Log-Concave Sequences in Coding Theory
Minjia Shi, Xuan Wang, Junmin An, Jon-Lark Kim
TL;DR
This work introduces and develops log-concavity for the nonzero weight distributions of linear codes, formalizing the condition $a_i^2 \ge a_{i-1}a_{i+1}$ and exploring its implications through weight enumerators connected by the MacWilliams identity. The authors prove that several important coding families—binary Hamming codes $\mathcal{H}_m$ (with $m=3$ or $m\ge 7$), their extended versions, and second order Reed–Muller codes $R(2,m)$—are log-concave (with some instances of 1-gap log-concavity for related families). They further show that moderate-length MDS codes are log-concave when the field size $q$ is large enough, determined by a quadratic-root threshold $q_0(n,k)$. By combining explicit weight enumerators with targeted inequalities, the paper establishes a bridge between log-concavity in combinatorics and classical coding theory, suggesting rich future directions for diverse code families and related weight-distribution sequences.
Abstract
We introduce the notion of logarithmically concave (or 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 call a linear code log-concave if its nonzero weight distribution is log-concave. Our main contribution is to show that all binary general Hamming codes of length $2^r -1$ ($r=3$ or $r \ge 5$), the binary extended Hamming codes of length $2^r ~(r \ge 3)$, and the second order Reed-Muller codes $R(2, m)~ (m \ge 2)$ are all log-concave while the homogeneous and projective second order Reed-Muller codes are either log-concave, or 1-gap log-concave. Furthermore, we show that any MDS $[n, k]$ code over $\mathbb F_q$ satisfying $3 \leqslant k \leqslant n/2 +3$ is log-concave if $q \geqslant q_0(n, k)$ which is the larger root of a quadratic polynomial. Hence, we expect that the concept of log-concavity in coding theory will stimulate many interesting problems.