Cyclic projective planes and binary, extended cyclic self-dual codes

AbstractIf P is a cyclic projective plane of order n, we give number theoretic conditions on n2 + n + 1 so that the binary code of P is contained in a binary cyclic code C whose extension is self-dual. When this containment occurs C does not contain any ovals of P. As a corollary to these conditions we obtain that the extended binary code of a cyclic projective plane of order 2s is contained in a binary extended cyclic self-dual code if and only if s is odd

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

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

Moderate-density parity-check codes from projective bundles

New constructions for moderate-density parity-check (MDPC) codes using finite geometry are proposed. We design a parity-check matrix for the main family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine the minimum distance and the dimension of these codes, and we show that they have a natural quasi-cyclic structure. We consider alternative constructions based on an incidence matrix of a Desarguesian projective plane and compare their error-correction performance with regards to a modification of Gallager’s bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance after one round of bit-flipping decoding given the parameters of the code’s parity-check matrix