Maximum scattered linear sets and MRD-codes
By Bence Csajbók, Giuseppe Marino, Olga Polverino, Ferdinando Zullo. Journal of Algebraic Combinatorics, 2017
76 citations recorded by the source corpus.
The rank of a scattered Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb F}_q$$\end{document}-linear set of PG(r-1,qn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{{PG}}}}(r-1,q^n)$$\end{document}, 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, n, q (rn even) for scattered Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb F}_q$$\end{document}-linear sets of rank rn / 2. In this paper, we prove that the bound rn / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb F}_q$$\end{document}-linear sets of PG(1,qn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{{PG}}}}(1,q^n)$$\end{document} of maximum rank n yield Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb F}_q$$\end{document}-linear MRD-codes with dimension 2n and minimum distance n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document}. We generalize this result and show that scattered Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb F}_q$$\end{document}-linear sets of PG(r-1,qn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{{PG}}}}(r-1,q^n)$$\end{document} of maximum rank rn / 2 yield Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb F}_q$$\end{document}-linear MRD-codes with dimension rn and minimum distance n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document}.
Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering
View Paper: https://www.semanticscholar.org/paper/f3fbe1f004cae4fc2dceae23f3d0919e278cbe6c
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Artificial Intelligence
In Discrete Mathematics and Combinatorics
In Electrical and Electronic Engineering
Citations in the corpus
- Exceptional scattered polynomials (2017, background): “On the other hand there are several recent constructions of MRD codes, which were proved to be inequivalent to Gabidulin codes [9, 12, 15, 24, 37, 41, 47].”
- New semifields and new MRD codes from skew polynomial rings (2018, background): “Motivation for many of these works arose due to the connection between MRD codes and maximum scattered subspaces; see, for example, [8, 44, 45].”
- MRD Codes: Constructions and Connections (2019, background): “Equivalence classes of maximum scattered subspaces with respect to the Desarguesian n -spread in V (2 n, q ) are in one-to-one correspondence with equivalence classes of F q n -linear MRD codes in M n × n ( F q ) with minimum distance n − 1 Theorem 10 ([17]) .”