Searcharxiv⌕ Search

arXiv subjects

Leo Storme

Publications and source records attributed to Leo Storme.

23 records · Page 2Linked to original sources
arXiv2012

On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual

In this paper, we study the p-ary linear code Ck(n, q), q = ph, p prime, h >= 1, generated by the incidence matrix of points and k-dimensional spaces in PG(n, q). For k >= n/2, we link codewords of Ck(n, q)\Ck(n, q) of weight smaller than 2q^k to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k < n/2, we present counterexamples to lemmas valid for k >= n/2. Next, we study the dual code of Ck(n, q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [12] to general dimension.

math.CO↗
arXiv2012

On codewords in the dual code of classical generalised quadrangles and classical polar spaces

In [9], the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q+(5, q) and H(5, q2), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q+(5, q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4, q), q even. To prove this, we show that a blocking set of Q(4, q), q even, of size q2 +1+r, where 0 < r < (q +4)/6, contains an ovoid of Q(4, q), improving on [5, Theorem 9].

math.CO↗
arXiv2012

A proof of the linearity conjecture for k-blocking sets in PG(n, p3), p prime

In this paper, we show that a small minimal k-blocking set in PG(n, q3), q = p^h, h >= 1, p prime, p >=7, intersecting every (n-k)-space in 1 (mod q) points, is linear. As a corollary, this result shows that all small minimal k-blocking sets in PG(n, p^3), p prime, p >=7, are Fp-linear, proving the linearity conjecture (see [7]) in the case PG(n, p3), p prime, p >= 7.

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↗