Stabilizer quantum codes from $J$-affine variety codes and a new Steane-like enlargement

TL;DR

The work introduces $J$-affine variety codes as a broad class of evaluation codes to construct stabilizer quantum codes, and develops both Euclidean and Hermitian duality frameworks to enable self-orthogonality and dual-containing constructions. It extends Steane-like enlargement via a new generalization and leverages subfield-subcodes with trace techniques to produce a wide array of quantum codes, including several records. The paper provides rigorous duality results (Euclidean and Hermitian), enlargement-based parameter guarantees, and extensive examples that surpass the Gilbert-Varshamov bound in many cases, including a notable $[[127,63,\ge 12]]_2$ code and a $[[63,45,\ge 6]]_4$ code. The findings expand the catalog of practical quantum codes and offer a systematic algebraic-geometric framework for designing high-rate, high-distance stabilizer codes for binary and nonbinary quantum systems.

Abstract

New stabilizer codes with parameters better than the ones available in the literature are provided in this work, in particular quantum codes with parameters $[[127,63, \geq 12]]_2$ and $[[63,45, \geq 6]]_4$ that are records. These codes are constructed with a new generalization of the Steane's enlargement procedure and by considering orthogonal subfield-subcodes --with respect to the Euclidean and Hermitian inner product-- of a new family of linear codes, the $J$-affine variety codes.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• proof
• Proposition 2
• Remark 1
• Theorem 4
• Theorem 5
• proof