Dihedral Quantum Codes
Nadja Willenborg, Martino Borello, Anna-Lena Horlemann, Habibul Islam
TL;DR
The paper addresses constructing short, nonabelian quantum CSS codes with moderate-density parity-check (MDPC) parity matrices by extending the lifted-product framework to dihedral group algebras $\mathbb{F}_{q}[D_{2n}]$. It develops a dihedral lifted product construction, provides an explicit dimension formula, presents induced-code techniques, and proves a distance bound, complemented by a concrete $D_{180}$ example. The results yield MDPC quantum codes with favorable decoding properties and potential cryptographic applications, highlighting that nonabelian MDPC quantum codes can be realized and analyzed within this algebraic framework. Overall, the work broadens the toolkit for quantum LDPC code design by leveraging dihedral group structure and concrete decomposition techniques to achieve short-block, efficiently decodable quantum codes with provable parameters.
Abstract
We establish dihedral quantum codes of short block length, a class of CSS codes obtained by the lifted product construction. We present the code construction and give a formula for the code dimension, depending on the two classical codes that the CSS code is based on. We also give a lower bound on the code distance and construct an example of short dihedral quantum codes.