Inhomogenous quantum codes (I): additive case

Abstract

In this paper, the quantum error-correcting codes are generalized to the inhomogenous quantum-state space $\mathbb{C}^{q_1 } \otimes \mathbb{C}^{q_2 } \otimes \cdots \otimes \mathbb{C}^{q_n }$, where q _i (1 ⩽ i ⩽ n) are arbitrary positive integers. By attaching an abelian group A _i of order q _i to the space Cqi $\mathbb{C}^{q_1 } \left( {1 \leqslant i \leqslant n} \right)$, we present the stabilizer construction of such inhomogenous quantum codes, called additive quantum codes, in term of the character theory of the abelian group A = A ₁⊕A ₂⊕...⊕ℂ _n . As usual case, such construction opens a way to get inhomogenous quantum codes from the classical mixed linear codes. We also present Singleton bound for inhomogenous additive quantum codes and show several quantum codes to meet such bound by using classical mixed algebraic-geometric codes.