A generalization of the normal rational curve in \(\mathrm {PG}(d,q^n)\) and its associated non-linear MRD codes
type: Item
instance of: scholarly article
title: A generalization of the normal rational curve in \(\mathrm {PG}(d,q^n)\) and its associated non-linear MRD codes
zbMATH Open document ID: 1393.51005
author: Giorgio Donati
author: Nicola Durante
published in: Designs, Codes and Cryptography
publication date: 2018-05-24T00:00:00Z
review text: A rank distance code is a subset of \(M_{m,n}(F_q)\) where the distance between two matrices is the rank of their difference. The size of an \((m,n,q;s)\)-rank distance code \(\chi\) satisfies the following Singleton type bound:\(|\chi | \leq q^{n(m-s)}\). Codes meeting this bound are known as maximum rank distance codes (MRD codes). Take two distinct points \(A,B\) of a \(d\)-dimensional projective space and a collineation \(\Phi\) between the stars of lines with these two points as vertices such that the map does not send the line containing the points to itself. The authors prove that if \(d=2\) or \(d \geq 3\) and the lines \(\Phi^{-1}(AB), AB, \Phi(AB)\) are not in a common plane, then the set \(C\) of points of intersection of corresponding lines under \(\Phi\) is the union of \(q-1\) scattered \(F_q\) linear sets of rank \(n\) together with \(\{A,B\}\). This set \(C\) is used to construct infinite families of non-linear \((d+1,n,q;d-1)\)-MRD codes, with \(d\leq n-1\).
reviewed by: Steven T. Dougherty
Mathematics Subject Classification ID: 51E99
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 6872536
zbMATH Keywords: linear sets
zbMATH Keywords: collineations
zbMATH Keywords: maximum rank distance codes (MRD codes)
MaRDI profile type: Publication
full work available at URL: s10623-017-0388-7
OpenAlex ID: W2726427535
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DOI: 10.1007/S10623-017-0388-7
description: scientific article; zbMATH DE number 6872536