Open problems in finite projective spaces

Hirschfeld, J. W. P.; Thas, J. A.

doi:10.1016/j.ffa.2014.10.006

Cited by 37 publications

(86 citation statements)

References 98 publications

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“…Problems connected with small complete arcs in PG (2, q) are considered in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]20,[23][24][25][26][27][28][30][31][32][33][34][35][36][37][38][39][40][41][42][44][45][46][47][48]50,51,[54][55][56][57][58][61][62][63]…”

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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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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“…If q is even and k i = 3 for some i, we may take n i = q +2 and take as C i an MDS code for the following reason. It was classically known the existence of a set S 2 ⊂ P 2 (F q ) such that #S 2 = q + 2 and no 3 of the point of S 2 are collinear (an arc in the terminology of [9,11]).…”

Section: Remark 23mentioning

confidence: 99%

Locally Recoverable Codes correcting many erasures over small fields

Ballico

2021

Des. Codes Cryptogr.

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We define linear codes which are s-Locally Recoverable Codes (or s-LRC), i.e. codes which are LRC in s ways, the case $$s=1$$ s = 1 roughly corresponding to the classical case of LRC codes. We use them to describe codes which correct many erasures, although they have small minimum distance. Any letter of a received word may be corrected using s different local codes. We use the Segre embedding of s local codes and then a linear projection.

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“…In order to study the problem of counting inequivalent linear MDS codes, Iampolskaia, Skorobogatov, and Sorokin give a formula for the number of ordered n-arcs in P 2 (F q ). For more on the connection between arcs and MDS codes, see [8]. There are 10 superfigurations on 9 points up to isomorphism, which we denote by 9 3 , .…”

Section: Definition 15 ([2]mentioning

confidence: 99%

Counting arcs in projective planes via Glynn’s algorithm

et al. 2017

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An n-arc in a projective plane is a collection of n distinct points in the plane, no three of which lie on a line. Formulas counting the number of n-arcs in any finite projective plane of order q are known for n ≤ 8. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted 9-arcs in the projective plane over a finite field of order q and showed that this count is a quasipolynomial function of q. We present a formula for the number of 9-arcs in any projective plane of order q, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as a special case. We obtain our formula from a new implementation of an algorithm due to Glynn; we give details of our implementation and discuss its consequences for larger arcs.

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Open problems in finite projective spaces

Cited by 37 publications

References 98 publications

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

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

Locally Recoverable Codes correcting many erasures over small fields

Counting arcs in projective planes via Glynn’s algorithm

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