Counting generalized Reed-Solomon codes

Peter Beelen; David Glynn; Tom Høholdt; Krishna Kaipa

doi:10.3934/amc.2017057

Article Contents

2017, Volume 11, Issue 4: 777-790. Doi: 10.3934/amc.2017057

This issue Previous Article On the performance of optimal double circulant even codes Next Article Quadratic residue codes over

$\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$

Counting generalized Reed-Solomon codes

1.
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, DK-2800 Kgs. Lyngby, Denmark
2.
School of Computer Science, Engineering and Mathematics, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia
3.
Department of Mathematics King Abdulaziz University, Jeddah 21859, Saudi Arabia
4.
Department of Mathematics, Indian Institute of Science Education and Research (IISER), Dr. Homi Bhabha Road, Pashan, Pune 411008, India

Received: April 30, 2016

Published: June 2018

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Abstract

In this article we count the number of $[n, k]$ generalized Reed-Solomon (GRS) codes, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of $[n, 3]$ MDS codes with $n=6, 7, 8, 9$.

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Mathematics Subject Classification: Primary: 94B27, 51E21.

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Table 1. Number of GRS and MDS codes of small length and dimension 3

$q$	$n$	$\#$ GRS	$\#$ MDS
$4$	$6$	$486$	$486$
$5$	$6$	$6144$	$6144$
$7$	$6$	$466560$	$1088640$
$7$	$7$	$5598720$	$5598720$
$7$	$8$	$33592320$	$33592320$
$8$	$6$	$2016840$	$6554730$
$8$	$7$	$42353640$	$141178800$
$8$	$8$	$592950960$	$2964754800$
$8$	$9$	$4150656720$	$41506567200$
$8$	$10$	$290545970400$	$290545970400$
$9$	$6$	$6881280$	$28901376$
$9$	$7$	$220200960$	$1604321280$
$9$	$8$	$5284823040$	$15854469120$
$9$	$9$	$84557168640$	$84557168640$
$9$	$10$	$676457349120$	$676457349120$

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