We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2, q). Each incidence structure is based on the various classes of points and lines created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distance. Some arguments involve the geometry of two conics in the plane. As a result, we prove, under mild conditions, the existence of two conics, one entirely internal or external to the other. We conclude with some simulation data to exhibit the effectiveness of our codes.

Abstract. For a field F and graph G of order n, the minimum rank of G over F is defined to be the smallest possible rank over all symmetric matrices A ∈ F n×n whose (i, j)th entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. It is shown that the minimum rank of a tree is independent of the field.

For an integer n ≥ 1, let [Formula: see text] and Sn be, respectively, the semigroup of partial transformations and the symmetric group on the set X = {1,…,n}. Then Sn is the group of units of [Formula: see text]. A subsemigroup S of [Formula: see text] is Sn-normal if for all a ∈ S and g ∈ Sn, g-1ag ∈ S. In 1976, Symons described the Sn-normal semigroups of full transformations of X. In 1995, Lipscomb and the second author determined the Sn-normal semigroups of partial injective transformations of X. In this paper, we complete the classification by describing all Sn-normal subsemigroups of [Formula: see text]. As a consequence of the classification theorem, we obtain a characterization of the automorphisms of any Sn-normal subsemigroup of [Formula: see text].