Puncturing Quantum Stabilizer Codes

TL;DR

This paper generalizes the puncturing technique for quantum stabilizer codes, enabling the construction of new codes with improved parameters and extending classical bounds to the quantum setting.

Contribution

It introduces a flexible puncturing method based on stabilizer matrices, allowing for new code constructions and a generalized proof of the Griesmer bound for quantum codes.

Findings

Constructed quantum codes with parameters exceeding previous bests

Developed a stabilizer matrix-based puncturing method

Extended the Griesmer bound to quantum stabilizer codes

Abstract

Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. For quantum codes, a form of puncturing is known, but its description is based on the code space rather than its generators. In this work, we generalize the puncturing procedure to allow more freedom in the choice of which coded states are kept and which are removed. We describe this puncturing by focusing on the stabilizer matrix containing the generators of the code. In this way, we are able to explicitly describe the stabilizer matrix of the punctured code given the stabilizer matrix of the original stabilizer code. The additional freedom in the procedure also opens up new ways to construct new codes from old, and we present several ways to utilize this for the search of codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known. Lastly, we generalize the proof of the Griesmer bound from the classical setting to stabilizer codes since the proof relies heavily on the puncturing technique.