Technical Program

Paper Detail

Paper Title On Dual Codes in the Doob Schemes
Paper IdentifierTH1.R9.2
Authors Denis S. Krotov, Sobolev Institute of Mathematics, Russia
Session Algebraic Coding Theory
Location Pontoise, Level 5
Session Time Thursday, 11 July, 09:50 - 11:10
Presentation Time Thursday, 11 July, 10:10 - 10:30
Manuscript  Click here to download the manuscript
Abstract The Doob scheme $D(m,n'+n'')$ is a metric association scheme defined on $E_4^m \times F_4^{n'}\times Z_4^{n''}$, where $E_4=GR(4^2)$ or, alternatively, on $Z_4^{2m} \times Z_2^{2n'} \times Z_4^{n''}$. 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 $C^\perp$ in the dual scheme. We note that in the case of a linear code $C$ in $E_4^m \times F_4^{n'}$, the weight distributions of $C$ and $C^\perp$ in the same scheme are also connected.