Related papers: New results on maximal partial line spreads in PG(…
For fixed integers p and q, let f(n,p,q) denote the minimum number of colors needed to color all of the edges of the complete graph K_n such that no clique of p vertices spans fewer than q distinct colors. A construction is given which…
In this paper we study a class of multishot network codes given by families of nested subspaces (flags) of a vector space $\mathbb{F}_q^n$, being $q$ a prime power and $\mathbb{F}_q$ the finite field of $q$ elements. In particular, we focus…
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…
Following the approach by R. K\"otter and F. R. Kschischang, we study network codes as families of k-dimensional linear subspaces of a vector space F_q^n, q being a prime power and F_q the finite field with q elements. In particular,…
In this paper we find new maximal cliques of size $\frac{q+1}{2}$ or $\frac{q+3}{2}$, accordingly as $q\equiv 1(4)$ or $q\equiv 3(4)$, in Paley graphs of order $q^2$, where $q$ is an odd prime power. After that we use new cliques to define…
Let $q$ be an odd prime power. Denote by $r(q)$ the value of $q$ modulo 4. In this paper, we establish a linear fractional correspondence between two types of maximal cliques of size $\frac{q+r(q)}{2}$ in the Paley graph of order $q^2$.
We study the maximum length of $q$-ary codes as a function of alphabet size, code size, and Singleton defect. For an $(n, M, d)_q$ code with dimension $\kappa = \log_q M \ge 2$ and Singleton defect $s = n - \lceil\kappa\rceil + 1 - d$, we…
Explicit constructions of infinite families of scattered ${\mathbb F}_q$--linear sets in $PG(r-1,q^t)$ of maximal rank $\frac{rt}2$, for $t$ even, are provided. When $q=2$ and $r$ is odd, these linear sets correspond to complete caps in…
In this paper we consider the density of maximal order elements in $\mathrm{GL}_n(q)$. Fixing any of the rank $n$ of the group, the characteristic $p$ or the degree $r$ of the extension of the underlying field $\mathbb{F}_q$ of size…
Two results are obtained that give upper bounds on partial spreads and partial ovoids respectively. The first result is that the size of a partial spread of the Hermitian polar space $\mathsf{H}(3, q^2)$ is at most $\left(\frac{2p^3+p}{3}…
We study how to formalize in the Coq proof assistant the smallest projective space PG(3,2). We then describe formally the spreads and packings of PG(3,2), as well as some of their properties. The formalization is rather straightforward,…
In this paper we present and analyze computational results concerning small complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$ and $N=4$ over the finite field of order $q$. The results have been obtained using…
In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left idealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of $\mathbb{F}_{q^6}[x]$, when…
The primary focus of this paper is overpartitions, a type of partition that plays a significant role in $q$-series theory. In 2006, Treneer discovered an explicit infinite family of congruences of overpartitions modulo $5$. In our research,…
2834 inequivalent spreads under the group $\mathrm{P}\Gamma\mathrm{L}(4,8)$. Therefore there are the same number of translation planes of order 64 with kernel containing $\mathrm{GF}(8)$, and we describe various properties of these planes.…
A small minimal k-blocking set B in PG(n, q), q = pt, p prime, is a set of less than 3(qk + 1)/2 points in PG(n, q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies…
An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J.…
Let N_q(g) the maximal number of points on a genus g curve over F_q. We prove that N_3(5)=13.
We study Q-Fano threefolds of large Fano index. In particular, we prove that the maximum of Fano index is attained for the weighted projective space P(3,4,5,7).
Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le…