This grant will support the PI and his collaborators to work on invariants of incidence matrices, difference sets, strongly regular graphs and the related error-correcting codes. Incidence matrices arise whenever one considers relations between two finite sets. Invariants of incidence matrices, such as p-ranks and Smith normal forms, can reveal a great deal of information of the underlying incidence structures. The PI plans to continue his research on various geometric incidence matrices, such as the subspace-inclusion matrices and their symplectic analogues. The Smith normal form results will have interesting geometric applications, for example, in classifying ovoids in three-dimensional projective space over a finite field of even order. Other topics of research include constructions and nonexistence proofs of difference sets and strongly regular graphs. The PI's investigation will make use of techniques from several different areas of mathematics, including representation theory, p-adic number theory and combinatorics.
Incidence matrices play an important role in many parts of discrete mathematics, including the theory of error-correcting codes. Most of the incidence matrices considered in this proposal can be used to generate low density parity check codes. Efficient error-correcting codes are used nowadays in our daily life, for example, in CD players, high speed modems, and cellular phones. Cyclic difference sets are the same objects as binary sequences with two-level periodic autocorrelation functions. Such sequences have many applications in radar, spread-spectrum communications and cryptography. This project will concentrate on the study of invariants of incidence matrices and the existence questions of difference sets.
