An invariant-theoretic approach to three weight enumerators of self-dual quantum codes
Abstract
This article is a continuation of our recent work (Yin Chen and Runxuan Zhang, Shape enumerators of self-dual NRT codes over finite fields. SIAM J. Discrete Math. 38 (2024), no. 4, 2841-2854) in the setting of quantum error-correcting codes. We use algebraic invariant theory to study three weight enumerators of formally self-dual quantum codes over arbitrary finite fields. We derive a quantum analogue of Gleason's theorem, demonstrating that the weight enumerator of a formally self-dual quantum code can be expressed algebraically by two polynomials. We also show that the double weight enumerator of a formally self-dual quantum code can be expressed algebraically by five polynomials. We explicitly compute the complete weight enumerators of some special self-dual quantum codes. Our approach illustrates the potential of employing algebraic invariant theory to compute weight enumerators of self-dual quantum codes.
Keywords
Cite
@article{arxiv.2409.03576,
title = {An invariant-theoretic approach to three weight enumerators of self-dual quantum codes},
author = {Yin Chen and Shan Ren and Runxuan Zhang},
journal= {arXiv preprint arXiv:2409.03576},
year = {2026}
}
Comments
Dr. Runxuan Zhang has joined the research team. The current version of the manuscript has been substantially revised and expanded to 21 pages, with four new examples of formally self-dual quantum codes and a new Proposition 2.4. The manuscript has been submitted for publication