arXiv subjects

Peter Sziklai

Publications and source records attributed to Peter Sziklai.

5 recordsLinked to original sources

arXiv2025

A note on short minimal codes from subgeometries

In a 2022, Bartoli, Cossidente, Marino, and Pavese proved that in the projective space ${\rm PG}(3,q^3)$, one can find three $\mathbb F_q$-subgeometries such that the union of their point sets is a strong blocking set. This proves the existence of linear minimal codes with parameters $[3(q^2+1)(q+1),4]_{q^3}$ for every prime power $q$. We give a short proof of this result for odd values of $q > 9$, using the theory of small blocking sets in projective planes.

math.CO↗

arXiv2023

Renitent lines

There are many examples for point sets in finite geometry, which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod $p$, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behaviour of these renitent lines.

math.CO↗

arXiv2023

Covering all but the low weight vertices of the unit cube

In this paper we discuss a result similar to the polynomial version of the Alon-Füredi theorem. We prove that if you want to cover the vertices of the $n$-dimensional unit cube, except those of weight at most $r$ then you need an algebraic surface of degree at least $n-r$.

math.CO↗

arXiv2012

A small minimal blocking set in PG(n,p^t), spanning a (t-1)-space, is linear

In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional space, are F_p-linear, hence confirming the linearity conjecture for blocking sets in this particular case.

math.CO↗

arXiv2012

An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n, q)

Let Ck(n, q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n, q), q = p^h, p prime, h >= 1. In this pa- per, we show that there are no codewords of weight in the open interval ] q^{k+1}-1/q-1, 2q^k[ in Ck(n, q) \ Cn-k(n, q) which implies that there are no codewords with this weight in Ck(n, q) \ Ck(n, q) if k >= n/2. In par- ticular, for the code Cn-1(n, q) of points and hyperplanes of PG(n, q), we exclude all codewords in Cn-1(n, q) with weight in the open interval ] q^n-1/q-1, 2q^n-1[. This latter result implies a sharp bound on the weight of small weight codewords of Cn-1(n, q), a result which was previously only known for general dimension for q prime and q = p2, with p prime, p > 11, and in the case n = 2, for q = p^3, p >= 7 ([4],[5],[7],[8]).

math.CO↗