Top

Designs, Codes and Cryptography

Published in:

01-06-2015

On small line sets with few odd-points

Author: Peter Vandendriessche

Published in: Designs, Codes and Cryptography | Issue 3/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper, we study small sets of lines in \({{\mathrm{PG}}}(n,q)\) and \({{\mathrm{AG}}}(n,q),\,q\) odd, that have a small number of odd-points. We fix a small glitch in the proof of an earlier bound in the affine case, we extend the theorem to the projective case, and we attempt to classify all the sets where equality is reached. For the projective case, we obtain a full classification. For the affine case, we obtain a full classification minus one open case where there is only a characterization.

previous article Triple arrays and Youden squares

next article The largest Erdős–Ko–Rado sets in designs

Assmus Jr. E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992).

Balister P., Bollobs B., Fredi Z., Thompson J.: Minimal symmetric differences of lines in projective planes, unpublished, available as arXiv:1303.4117 [math.CO].

Frumkin A., Yakir A.: Rank of inclusion matrices and modular representation theory. Israel J. Math. 71, 309–320 (1990).

Hellerstein L., Gibson G., Karp R., Katz R., Patterson D.: Coding techniques for handling failures in large disk arrays. Algorithmica 12, 18–208 (1994).

Hirschfeld J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York. ISBN: 0-19-853536-8 (1985).

Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces: update 2001. In: Blokhuis A., Hirschfeld J.W.P., Jungnickel D., Thas J.A. (eds.), Developments in Mathematics, Finite Geometries, Proceedings of the Fourth Isle of Thorns Conference, Chelwood Gate, July 16–21, 2000, vol. 3, pp. 201–246. Kluwer, Dordrecht (2000).

Müller M., Jimbo M.: Erasure-resilient codes from affine spaces. Discret. Appl. Math. 143, 292–297 (2004).

Segre B.: Ovals in a finite projective plane. Can. J. Math. 7, 414–416 (1955).

Title: On small line sets with few odd-points
Author: Peter Vandendriessche
Publication date: 01-06-2015
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 3/2015
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-014-9920-1

Springer Professional

On small line sets with few odd-points

Abstract

Premium Partner

Springer Professional

Abstract

Please log in to get access to your license.

Other articles of this Issue 3/2015

Construction of minimal non-abelian left group codes

Hanani triple packings and optimal -ary codes of constant weight three

The largest Erdős–Ko–Rado sets in designs

Triple arrays and Youden squares

Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

Difference matrices related to Sophie Germain primes using functions on the fields

Premium Partner