Csajbok Bence: On the minimal number of odd-secants of small point sets in PG(2,q), q odd
The weights of codewords in the p-ary linear code defined by the points and lines of PG(2,q), q=p^h, p prime, have been studied by many authors.
Recently, the binary code has been investigated also for q odd, in the following sense.
What is the minimal size of the symmetric difference of r lines in PG(2,q), q odd? Equivalently, given a line set R of size r, what is the minimal number
of points incident with an odd number of lines in R?
Dually, for an r-set S in PG(2,q), q odd, what is the minimal number of odd-secants of S? Denote this number by f(r).
In [2], Vandendriessche classified r-sets with at most 2q+2-r odd-secants. In [1], Balister et al. determined the exact value of f(r) for r < q+2 and
gave lower and upper bounds on f(q+2). The investigation of f(q+3) was proposed in [1], where the authors presented exact values for q < 11 and an
estimate for q=11.
In this talk we present a lower bound on f(q+3), improve the lower bound on f(q+2) and answer a related question of Vandendriessche.
[1] P. Balister, B. Bollobas, Z. Furedi and J. Thompson, Minimal Symmetric Differences of Lines in Projective Planes, in press, J. Combin. Des., DOI:
10.1002/jcd.21390, published online 13 February 2014.
[2] P. Vandendriessche, On small line sets with few odd-points, in press, Des. Codes Cryptogr., DOI 10.1007/s10623-014-9920-1, published online February
2014.