Title: What codes have log-concave sequences?

Speaker: Jon-Lark Kim (Sogang University, S. Korea)

Date/Time: Oct. 24(Thursday), 3-4pm

Venue: 5307

Abstract:

The notion of logarithmically concave (or log-concave) sequences has been actively studied in Algebra, Combinatorics, and Analysis.

We introduce log-concave sequences in Coding Theory. A sequence $a_0, a_1, \dots, a_n$ of real numbers is called log-concave if $a_i^2 \ge a_{i-1}a_{i+1}$ for all $1 \le i \le n-1$. A natural sequence of positive numbers in Coding Theory is the weight distribution of a linear code consisting of the nonzero values among $A_i$'s where $A_i$ denotes the number of codewords of weight $i$. We show that all binary Hamming codes except one length have log-concave nonzero weight distributions and the second order Reed-Muller codes also have log-concave nonzero weight distributions. This is a joint work with Minjia Shi, Xuan Wang, and Junmin An.