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Log-Concave Sequences in Coding Theory
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DOI：10.1109/TIT.2025.3613570.png)
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Abstract
En 中文
We introduce the notion of logarithmically con-cave (or log-concave) sequences in coding theory. A sequence as a(0),a(1)..., a(n), of real numbers is called log-concave if a(i)(2) >= a(i-1)a(i+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's where A; 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 (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, satisfying 3 <= k <= n / 2 + 3 is log-concave if q >= q(0)(n,k) which is the larger 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.
Keywords:
Codes
Linear codes
Polynomials
Vectors
Information theory
Reed-Muller codes
Generators
Upper bound
Training
Standards
Log-concave
weight distribution
linear code
Hamming code
Reed-Muller code
Journal
I
IF:
2.9
Papers:
317
Citations:
0