# On dual codes in the Doob schemes

## TL;DR

This paper establishes MacWilliams identities for dual codes in Doob schemes, linking their weight distributions and defining dual schemes with different metrics, extending classical coding theory results.

## Contribution

It introduces dual schemes with different metrics in Doob schemes and proves MacWilliams identities connecting weight distributions of codes and their duals.

## Key findings

- MacWilliams identities for additive and linear codes in Doob schemes
- Dual schemes with different metrics are characterized
- Weight distributions of codes and their duals are related

## 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.