Quantum Anticodes

TL;DR

This paper develops a symplectic framework for quantum error correcting codes using anticodes, offering new invariants, operations, and insights into code structure and error correction phenomena.

Contribution

It introduces a symplectic anticode perspective for quantum codes, unifying various code families and extending classical concepts to quantum error correction.

Findings

Provides a symplectic framework encompassing stabilizer and subsystem codes.

Introduces new invariants capturing local algebraic and combinatorial features.

Defines operations like puncturing and shortening with algebraic interpretations.

Abstract

This work introduces a symplectic framework for quantum error correcting codes in which local structure is analyzed through an anticode perspective. In this setting, a code is treated as a symplectic space, and anticodes arise as maximal symplectic subspaces whose elements vanish on a prescribed set of components, providing a natural quantum analogue of their classical counterparts. This framework encompasses several families of quantum codes, including stabilizer and subsystem codes, provides a natural extension of generalized distances in quantum codes, and yields new invariants that capture local algebraic and combinatorial features. The notion of anticodes also naturally leads to operations such as puncturing and shortening for symplectic codes, which in turn provide algebraic interpretations of key phenomena in quantum error correction, such as the cleaning lemma and complementary recovery and yield new descriptions of weight enumerators.