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

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.