A characterization of \(\mathbb F_q\)-linear subsets of affine spaces \(\mathbb F_{q^2}^n\)

Edoardo Ballico

doi:10.4067/S0719-06462022000100095

DOI:

https://doi.org/10.4067/S0719-06462022000100095

Abstract

Let \(q\) be an odd prime power. We discuss possible definitions over \(\mathbb F_{q^2}\) (using the Hermitian form) of circles, unit segments and half-lines. If we use our unit segments to define the convex hulls of a set \(S\subset \mathbb F_{q^2}^n\) for \(q\notin \{3,5,9\}\) we just get the \(\mathbb F_q\)-affine span of \(S\).

Keywords

Finite field , Hermitian form

Edoardo Ballico Department of Mathematics, University of Trento, 38123 Povo (TN), Italy.

Pages: 95–103
Date Published: 2022-04-04
Vol. 24 No. 1 (2022)

E. Ballico, “On the numerical range of matrices over a finite field”, Linear Algebra Appl., vol. 512, pp. 162–171, 2017.

E. Ballico, Corrigendum to “On the numerical range of matrices over a finite field” [Linear Algebra Appl., vol. 512, pp. 162–171, 2017], Linear Algebra Appl., vol. 556, pp. 421–427, 2018.

E. Ballico, “The Hermitian null-range of a matrix over a finite field”, Electron. J. Linear Algebra, vol. 34, pp. 205–216, 2018.

E. Ballico, “A numerical range characterization of unitary matrices over a finite field”, Asian-European Journal of Mathematics (AEJM) (to appear). doi: 10.1142/S1793557122500498

F. F. Bonsall and J. Duncan, Numerical ranges II, London Mathematical Society Lecture Note Series, no. 10, New York-London: Cambridge University Press, 1973.

K. Camenga, B. Collins, G. Hoefer, J. Quezada, P. X. Rault, J. Willson and R. J. Yates, “On the geometry of numerical ranges over finite fields”, Linear Algebra Appl., vol. 628, pp. 182–201, 2021.

Ch. Chorianopoulos, S. Karanasios and P. Psarrakos, “A definition of numerical range of rectangular matrices”, Linear Multilinear Algebra, vol. 57, no. 5, pp. 459–475, 2009.

J. I. Coons, J. Jenkins, D. Knowles, R. A. Luke and P. X. Rault, “Numerical ranges over finite fields”, Linear Algebra Appl., vol. 501, pp. 37–47, 2016.

K. E. Gustafson and D. K. M. Rao, Numerical range, Universitext, New York: Springer-Verlag, 1997.

J. W. P. Hirschfeld, Projective geometries over finite fields, Oxford Mathematical Monographs, New York: The Clarendon Press, Oxford University Press, 1979.

J. W. P. Hirschfeld and J. A. Thas, General Galois geometries, Oxford Mathematical Monographs, New York: The Clarendon Press, Oxford University Press, 1991.

R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge: Cambridge University Press, 1985.

R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge: Cambridge University Press, 1991.

L. Jin, “Quantum stabilizer codes from maximal curves”, IEEE Trans. Inform. Theory, vol. 60, no. 1, pp. 313–316, 2014.

K. Ireland and M. Rosen, A classical introduction to modern number theory, Second Edition, Graduate Texts in Mathematics, 84, New York: Springer-Verlag, 1990.

R. Ke, W. Li and M. K. Ng, “Numerical ranges of tensors”, Linear Algebra Appl., vol. 508, pp. 100–132, 2016.

J.-L. Kim and G. L. Matthews, “Quantum error-correcting codes from algebraic curves”, in Advances in algebraic geometry codes, Ser. Coding Theory Cryptol., vol. 5, Hackensack, NJ: World Sci. Publ., 2008, pp. 419–444.

R. Lidl and H. Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, 20, Cambridge: Cambridge University Press, 1997.

R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, Cambridge: Cambridge University Press, 1994.

C. Munuera, W. Tenório and F. Torres, “Quantum error-correcting codes from algebraic geometry codes of Castle type”, Quantum Inf. Process., vol. 15, no. 10, pp. 4071–4088, 2016.

P. J. Psarrakos and M. J. Tsatsomeros, “Numerical range: (in) a matrix nutshell”, National Technical University, Athens, Greece, Notes, 2004.

C. Small, Arithmetic of finite fields, Monographs and Textbooks in Pure and Applied, 148, New York: Marcel Dekker, Inc., 1991.