d -1, d, 2d)_q$. This is the q-analog of a result in classical coding theory and it may be seen as a first step to prove a q-analog of the famous Assmus–Mattson Theorem.

Abstract

In this paper we prove that rank metric codes with special properties imply the existence of $q$-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a [2$d$,$d$, $d$]dually almost MRD code $C \leq \mathbb{F}^{2d}_{g^m} (2d \leq m )$ which has no code words of rank weight $a + 1$ form a $q$-Steiner system $S (d -1, d, 2d)_q$. This is the q-analog of a result in classical coding theory and it may be seen as a first step to prove a q-analog of the famous Assmus–Mattson Theorem.

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Additional indexing

Item Type:Journal Article, not_refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Theoretical Computer Science, Discrete Mathematics and Combinatorics
Language:English
Date:1 October 2018
Deposited On:23 Jan 2019 15:19
Last Modified:22 Feb 2020 08:47
Publisher:Elsevier
ISSN:0012-365X
OA Status:Closed
Publisher DOI:https://doi.org/10.1016/j.disc.2018.06.034
Project Information:
  • : FunderSNSF
  • : Grant ID200020_169510
  • : Project TitleAlgebraic Constructions and Decoding of Rank Metric Codes with Applications to Network Coding and Code based Cryptography

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