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Rank Duality of Circulant Matrices from Primitive Roots

Kenichi Takemura

Abstract

We investigate the construction of circulant matrices derived from primitive roots over finite fields. Our approach reduces exponential sums to Jacobi sums, thereby establishing explicit connections between character theory and matrix structures. The results provide new insights into the interaction between additive and multiplicative characters, and demonstrate how circulant configurations encode arithmetic information in a highly symmetric form. These findings contribute to a deeper understanding of structured matrices in finite fields and open further directions for applications in number theory and combinatorics.

Rank Duality of Circulant Matrices from Primitive Roots

Abstract

We investigate the construction of circulant matrices derived from primitive roots over finite fields. Our approach reduces exponential sums to Jacobi sums, thereby establishing explicit connections between character theory and matrix structures. The results provide new insights into the interaction between additive and multiplicative characters, and demonstrate how circulant configurations encode arithmetic information in a highly symmetric form. These findings contribute to a deeper understanding of structured matrices in finite fields and open further directions for applications in number theory and combinatorics.
Paper Structure (14 sections, 4 theorems, 36 equations, 1 table)

This paper contains 14 sections, 4 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction
  2. Definition of Circulant Matrices from Primitive Roots
  3. Multiplicative characters, Gauss sums and the first moment
  4. (A) The parity vanishing for nontrivial even characters.
  5. (B) The nonvanishing and explicit formula for odd characters.
  6. Remark on references and nonvanishing facts.
  7. Main Theorem
  8. Numerical Examples and verification
  9. Applications
  10. Coding theory
  11. Spectral graph theory
  12. Generalizations and further directions
  13. On the naming "Circulant Matrices from Primitive Roots"
  14. Conclusion

Key Result

Lemma 1

Let notation be as above. Then

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1: Evaluation of $S(\chi_k)$
  • proof
  • Corollary 1
  • Theorem 1: Rank duality of Circulant Matrices from Primitive Roots
  • proof
  • Lemma 2: Jacobi sums in terms of Gauss sums
  • proof