Construction of T-matrices of order 6m + 1

Abstract

In this paper, we prove the necessary and sufficient condition for an integer n to equal a2 + 3b2. Consequently, every prime power 6m + 1 has a representation of the form a2 + 3b2. 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 integers {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 ≠ 9, we give concrete constructions.