 # Construction of T-matrices of order 6m+1

Journal Article

### 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.

• 2017

### Citation

• 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.

### Scopus Eid

• 2-s2.0-85018437041

• 18

• 1731

• 1749

• 101

• 8

• India

### 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.

• 2017

### Citation

• 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.

### Scopus Eid

• 2-s2.0-85018437041

• 18

• 1731

• 1749

• 101

• 8

• India