# A Note on Linear Complementary Pairs of Group Codes

## TL;DR

This paper provides a simple proof that in linear complementary pairs of group codes, one code uniquely determines the other and their duals are permutation equivalent, simplifying previous complex polynomial-based proofs.

## Contribution

It introduces an elementary proof for the relationship between complementary group codes and their duals, extending earlier results with a more straightforward approach.

## Key findings

- D code is uniquely determined by C in a complementary pair
- D^ot is permutation equivalent to C
- Simplifies previous polynomial-based proofs

## Abstract

We give a short and elementary proof of the fact that for a linear complementary pair $(C,D)$, where $C$ and $D$ are $2$-sided ideals in a group algebra, $D$ is uniquely determined by $C$ and the dual code $D^\perp$ is permutation equivalent to $C$. This includes earlier results of Carlet et al. and G\"uneri et al. on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.