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 Guneri et al. on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.

中文翻译：

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关于群码线性互补对的注记

我们给出了一个简短的基本证明，即对于线性互补对 $(C,D)$，其中 $C$ 和 $D$ 是群代数中 $2$ 边的理想，$D$ 唯一确定为$C$ 和双码 $D^\perp$ 是等价于 $C$ 的排列。这包括 Carlet 等人的早期结果。和 Guneri 等人。在多项式空间中通过微妙而冗长的计算证明的 nD 循环码。