2009/01/14 by Yanling Chen, Chen, Yanling, A. J. Han Vinck +2
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #graph theory and CDMA systems #math.IT
paper · pdf · doi:10.48550/arxiv.0901.2062
openalex publication_date 2009/01/14 · arxiv created 2009/04/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider the Reed-Muller (RM) codes. For the first order RM code, we prove that it is unique in the sense that any linear code with the same length, dimension and minimum distance must be the first order RM code; For the second order RM code, we give a constructive linear sub-code family for the case when m is even. This is an extension of Corollary 17 of Ch. 15 in the coding book by MacWilliams and Sloane. Furthermore, we show that the specified sub-codes of length <= 256 have minimum distance equal to the upper bound or the best known lower bound for all linear codes of the same length and dimension. As another interesting result, we derive an additive commutative group of the symplectic matrices with full rank.