A Construction of Quantum Stabilizer Codes from Classical Codes and Butson Hadamard Matrices

In this paper, we give a constructive proof to show that if there exist a classical linear code C is a subset of F_q^n of dimension k and a classical linear code D is a subset of F_q^k^m of dimension s, where q is a power of a prime number p, then there exists an [[nm, ks, d]]_q quantum stabilizer code with d determined by C and D by identifying the stabilizer group of the code. In the construction, we use a particular type of Butson Hadamard matrices equivalent to multiple Kronecker products of the Fourier matrix of order p. We also consider the same construction of a quantum code for a general normalized Butson Hadamard matrix and search for a condition for the quantum code to be a stabilizer code.