The eigenstructure of finite field trigonometric transforms
Let \(\mathrm{GF}(q)\) denote a finite field with \(q\) elements. Consider the set \(\mathrm{GI}(p)=\{a+jb\,|\,a,b\in\mathrm{GF}(p)\}\), where \(p\) is a prime such that \(j^2\equiv -1\,(\mathrm{mod}\,p)\) is a quadratic nonresidue over~\(\mathrm{GF}(p)\), i.e., \(p\equiv 3\,(\mathrm{mod}\,4)\). Let \(\zeta=a+jb\in\mathrm{GI}\,(p)\) satisfy \(a^2+b^2\equiv 1\,(\mathrm{mod}\,p)\), and let \(\mathrm{ord}\,(\zeta)\) denote its multiplicative order. Originating from \textit{R.~M.~Campello de Souza, H.~M.~de Oliveira, A.~N.~Kauffman} and \textit{A.~J.~A.~Paschoal} [Proc. IEEE Internat. Sympos. Informat. Theory, p.~293 (1998)], define \[ \cos_\zeta(x)=\frac{\zeta^x+\zeta^{-x}}{2},\quad \sin_\zeta(x)=\frac{\zeta^x-\zeta^{-x}}{2j},\quad x= 0,1,\dots,\mathrm{ord}\,(\zeta)-1. \] The authors present various types of trigonometric transformations that change a vector with entries in~\(\mathrm{GF}(p)\) into one with entries in~\(\mathrm{GI}(p)\). They find the eigenvalues of the transformation matrices and study their multiplicities. They also propose procedures for constructing corresponding eigenvectors. Finally, the authors suggest applications to multiuser communication systems and error-correcting codes.
scientific article; zbMATH DE number 5924562