Geometric structures and linear codes related to conics in classical projective planes of odd orders.

摘要

Let C be a given conic in the classical projective plane PG(2, q), where q is an odd prime power. We consider several problems regarding various incidence structures related to the conic C . We give upper bounds on the 2-ranks of the incidence matrix of internal points versus their polars and the incidence matrix of external points versus their polars, when q takes certain special forms. The techniques used include basic combinatorial counting, group actions, and modular representation theory of projective special linear group PSL(2, q).The above two families of incidence matrices define parity check matrices for two families of LDPC (Low-Density Parity-Check) codes. The study of LPDC codes is one of the hottest topics in coding theory today. The dimensions of the aforementioned codes are very important parameters of these codes. The upper bounds on the 2--ranks of the above two families of incidence matrices give the lower bounds on the dimensions of corresponding LDPC codes over F2 for some special forms of the prime power q.