2007/01/16 by Maximilien Gadouleau, Zhiyuan Yan, Gadouleau, Maximilien +1
Computer Science · Mathematics · #Cellular Automata and Applications #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.cs/0701097
5 pages, submitted to IEEE ISIT 2007
arxiv created 2007/01/16 · openalex publication_date 2007/01/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper investigates the relationship between the rank weight distribution of a linear code and that of its dual code. The main result of this paper is that, similar to the MacWilliams identity for the Hamming metric, the rank weight distribution of any linear code can be expressed as an analytical expression of that of its dual code. Remarkably, our new identity has a similar form to the MacWilliams identity for the Hamming metric. Our new identity provides a significant analytical tool to the rank weight distribution analysis of linear codes. We use a linear space based approach in the proof for our new identity, and adapt this approach to provide an alternative proof of the MacWilliams identity for the Hamming metric. Finally, we determine the relationship between moments of the rank distribution of a linear code and those of its dual code, and provide an alternative derivation of the rank weight distribution of maximum rank distance codes.