An invariant-theoretic approach to three weight enumerators of self-dual quantum codes

TL;DR

The paper develops an invariant-theoretic framework to study three weight enumerators of formally self-dual quantum codes by translating MacWilliams identities into group actions. It proves a quantum Gleason-type result: the Shor-Laflamme weight enumerator $B(x,y)$ is determined by two basic invariants, and the double weight enumerator $C(x,y,z,w)$ is governed by five invariants subject to a single relation. Two explicit low-dimensional examples ($q=2$ and $q=3$) for complete weight enumerators illustrate that the invariant ring is polynomial in the binary case and a complete intersection in the ternary case, with the Bell state code acting as a key exemplar. The work demonstrates how algebraic invariant theory provides both structural insights and computational tools for weight enumerators of self-dual quantum codes, potentially enabling systematic computation and bounds via invariant rings. Overall, the approach connects quantum coding theory with classical invariant theory, highlighting a path to explicitly characterize weight enumerators through finite-group actions.

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.

Theorems & Definitions (27)

• Example 2.1: Bell state code
• Example 2.2
• Example 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Example 3.1
• Theorem 3.3
• Remark 3.4
• proof : Proof of Theorem \ref{['thm3.3']}