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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.

Publication Date


  • 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

Number Of Pages


  • 18

Start Page


  • 1731

End Page


  • 1749

Volume


  • 101

Issue


  • 8

Place Of Publication


  • 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.

Publication Date


  • 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

Number Of Pages


  • 18

Start Page


  • 1731

End Page


  • 1749

Volume


  • 101

Issue


  • 8

Place Of Publication


  • India