Asymmetric Quantum Codes from τ-Paired Matrix-Product Codes

Asymmetric quantum codes are useful for quantum channels in which phase and bit errors occur with different probabilities, since the two distances, dz and dx, can be controlled separately. We develop a permutation-paired matrix-product construction for such codes over Fq. The main task is to build classical code pairs C,D⊆Fq2kn satisfying the Hermitian inclusion D⊥H⊆C, while keeping explicit dimension and distance bounds. Let A∈Fq2k×k be a non-singular-by-columns (NSC) matrix with AA†=DPτ, where D is an invertible diagonal and Pτ corresponds to an involution τ. For C=[C1,…,Ck]A and D=[D1,…,Dk]A, we prove D⊥H=[Dτ(1)⊥H,…,Dτ(k)⊥H]A. Thus, the global inclusion D⊥H⊆C is equivalent to the shorter paired inclusions Dτ(i)⊥H⊆Ci. This yields asymmetric quantum codes with parameters [[kn,∑i=1k(ri+si)−kn,dz/dx]]q, where the bounds for dz and dx follow from NSC matrix-product distance estimates. For nested maximum distance separable (MDS) constituents, the paired conditions reduce to ri+sτ(i)≥n, giving explicit infinite families. Concrete τ-OD matrices and numerical examples show that nontrivial permutations can increase the quantum dimension while preserving prescribed lower bounds for dz and dx.