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

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 PrL(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. © 2005 Wiley Periodicals, Inc.