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

Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

Johnson type bounds for mixed dimension subspace codes

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.Comment: 16 pages, typos correcte

Classification of large partial plane spreads in PG(6,2)PG(6,2) and related combinatorial objects

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$.Comment: 31 pages, 9 table