Twisted Fiber Bundle Codes over Group Algebras

We introduce a twisted fiber bundle construction of quantum CSS codes over group algebras $R={\mathbb{F}}_2[G]$, where each base generator carries a generator-dependent $R$-linear fiber twist satisfying a flatness condition. This construction extends the untwisted lifted product code, recovered when all twists are identities. We show that invertible twists (satisfying a flatness condition) give a complex chain-isomorphic to the untwisted one, so the resulting binary CSS codes have the same blocklength $n$ and encoded dimension $k$. In contrast, singular chain-compatible twists can lower boundary ranks and increase the number of logical qubits. Examples over $R={\mathbb{F}}_2[D_3]$ show that singular chain-compatible twists can increase the encoded dimension $k$ at fixed blocklength $n$, and in these finite examples the minimum distance $d$ remains unchanged. This provides evidence that singular twisting enlarges the design space beyond the ordinary lifted product construction.