Log-Concave Sequences in Coding Theory

We introduce the notion of logarithmically concave (or log-concave) sequences in coding theory. A sequence a₀, a₁, . . . , a_n of real numbers is called log-concave if a²_i ⩾ a_i−1^ai+1 for all 1 ⩽ i ⩽ 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 Ai 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 ⩾ 5), the binary extended Hamming codes of length 2^r (r ⩾ 3), and the second order Reed-Muller codes R(2,m) (m ⩾ 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 F_q satisfying 3 ⩽ k ⩽ n/2 + 3 is log-concave if q ⩾ q₀(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.