Skip to main content
SpringerLink
Log in

Non-symmetric association schemes of symmetric matrices

  • Published:
Acta Mathematicae Applicatae Sinica Aims and scope Submit manuscript

Abstract

LetX n be the set ofn ×n symmetric matrices over a finite fieldF q, whereq is a power of an odd prime. ForS 1,S 2 εX n , we define (S 1,S 2) εR 0 iffS 1 =S 2; (S 1,S 2) εR (r,ξ) iffS 1 -S 2 is congruent to

$$\left( {\begin{array}{*{20}c} {I^{(r - 1)} } & {} & {} \\ {} & \xi & {} \\ {} & {} & {0^{(n - r)} } \\ \end{array} } \right),$$

where ξ = 1 orz,z being a fixed non-square element ofF q . ThenX n = (X n , {R 0,R (r,ξ) | 1 ≤r ≤n, ξ = 1 orz}) is a non-symmetric association scheme of class 2n onX n . The parameters ofX n have been computed. And we also prove thatX n is commutative.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bannai, Eiichi and Ito, Tatsuro, Algebraic Combinatorics I Association Schemes, Benjamin/Cummings, 1984.

  2. Egawa, Yashimi, Association Schemes of Quadratic Forms,J. of Combinatorial Theory, Ser A,38 (1985), 1–14.

    Google Scholar 

  3. Hall, Marshell, Combinatorial Theory, 2nd ed., John Wiley and Sons, 1986.

  4. Huo Yuanji and Zhu Xueli, Association Schemes with Several Associate Classes of Symmetric Matrices,Acta Mathematicae Applicatae Sinica,10 (1987), 257–266 (in Chinese).

    Google Scholar 

  5. Wan Zhe-xian, Notes on Finite Geometries and the Construction of PBIB Designs VI Some Association Schemes and PBIB Designs Based on Finite Geometries,Acta Scientia Sinica,14 (1965), 1872–1876.

    Google Scholar 

  6. Wan Zhe-xian, Finite Geometries and Block Designs, Sankhza:The Indian Journal of Statistics, Series A,54 (1991).

  7. Wan Zhe-xizn, Anzahl Theorems in Finite Singular Symplectic, Unitary and Orthogonal Geometries, accepted for Publication inDiscret Mathematics.

  8. Wan Zhe-xian, Dai Zong-duo, Feng Xu-ning and Yang Ben-fu, Studies on Finite Geometries and the Construction of Block Designs, Science Press, Beijing, 1966 (in Chinese).

    Google Scholar 

  9. Wang Yang-xian, Association Schemes with Several Associate Classes of Matrices,Acta Mathematicae Applicate Sinica,3 (1980), 97–105 (in Chinese).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Zhangjiakou Teachers College, 075028, Zhangjiakou, China

    Huo Yuanji

  2. Institute of Systems Science, Academia Sinica, 100080, Beijing, China

    Wan Zhexian

Authors
  1. Huo Yuanji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wan Zhexian
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huo, Y., Wan, Z. Non-symmetric association schemes of symmetric matrices. Acta Mathematicae Applicatae Sinica 9, 236–255 (1993). https://doi.org/10.1007/BF02032918

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02032918

Keywords

Search

Navigation