Log-Concave Sequences in Coding Theory

초록

We introduce the notion of logarithmically concave (or log-concave) sequences in coding theory. A sequence a<inf>0</inf>, a<inf>1</inf>, . . ., a<inf>n</inf> of real numbers is called log-concave if a<inf>i</inf>2 ⩾ a<inf>i-1</inf>a<inf>i+1</inf> 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<inf>i</inf>’s where A<inf>i</inf> 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 2r - 1 (r = 3 or r ⩾ 5), the binary extended Hamming codes of length 2r (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<inf>q</inf> satisfying 3 ⩽ k ⩽ n/2 + 3 is log-concave if q ⩾ q<inf>0</inf>(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. © 1963-2012 IEEE.

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