Block components of generalized quaternion group codes
Nadja Willenborg
TL;DR
The paper analyzes codes in the semisimple group algebra $\mathbb{F}_q[Q_{4n}]$ under the assumption $(4n,q)=1$ and shows that each code can be decomposed into central irreducible blocks via the Wedderburn decomposition. Using a generating idempotent $\lambda$, the authors define auxiliary components $\phi_j$, $\tilde\phi_j$, $\psi_i$, $\tilde\psi_i$ and prove that the image under the Wedderburn isomorphism $\rho(C)$ splits as a direct sum of blocks $C_j$ and $C_i'$ whose explicit forms depend on these generators; the result covers parity-sensitive cases through $\delta(n)$ and $\mu(n)$. The paper then specializes to induced codes from cyclic subgroups, showing how to compute the blocks of $C$ from the factorization of $\hat x^{2n/\ell}-1$ and an embedding into $\mathbb{F}_q[Q_{4n}]$, yielding a practical procedure for block reconstruction. The approach provides a framework for constructing lifted product codes and lays groundwork for analogous analysis of dihedral codes via explicit generating idempotents, highlighting both the algebraic structure and potential applications in communications and post-quantum contexts.
Abstract
Codes in the generalized quaternion group algebra $\mathbb{F}_q[Q_{4n}]$ are considered. Restricting to char$\mathbb{F}_q \nmid 4n$ the structure of an arbitrary code $C \subseteq \mathbb{F}_q[Q_{4n}]$ is described via the Wedderburn decomposition. Moreover it is known that in this case every code $C \subseteq \mathbb{F}_q[Q_{4n}]$ has a generating idempotent $λ\in \mathbb{F}_q[Q_{4n}]$. Given the generating idempotent of a code $C$ we determine the different components in its decomposition $C \cong \bigoplus_{j=1}^{r+s}C_j \oplus \bigoplus_{i=1}^{k+t}C'_{i}.$ Afterwards we apply this result to describe the blocks of codes induced by cyclic group codes.