Classical and quantum cyclic redundancy check codes
Simeon Ball, Ricard Vilar
TL;DR
The paper addresses the design of quantum error-correcting codes capable of correcting burst-like errors by transferring the cyclic redundancy check (CRC) framework from classical to quantum settings. It develops a stabilizer-code construction from classical CRC codes, leveraging the symplectic structure and the map $\tau$ to relate Pauli errors to additive codes over ${\mathbb F}_p$, and establishes that classical Reiger-bound-achieving CRCs yield quantum codes meeting the quantum Reiger bound $n-k \ge 4\ell$. It proves that quantum CRC codes can, under a $c$-property, correct burst errors of length $\ell$ with a corresponding linear-time decoding algorithm for a specific family with parameters $\,[\![mk,k]\!]$, $n=mk$, and $l=ck$, where $m$ is bounded below by the Reiger constraint. Simulations on a Markovian correlated depolarizing channel demonstrate superior entanglement fidelity of quantum CRC codes relative to prior constructions, particularly under correlation, indicating practical impact for robust quantum communication and storage in bursty noise environments.
Abstract
We prove that certain classical cyclic redundancy check codes can be used for classical error correction and not just classical error detection. We extend the idea of classical cyclic redundancy check codes to quantum cyclic redundancy check codes. This allows us to construct quantum stabiliser codes which can correct burst errors where the burst length attains the quantum Reiger bound. We then consider a certain family of quantum cyclic redundancy check codes for which we present a fast linear time decoding algorithm.