Classical arcs in PG(r, q) for 11 ⩽ q ⩽ 19

Chao, Jia; Kaneta, Hitoshi

doi:10.1016/s0012-365x(96)00319-6

Cited by 10 publications

(13 citation statements)

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“…As regards the structure and the properties of classical superregular matrices, we refer to [4], [5], [22], and [23]. Our search proves that NCSR matrices of size k Â l where k; l !…”

Section: Theoremmentioning

confidence: 91%

“…It is of type 3 if and only if q is even, k ¼ 3 and [4,12]). The number of the nonequivalent classical n-arcs in PGðr; qÞ does not depend on the dimension r, and it equals to the number of non-equivalent n-point sets in PGð1; qÞ (see Theorem 5).…”

Section: Definitionmentioning

confidence: 99%

“…For 11 q 32, the spectrum of the sizes of complete arcs in PGð2; qÞ was searched by Chao and Kaneta [3,4,5] and Faina, Marcugini, Milani, and Pambianco [7,8,9,17,18,19,20]. For the range q 19, the number of complete arcs of different lengths in PGð2; qÞ has been tabulated in [12].…”

Section: Types Of Superregular Matricesmentioning

confidence: 99%

“…To save space, all cases where k þ l > m 0 ðk À 1; qÞ are omitted from the basic listing according the types, with the exception of the following two possibilities: k ¼ 3 for an even q; q ¼ 9 and k ¼ l ¼ 5. It is known that in the specified cases, only classical n-arcs exist in PGðk À 1; qÞ if q 19 (see e.g., [4]). The results that apply to the exceptional cases are contained in the last continuation of Table V Table V …”

Section: Theoremmentioning

confidence: 99%

“…The question about the number and types of complete MDS codes is also examined and completely (partially) answered for the same pairs of q and k. From this, the known results on the number and structure of complete MDS codes that can be found in [1], [2], [3], [4], [5], [6], [7], [8], [9], [11], [12], [13], [17], [18], [19], [20], and [21] are extended to greater dimensions and to some particular MDS codes of given sizes. Besides, a considerable number of already known results are confirmed.…”

Section: Introductionmentioning

confidence: 99%

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Types of superregular matrices and the number ofn‐arcs and completen‐arcs in PG (r, q)

Kéri

2005

J of Combinatorial Designs

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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 À 2 and arbitrary n, (ii) for 23 q 32, r ¼ 2 and n ! q À 8. The equivalence classes over both PGL (k, q) and PÀ ÀL(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 0 (r, q)-the smallest and the second largest size for complete arcs in PG(r, q)-are also reported, stating that m 0 (2, 31) ¼ 22, m 0 (2, 32) ¼ 24, t(3, 23) ¼ 10, and m 0 (3, 23) ¼ 16. #

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“…As regards the structure and the properties of classical superregular matrices, we refer to [4], [5], [22], and [23]. Our search proves that NCSR matrices of size k Â l where k; l !…”

Section: Theoremmentioning

confidence: 91%

Section: Definitionmentioning

confidence: 99%