M. D. S. codes and arcs in projective spaces: a survey

Le Matematiche. 1992;47(2):315-328

 

Journal Homepage

Journal Title: Le Matematiche

ISSN: 0373-3505 (Print); 2037-5298 (Online)

Publisher: Università degli Studi di Catania

Society/Institution: University of Catania

LCC Subject Category: Science: Mathematics

Country of publisher: Italy

Language of fulltext: Italian, French, English

Full-text formats available: PDF

 

AUTHORS

Joseph A. Thas

EDITORIAL INFORMATION

Peer review

Editorial Board

Instructions for authors

Time From Submission to Publication: 53 weeks

 

Abstract | Full Text

<p><span style="font-style: normal;"><span>Let </span></span><em><span>C</span></em><span style="font-style: normal;"><span> be a code of length </span></span><em><span>k</span></em><span style="font-style: normal;"><span> over an alphabet </span></span><em><span>A</span></em><span style="font-style: normal;"><span> of size </span></span><em><span>q </span></em><span>greather or equal</span><em><span> 2</span></em><span style="font-style: normal;"><span>. Having chosen </span></span><em><span>m</span></em><span style="font-style: normal;"><span> with </span></span><em><span>2 m  k</span></em><span style="font-style: normal;"><span> we impose the following condition on </span></span><em><span>C</span></em><span style="font-style: normal;"><span>: no two words agree in as many as </span></span><em><span>m</span></em><span style="font-style: normal;"><span> positions. It then follows that </span></span><em><span>|C| q</span></em><sup><em><span>m</span></em></sup><span style="font-style: normal;"><span>. If </span></span><em><span>|C|=q</span></em><sup><em><span>m</span></em></sup><span style="font-style: normal;"><span>, then </span></span><em><span>C</span></em><span style="font-style: normal;"><span> is called a Maximum Distance Separable code (M.D.S. code). A </span></span><em><span>k</span></em><span style="font-style: normal;"><span>-arc in </span></span><em><span>PG(n,q)</span></em><span style="font-style: normal;"><span> is a set </span></span><em><span>K</span></em><span style="font-style: normal;"><span> of </span></span><em><span>k</span></em><span style="font-style: normal;"><span> points with </span></span><em><span>k </span></em><span>at least</span><em><span> n+1</span></em><span style="font-style: normal;"><span> such that no </span></span><em><span> n+1</span></em><span style="font-style: normal;"><span> points lie in a hyperplane. It can be shown that arcs and linear M.D.S. codes are equivalent objects. Here we give a survey of important results on </span></span><em><span style="text-decoration: none;"><span>k</span></span></em><span style="font-style: normal;"><span style="text-decoration: none;"><span>-arcs, in particular we survey the answers to three fundamental problems on arcs posed by B. Segre in 1955.</span></span></span></p>