Butson-Hadamard Codes and Related Quantum Codes

Abstract

A Butson-Hadamard (BH) matrix H is a square matrix of dimension n whose entries are complex roots of unity such that HH∗ = nI. In the first part of this thesis, we deal with codes obtained from BH matrices, called BH codes, focusing on their minimum distances. We first consider the usual Hamming distance and find lower bounds for distances of BH codes. Then we turn our attention to homogeneous weights, and search for distances of BH code families under these weights. Next, we introduce the notion of quasi-homogeneous weights as a generalization of homogeneous weights and show that certain BH codes equipped with quasi-homogeneous weights are Plotkin optimal. In addition, we obtain distances of BH codes under certain quasi-homogeneous weights. Our results are applied to determine parameters of p-ary codes projected under Gray isometries from BH codes over Z_{p^e} , where p is a prime number and e ≥ 2 is an integer. In the second part of this thesis, we study quantum stabilizer codes and give two constructions. In particular, we give a constructive proof to show that if there exist a classical linear code C ⊆ (F_q)^n of dimension k and a classical linear code D ⊆ (F_q^k)^m of dimension s, where q is a power of a prime number p, then there exists an [[nm, ks, δ]]_q quantum stabilizer code with δ 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.