2019/01/31 by Denis S. Krotov, Krotov, Denis S.
Computer Science · Engineering · Mathematics · #94B05 #Cellular Automata and Applications #Coding theory and cryptography #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.DM #cs.IT #graph theory and CDMA systems #math.CO #math.IT #msc:94B05
paper · pdf · doi:10.48550/arxiv.1902.00020
arxiv created 2019/01/31 · openalex publication_date 2019/01/31 · arxiv updated 2019/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Doob scheme D(m,n'+n'') is a metric association scheme defined on E4m × F4n'× Z4n'', where E4=GR(42) or, alternatively, on Z42m × Z22n' × Z4n''. We prove the MacWilliams identities connecting the weight distributions of a linear or additive code and its dual. In particular, for each case, we determine the dual scheme, on the same set but with different metric, such that the weight distribution of an additive code C in the Doob scheme D(m,n'+n'') is related by the MacWilliams identities with the weight distribution of the dual code C^⊥ in the dual scheme. We note that in the case of a linear code C in E4m × F4n', the weight distributions of C and C^⊥ in the same scheme are also connected.