Dimensions of some binary codes arising from a conic in PG(2,q)

AbstractLet O be a conic in the classical projective plane PG(2,q), where q is an odd prime power. With respect to O, the lines of PG(2,q) are classified as passant, tangent, and secant lines, and the points of PG(2,q) are classified as internal, absolute and external points. The incidence matrices between the secant/passant lines and the external/internal points were used in Droms et al. (2006) [6] to produce several classes of structured low-density parity-check binary codes. In particular, the authors of Droms et al. (2006) [6] gave conjectured dimension formula for the binary code L which arises as the F2-null space of the incidence matrix between the secant lines and the external points to O. In this paper, we prove the conjecture on the dimension of L by using a combination of techniques from finite geometry and modular representation theory

On Binary Codes from Conics in PG(2,q)

Let A be the incidence matrix of passant lines and internal points with respect to a conic in PG(2, q), where q is an odd prime power. In this article, we study both geometric and algebraic properties of the column null space L of A over the finite field of 2 elements. In particular, using methods from both finite geometry and modular presentation theory, we manage to compute the dimension of L, which provides a proof for the conjecture on the dimension of the binary code generated by L

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

Classes of codes from quadratic surfaces of PG(3,q)

We examine classes of binary linear error correcting codes constructed from certain sets of lines defined relative to one of the two classical quadratic surfaces in $PG(3,q)$. We give an overview of some of the properties of the codes, providing proofs where the results are new. In particular, we use geometric techniques to find small weight codewords, and hence, bound the minimum distance

On Mathon's construction of maximal arcs in Desarguesian planes. II

In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when $m\geq 5$ and $m\neq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,1}-map is (\floor {m/2} +1). This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $m\geq 7$ and $m\neq 9$, then the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,q}-map is either \floor {m/2} +1 or \floor{m/2} +2.Comment: 21 page