On Saturating Sets in Small Projective Geometries

AbstractA set of points, S⊆PG(r, q), is said to be ϱ -saturating if, for any point x∈PG(r, q), there exist ϱ+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,ϱ ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3 q+ 1 forq= 2, q≥ 4. We further give an upper bound onk (ϱ+ 1, pm, ϱ)

Applications of finite geometry in coding theory and cryptography

We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory

Binary and Ternary Quasi-perfect Codes with Small Dimensions

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.Comment: 4 page

Constructing saturating sets in projective spaces using subgeometries

A $\varrho$-saturating set of $\text{PG}(N,q)$ is a point set $\mathcal{S}$ such that any point of $\text{PG}(N,q)$ lies in a subspace of dimension at most $\varrho$ spanned by points of $\mathcal{S}$. It is generally known that a $\varrho$-saturating set of $\text{PG}(N,q)$ has size at least $c\cdot\varrho\,q^\frac{N-\varrho}{\varrho+1}$, with $c>\frac{1}{3}$ a constant. Our main result is the discovery of a $\varrho$-saturating set of size roughly $\frac{(\varrho+1)(\varrho+2)}{2}q^\frac{N-\varrho}{\varrho+1}$ if $q=(q')^{\varrho+1}$, with $q'$ an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of $\varrho$-saturating sets if $\varrho<\frac{2N-1}{3}$. As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes. To prove that this construction is a $\varrho$-saturating set, we observe that the affine parts of $q'$-subgeometries of $\text{PG}(N,q)$ having a hyperplane in common, behave as certain lines of $\text{AG}\big(\varrho+1,(q')^N\big)$. More precisely, these affine lines are the lines of the linear representation of a $q'$-subgeometry $\text{PG}(\varrho,q')$ embedded in $\text{PG}\big(\varrho+1,(q')^N\big)$.Comment: [v1] 25 pages, 1 figure [v2] 30 pages, 1 figure: added translation of the main results to the coding theoretical context and made a more thorough comparison with the existing literature [v3] 30 pages, 1 figure: fixed some details and minor grammar and spelling mistake