Construction of T-matrices of order 6m+1

Authors

Mingyuan Xia, Central China Normal University, Huazhong Normal UniversityFollow
Tianbing Xia, University of WollongongFollow
Jennifer Seberry, University of WollongongFollow
Hong Qin, Central China Normal UniversityFollow

RIS ID

114789

Publication Details

Xia, M., Xia, T., Seberry, J. & Qin, H. (2017). Construction of T-matrices of order 6m+1. Far East Journal of Mathematical Sciences, 101 (8), 1731-1749.

Abstract

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.