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

✍ Scribed by Gerzson Kéri

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 182 KB
• Volume: 14
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20091

✦ Synopsis

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 PΓL(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. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 363–390, 2006