200 research outputs found
On hyperovals of polar spaces
We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)
LDPC codes associated with linear representations of geometries
We look at low density parity check codes over a finite field K associated with finite geometries T*(2) (K), where K is any subset of PG(2, q), with q = p(h), p not equal char K. This includes the geometry LU(3, q)(D), the generalized quadrangle T*(2)(K) with K a hyperoval, the affine space AG(3, q) and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of K
A study of (x(q+1),x;2,q)-minihypers
In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x <= q(2) - q, not decomposable in the sum of another minihyper and a line, a (j (q + 1), j; 2, q)-minihyper, where j = q(2) - q-x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44: 169-196, 2007), giving further results on these minihypers
Pseudo-ovals in even characteristic and ovoidal Laguerre planes
Pseudo-arcs are the higher dimensional analogues of arcs in a projective
plane: a pseudo-arc is a set of -spaces in
such that any three span the whole space. Pseudo-arcs of
size are called pseudo-ovals, while pseudo-arcs of size are
called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from
applying field reduction to an arc in .
We explain the connection between dual pseudo-ovals and elation Laguerre
planes and show that an elation Laguerre plane is ovoidal if and only if it
arises from an elementary dual pseudo-oval. The main theorem of this paper
shows that a pseudo-(hyper)oval in , where is even and
is prime, such that every element induces a Desarguesian spread, is
elementary. As a corollary, we give a characterisation of certain ovoidal
Laguerre planes in terms of the derived affine planes
- …