Group codes over crystallographic point groups

Consider a finite field \({\mathbb {F}}_{q}\) ${\mathbb{F}}_q$ with characteristic p, where G is a crystallographic point group satisfying \(p \not \mid |G|\) $p\nmid |G|$ and \(q=p^n\) $q=p^n$ . In this paper, we propose studying group codes in the crystallographic point group algebras \({\mathbb {F}}_{q}G\) ${\mathbb{F}}_{q}G$ for the point groups \(C_{2h}\) $C_{2h}$ , \(C_{6v}\) $C_{6v}$ , and \(D_{6h}\) $D_{6h}$ . We compute the unique (linear and nonlinear) idempotents of \({\mathbb {F}}_{q}G\) ${\mathbb{F}}_{q}G$ that correspond to the characters of the crystallographic point groups. These idempotents play a crucial role in characterizing the properties of the group codes. Based on the above results, we characterize the minimum distances and dimensions of the group codes. This provides valuable information about the error-correcting capabilities and the amount of information that can be transmitted through these codes. Furthermore, we construct MDS (Maximum Distance Separable) group codes and almost MDS group codes.