Types of superregular matrices and the number of n-arcs and complete n-arcs in PG(r, q)

Kéri, Gerzson (2006) Types of superregular matrices and the number of n-arcs and complete n-arcs in PG(r, q). Journal of Combinatorial Designs, 14 (5). pp. 363-390.

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Abstract

Based on the classification of superregular matrices, the numbers of non-equivalent n-arcs and complete n-arcs in PG(r, q) are determined (i) for 4<=q<=19, 2<=r<=q-2 and arbitrary n, (ii) for 23<=q<=32, r=2 and n>=q-8. The equivalence classes over both PGL (k, q) and PGammaL(k, q) are considered throughout the examinations and computations. For the classification, an n-arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non-singular square submatrices. Four types of superregular matrices are studied and the non-equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m'(r, q) - the smallest and the second largest size for complete arcs in PG(r, q) - are also reported, stating that m'(2, 31) = 22, m'(2, 32) = 24, t(3, 23) = 10, and m'(3, 23) = 16.

Item Type: ISI Article
Uncontrolled Keywords: complete $n$-arc, MDS code, superregular matrix
Subjects: Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány
Depositing User: Eszter Nagy
Date Deposited: 11 Dec 2012 15:20
Last Modified: 11 Dec 2012 15:20
URI: http://eprints.sztaki.hu/id/eprint/4148

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