MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes

TL;DR

This paper introduces new quantum error-correcting codes derived from generalized monomial-Cartesian codes, demonstrating their optimality and superiority over existing codes through explicit formulas and bounds.

Contribution

It constructs novel stabilizer quantum codes using an explicit twist vector from generalized monomial-Cartesian codes, providing formulas for their parameters and showing they outperform known codes.

Findings

Codes are MDS when m=1.

Codes are Hermitian Almost MDS when m=2 with minimum distance ≥ 3.

Infinite family of codes beat the Gilbert-Varshamov bound.

Abstract

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in $m$ variables. When $m=1$ our codes are MDS, and when $m=2$ and our lower bound for the minimum distance is $3$ the codes are at least Hermitian Almost MDS. For an infinite family of parameters when $m=2$ we prove that our codes beat the Gilbert-Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.