$ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $

Dean Crnković; Nina Mostarac; Bernardo G. Rodrigues; Leo Storme

doi:10.3934/amc.2020075

Article Contents

2021, Volume 15, Issue 3: 423-440. Doi: 10.3934/amc.2020075

This issue Previous Article The $[46, 9, 20]_2$ code is unique Next Article Finding small solutions of the equation

$ \mathit{{Bx-Ay = z}} $

and its applications to cryptanalysis of the RSA cryptosystem

$ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $

1.
Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia
2.
Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0002, Pretoria, South Africa
3.
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Ghent, Belgium

Received: September 2019

Early access: April 2020

Published: August 2021

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Abstract

In this paper we construct $ 2 $-PD-sets of $ 16 $ elements for codes from the Desarguesian projective planes $ \mathrm{PG}(2,q) $, where $ q = 2^h $ and $ 5\leq h \leq 9 $. We also construct $ 3 $-PD-sets of $ 75 $ elements for the code from the Desarguesian projective plane $ \mathrm{PG}(2,q) $, where $ q = 2^9 $. These $ 2 $-PD-sets and $ 3 $-PD-sets can be used for partial permutation decoding of codes obtained from the Desarguesian projective planes.

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Mathematics Subject Classification: Primary: 51E20, 94B05.

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Table 1. Codes of $ \mathrm{PG}(2,q) $: lower bounds on sizes of PD-sets and $ 2 $-PD-sets

$ q $	Code	$ t $	$ r $	$ b $	$ b_2 $
$ 32 $	[1057,244, 33]	16	813	180	3
$ 64 $	[4161,730, 65]	32	3431	1623	3
$ 128 $	[16513, 2188,129]	64	14325	40696	3
$ 256 $	[65793, 6562,257]	128	59231	3965945	3
$ 512 $	[262657, 19684,513]	256	242973	3625171287	3

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References

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