KAIST Discrete Math Seminar

We introduce a new class of rate one half codes, called complementary information set codes. A binary linear code of length 2n and dimension n is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune functions of use in the security of hardware implementations of  cryptographic primitives. In this talk, we give optimal or best known CIS codes of length <132. We  derive general constructions based on cyclic codes, double circulant codes, strongly regular graphs, and doubly regular tournaments. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths up to 12 by the building up construction. This is a joint work with Claude Carlet, Philippe Gaborit, and Patrick Sole.

Self-dual codes have become one of the most active research areas in coding theory due to their rich mathematical theories. In this talk, we start with an introduction to coding theory. Then we describe some recent results on the constructions of self-dual codes over rings, and applications to lattices and network coding theory. We conclude the talk with some open problems.