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

Kéri, Gerzson

doi:10.1002/jcd.20091

Cited by 27 publications

(28 citation statements)

References 16 publications

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“…In Table 2 we give the updated list of the known sizes of complete arcs in PG (2, q). Note also that in [59] it is shown that m 2 (2, 31) = 22, m 2 (2, 32) = 24.…”

Section: Construction Cmentioning

confidence: 99%

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

et al. 2009

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More than thirty new upper bounds on the smallest size t2(2, q) of a complete arc in the plane PG(2, q) are obtained for 169 ≤ q ≤ 839. New upper bounds on the smallest size t2(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m2(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m2(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t2(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ( 1 2 (q + 3) + δ)-arcs other than conics that share 1 2 (q + 3) points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), q ≡ 2 (mod 3) odd, we propose new constructions of 1 2 (q + 7)-arcs and show that they are complete for q ≤ 3701. Mathematics Subject Classification (2000). 51E21, 51E22, 94B05.

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“…In Table 2 we give the updated list of the known sizes of complete arcs in PG (2, q). Note also that in [59] it is shown that m 2 (2, 31) = 22, m 2 (2, 32) = 24.…”

Section: Construction Cmentioning

confidence: 99%

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

et al. 2009

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“…In particular, a complete arc in a plane PG (2, q), the points of which are treated as 3-dimensional q-ary columns, defines a parity check matrix of a q-ary linear code with codimension 3, Hamming distance 4, and covering radius 2. Arcs can be interpreted as linear maximum distance separable (MDS) codes [74,Sec.7], [76] and they are related to optimal coverings arrays [43], superregular matrices [49], and quantum codes [19].…”

Section: Introduction the Main Resultsmentioning

confidence: 99%

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

et al. 2015

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In the projective plane PG(2, q), upper bounds on the smallest size t2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤ 301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t2(2, q) valid in this region, in particular t2(2, q) < 0.998 3q ln q for 7 ≤ q ≤ 160001;t2(2, q) < 1.05 3q ln q for 7 ≤ q ≤ 301813;t2(2, q) < √ q ln 0.7295 q for 109 ≤ q ≤ 160001;t2(2, q) < √ q ln 0.7404 q for 160001 < q ≤ 301813.The new upper bounds are obtained by finding new small complete arcs in PG(2, q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q's with q ≤ 430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥ 109. Finally, random complete arcs in PG(2, q), q ≤ 46337, q prime, are considered. The random complete arcs

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“…Superregular matrices with entries in a finite field can be obtained from Cauchy matrices or Vandermonde matrices (see, for example, [15,16,19,20]). …”

Section: Resultsmentioning

confidence: 99%

A construction of MDS array codes

Cardell

Climent

Requena

2013

WIT Transactions on Information and Communication Technologies

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

Cited by 27 publications

References 16 publications

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

A construction of MDS array codes

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