CONSTRUCTION OF T-MATRICES OF ORDER 6m+1

In this paper, we prove the necessary and sufficient condition for an integer n = a^2+3b^2. Consequently, every prime power 6m+1 has a representation of the form a^2+3b^2. Then we show how to construct T-matrices of order 6m+1 by using 4 sequences of lengths r, r, 2m-r, 2m-r with r=m-2 or r=m in which the first is a subset of the integeres {0, 1, ..., 2m-1} with size r, the second and third sequences are of (1, -1), and every component of the last sequence belongs to the set {0, 1, 2}. For m <=13 and m not equals to 9, we give concrete constructions.