🤖 AI Summary
Long-distance, high-rate quantum error-correcting codes (QECCs) remain scarce, and the Hull structure of projective Reed–Muller (PRM) codes—critical for constructing symmetric/asymmetric QECCs and entanglement-assisted (EA) codes—is poorly understood.
Method: This work systematically investigates PRM codes for quantum code construction, unifying CSS, Hermitian, and entanglement-assisted quantum error-correcting code (EAQECC) frameworks. It leverages PRM codes and their subfield subcodes to derive both symmetric and asymmetric quantum codes, as well as EAQECCs with tunable entanglement consumption.
Contribution/Results: First, it fully characterizes the Hull variation of PRM codes, enabling precise optimization of code parameters—including minimum distance and rate. Second, it constructs multiple new families of quantum codes achieving known bounds (e.g., Singleton and Hamming bounds). These advances significantly broaden the applicability and design flexibility of projective geometric codes in fault-tolerant quantum computation.
📝 Abstract
Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed-Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed-Muller codes.
