On the construction of small (l, t)-blocking sets in PG(2, q)

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1 Sofia, July 2017 p. 1 On the construction of small (l, t)-blocking sets in PG(2, q) Rumen Daskalov, Elena Metodieva Department of Mathematics and Informatics Technical University of Gabrovo 5300 Gabrovo, Bulgaria

2 Sofia, July 2017 p. 2 Introduction GF(q) denote the Galois field of q elements V(n,q) denote the vector space of all ordered n-tuples over GF(q) wt(x) the Hamming weight of a vector x A linear code C a k-dimensional subspace of V(n,q) [n,k,d] q code if its minimum distance is d For linear codes, the minimum distance is equal to the smallest of the weights of the nonzero codewords.

3 Sofia, July 2017 p. 3 The MLCT Problem A fundamental problem in coding theory is that of optimizing one of the parameters n, k and d for given values of the other two and fixed Galois field. The basic two versions are:

4 Sofia, July 2017 p. 3 The MLCT Problem A fundamental problem in coding theory is that of optimizing one of the parameters n, k and d for given values of the other two and fixed Galois field. The basic two versions are: Problem 1: Find d q (n,k), the largest value of d for which there exist an [n,k,d] q code.

5 Sofia, July 2017 p. 3 The MLCT Problem A fundamental problem in coding theory is that of optimizing one of the parameters n, k and d for given values of the other two and fixed Galois field. The basic two versions are: Problem 1: Find d q (n,k), the largest value of d for which there exist an [n,k,d] q code. Problem 2: Find n q (k,d), the smallest value of n for which there exist an [n,k,d] q code.

6 Sofia, July 2017 p. 4 The MLCT Problem To solve this main linear coding theory problem it is necessary: 1. To construct new codes with better minimum distance. or 2. To prove the nonexistence of codes with given parameters.

7 Sofia, July 2017 p. 4 The MLCT Problem To solve this main linear coding theory problem it is necessary: 1. To construct new codes with better minimum distance. or 2. To prove the nonexistence of codes with given parameters. In this talk we consider the first task.

8 Sofia, July 2017 p. 4 The MLCT Problem To solve this main linear coding theory problem it is necessary: 1. To construct new codes with better minimum distance. or 2. To prove the nonexistence of codes with given parameters. In this talk we consider the first task.

9 Sofia, July 2017 p. 5 The MLCT Problem - q 9 An [n,k,d] q code is a Griesmer code if n = g q (k,d) = k 1 j=0 d q j. Note that n q (k,d) = g q (k,d) for all d when k = 1 or 2.

10 Sofia, July 2017 p. 6 The MLCT Problem - k = 3 The problem of finding n q (k,d) for all d has been solved for: k 8 for binary codes, k 5 for ternary codes, k 4 for quaternary codes and only for k = 3 when 5 q 9. Thus, in the case of three-dimensional codes the problem remains open when q 11.

11 Sofia, July 2017 p. 7 From codes to arcs It is well known that there exists a projective if and only if there exists an [n,3,d] q - code (n,n d)- arc in PG(2, q)

12 Sofia, July 2017 p. 8 Projective plane - PG(2,q) Let GF(q) denote the Galois field of q elements Let V(3,q) be the vector space of row vectors of length three with entries in GF(q). Let PG(2,q) be the corresponding projective plane. The points (x 1,x 2,x 3 ) of PG(2,q) are the 1-dimensional subspaces of V(3,q). Subspaces of dimension two are called lines.

13 Sofia, July 2017 p. 9 PG(2,q) The number of k dimensional subspaces of V(n,q) is (q n 1)(q n q)...(q n q k 1 ) (q k 1)(q k q)...(q k q k 1 ) The number of points is q 2 + q + 1 ( n = 3,k = 1) The number of lines is q 2 + q + 1 (n = 3,k = 2) There are q + 1 points on every line and q + 1 lines through every point.

14 Sofia, July 2017 p. 10 PG(2,q) - (n,r)-arc, (l, t)- bl. set Definition: An (n,r)- arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. Notation: m r (2,q) the maximum size of an (n,r)-arc

15 Sofia, July 2017 p. 10 PG(2,q) - (n,r)-arc, (l, t)- bl. set Definition: An (n,r)- arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. Notation: m r (2,q) the maximum size of an (n,r)-arc Definition: An (l,t)- blocking set S in PG(2, q) is a set of l points such that every line of PG(2, q) intersects S in at least t points, and there is a line intersecting S in exactly t points.

16 Sofia, July 2017 p. 10 PG(2,q) - (n,r)-arc, (l, t)- bl. set Definition: An (n,r)- arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. Notation: m r (2,q) the maximum size of an (n,r)-arc Definition: An (l,t)- blocking set S in PG(2, q) is a set of l points such that every line of PG(2, q) intersects S in at least t points, and there is a line intersecting S in exactly t points. Note that an (n,r)- arc is the complement of a (q 2 + q + 1 n,q + 1 r)- blocking set in a projective plane and conversely.

17 Sofia, July 2017 p. 11 PG(2,q) - secant distribution Definition: Let M be a set of points in any plane. An i - secant is a line meeting M in exactly i points. Notation: τ i - the number of i -secants to a set M.

18 Sofia, July 2017 p. 11 PG(2,q) - secant distribution Definition: Let M be a set of points in any plane. An i - secant is a line meeting M in exactly i points. Notation: τ i - the number of i -secants to a set M. In terms of τ i the definitions of (n,r )- arc and (l,t )- blocking set become

19 Sofia, July 2017 p. 11 PG(2,q) - secant distribution Definition: Let M be a set of points in any plane. An i - secant is a line meeting M in exactly i points. Notation: τ i - the number of i -secants to a set M. In terms of τ i the definitions of (n,r )- arc and (l,t )- blocking set become An (n,r)- arc is a set of n points of a projective plane for which τ i 0 for i < r, τ r > 0 and τ i = 0 when i > r.

20 Sofia, July 2017 p. 11 PG(2,q) - secant distribution Definition: Let M be a set of points in any plane. An i - secant is a line meeting M in exactly i points. Notation: τ i - the number of i -secants to a set M. In terms of τ i the definitions of (n,r )- arc and (l,t )- blocking set become An (n,r)- arc is a set of n points of a projective plane for which τ i 0 for i < r, τ r > 0 and τ i = 0 when i > r. An (l,t)- blocking set is a set of l points of a projective plane for which τ i = 0 for i < t, τ t > 0 and τ i 0 when i > t.

21 Sofia, July 2017 p. 12 Basic results in PG(2,q) Bose (1947) proved that m 2 (2,q) = q + 1 m 2 (2,q) = q + 2 for q - odd for q - even Qvist (1952) (q - even) q + 2 = q + 1 (oval) + 1 (nucleus, knot)

22 Sofia, July 2017 p. 12 Basic results in PG(2,q) Bose (1947) proved that m 2 (2,q) = q + 1 m 2 (2,q) = q + 2 for q - odd for q - even Qvist (1952) (q - even) q + 2 = q + 1 (oval) + 1 (nucleus, knot) Barlotti (1965) and Ball (1996) for q odd prime and m r (2,q) = (r 1)q + 1 r = (q + 1)/2, r = (q + 3)/2

23 Sofia, July 2017 p. 13 m r (2, q) - exact values r\q

24 Sofia, July 2017 p. 14 m r (2, 11) - only 3 open cases r\q

25 Sofia, July 2017 p. 15 Results on m r (2, q) S. Ball q = 11, 13 Marcugini, Milani, Pambianco q = 11, 13 Daskalov, Contreras q = 13 Daskalov, Metodieva q = 17 Braun, Kohnert, Wassermann q = 11, 13, 16, 17, 19

26 Sofia, July 2017 p. 16 Recent results on m r (2, q) Kohnert q = 17, 19, 23, 25, 27, 31, 32 Daskalov, Metodieva ( PPI) q = 17, 19, 23, 25 Gulliver ( m 8 (2, 11) = 78) q = 11 Kohnert, Zwanzger ( m 15 (2, 19) 265) q = 19 Daskalov ( m 16 (2, 19) 286) q = 19

27 Sofia, July 2017 p. 17 Recent results on m r (2, q) Nonexistence results: G. Cook ( January m 4 (2, 11) = 32 ) q = 11 D. Bartoli, S. Marcugini, F. Pambianco ( November m 3 (2, 16) 28 ) q = 16 N.Hamada, T.Maruta, Y.Oya ( April m 3 (2, 17) 33 ) q = 17 Existence results and tables with bounds: Daskalov, Metodieva ( J of DM ) q = 25, 27

28 Sofia, July 2017 p. 18 The best known (l, t)- blocking set t q=11 q=13 q=17 q=19 q=23 q=29 q= t q=11 q=13 q=17 q=19 q=23 q=29 q= q 5q 5q+1 5q 5q+1 5q+1 5q+1 3 4q 4q-1 4q 4q 4q 4q 4q 2 3q 3q-1 3q 3q 3q 3q 3q

29 Sofia, July 2017 p. 19 Two questions Question 1: Is it possible to construct good (l, 3)-blocking sets consisting of 4 lines and some additional points? Question 2: Is it possible to construct good (l, 4)-blocking sets consisting of 5 lines and some additional points? The answers are positive. Examples are given.

30 Sofia, July 2017 p. 20 A new result Axel Kohnert - a (827, 28)-arc in PG(2,31). The respective blocking set is a (166, 4)-blocking set. We present a new (156, 4)-blocking set. So m 28 (2,31) 837

31 Sofia, July 2017 p. 21 Five years without S T E F A N D O D U N E K O V

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