New Quantum Stabilizer Codes from generalized Monomial-Cartesian Codes constructed using two different generalized Reed-Solomon codes
Oisin Campion, Fernando Hernando, Gary McGuire
TL;DR
This work introduces Generalized Monomial Cartesian Codes (GMC) built from two generalized Reed-Solomon codes to enable Hermitian self-orthogonality and hence quantum stabilizer codes. By exploiting separable GMCs (Q(i,j)=v_i w_j) and footprint-based orthogonality criteria, the authors derive explicit conditions and the maximum self-orthogonal distances T^* for both two-point and multi-point constructions in the Y-variable. They obtain large families of quantum codes with parameters [[n, n-2|Δ_t|, t]]_q that surpass the quantum Gilbert–Varshamov bound, including infinite families and concrete examples across various q, and provide three explicit code families with long length and strong distance properties. The results significantly broaden the landscape of quantum code constructions beyond MDS limitations, offering practical families with beatable GV bounds and rich combinatorial structure.
Abstract
In this work, we define Generalized Monomial Cartesian Codes (GMCC), which constitute a natural extension of generalized Reed-Solomon codes. We describe how two different generalized Reed-Solomon codes can be combined to construct one GMCC. We further establish sufficient conditions ensuring that the GMCC are Hermitian self-orthogonal, thus leading to new constructions of quantum codes.