Log-Concave Sequences in Coding Theory

Minjia Shi, Xuan Wang, Junmin An, Jon-Lark Kim

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 $a_{i}^{2} \geqslant a_{i-1}a_{i+1}$ for all $1 \leqslant i \leqslant n-1$ . 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 $r \geqslant 5$ ), the binary extended Hamming codes of length $2^{r} ~(r \geqslant 3)$ , and the second order Reed-Muller codes $R(2, m)~(m \geqslant 2)$ 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. We also show that most of QR codes, BCH codes and Roth-Lempel NMDS codes are not log-concave. Hence, we expect that the concept of log-concavity in coding theory will stimulate many interesting problems.

External IDs:dblp:journals/tit/ShiWAK25