Lecture: The Kerdock, Preparata and related codes, and their applications

Tentative Contents of  Talk:

-    Nonlinear codes.
-    Applications in real life.
-    Linearity over Z4, duality.
-    Bent functions.
-    Low correlation sequences.
-    Cryptography.
-    Association schemes and spherical codes.
-    Other relations (to algebra, etc)

Name of Speaker:

Dr Kanat Abdukhalikov

Lecture Kind:

o    Research
o    Awareness
o    General

Abstract:

.
The Kerdock and Preparata codes are famous nonlinear binary codes
that are better than any comparable  linear binary codes.In this talk we will speak on Kerdock, Preparata and related nonlinear codes,their linear constructions over rings, and applications in various branches of mathematics (algebra,combinatorics, cryptography, computer science).

Date & Venue

4 April, 2010 at Center of  Excellence in Information Assurance, College of  Computer &Information Sciences, king Saud University

Time &Duration

10:30 am
(1 hour + duration )

Outlines about the Lecture Presenter:

Dr. Abdukhalikov’s research is mainly related to integral positive definite lattices, codes and representations of finite groups. From the viewpoint of integral lattices theory invariant lattices which are even unimodular are of special interest. The point is that in spite of the abundance of such lattices, most of them have trivial automorphism groups. Therefore, it is of some interest to produce families of positive-definite, even, unimodular lattices with large automorphism groups and unbounded ranks. Lattices in small dimension are studied in details. In the general case, apart from root lattices, only few explicit constructions of infinite series of even unimodular Euclidean lattices are known. Interest in such kind of research was arisen after Thompson's construction of unimodular lattice of rank 248 and related sporadic Thompson group.   
Abdukhalikov studied and described lattices invariant under some classes of groups and their natural modules (doubly transitive groups, finite groups of Lie types and their Steinberg modules, finite affine groups etc). These constructions give examples of series of (even unimodular) root-free lattices with interesting properties. Another series of good lattices is obtained from orthogonal decompositions of Lie algebras of type A.
Unimodular Hermitian lattices over Eisenstein numbers in dimension at most 12 were described in 1978 by W. Feit. Abdukhalikov and Scharlau extended classification of such unimodular lattices up to dimension 15.
Next part of his research concerns codes. He applied group theoretic approach to codes over ring of integers modulo p^d, p prime. There has been much interest in Z_4 codes as they have been shown to be a systematic way of constructing very good binary codes. For example, famous Kerdock and Preparata codes can be very simply constructed as binary images of linear quaternary codes. This fact stimulated a production of a huge number of papers on linear quaternary codes. The quaternary Kerdock and Preparata codes are analogs of classical Reed-Muller codes: they have prime power dimension and they are invariant under some affine group. Abdukhalikov determined necessary and sufficient conditions for codes over ring of integers modulo p^d to be invariant under arbitrary affine group, and solved this problem in maximum generality.
The most importance of inventing quaternary codes was an explanation of formal duality between nonlinear Kerdock and Preparata codes. In 1999 van Dam and de Caen discovered two formally dual association schemes. It turns out that this duality can be also explained in terms of quaternary Kerdock-Preparata codes.  It was done by Abdukhalikov, Bannai and Suda. Moreover, based on quaternary Kerdock-Preparata codes, they presented a series of association schemes, which leads to universally optimal configurations, that is, to certain spherical point configurations in Euclidean space. These configurations are also connected to the notion of mutually unbiased bases which is studied in quantum physics, quantum computing and quantum cryptography. Finally, this sort of problems is connected to orthogonal decompositions of Lie algebras, and some number theory questions. It should be pointed out that the study of binary nonlinear Kerdock codes is closely related to the study of bent functions, the subject which is under very intensive investigations last ten years due to their applications to cryptography and modern mobile telephone technology.