On the Hermitian curve and its intersections with some conics

We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In particular, we provide explicit counting formulas that have also applications to some Hermitian codes.

[1]  Chaoping Xing.  On Automorphism Groups of the Hermitian Codes , 1995, IEEE Trans. Inf. Theory.

[2]  H. Niederreiter,et al.  Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[3]  Nicola Durante,et al.  On the intersection of a Hermitian curve with a conic , 2010, Des. Codes Cryptogr..

[4]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[5]  E. Ballico,et al.  On Goppa codes on the Hermitian curve , 2012 .

[6]  Henning Stichtenoth,et al.  A characterization of Hermitian function fields over finite fields. , 1994 .

[7]  Ian F. Blake,et al.  Algebraic-Geometry Codes , 1998, IEEE Trans. Inf. Theory.

[8]  Olav Geil,et al.  On codes from norm-trace curves , 2003 .

[9]  Alain Couvreur,et al.  The dual minimum distance of arbitrary dimensional algebraic--geometric codes , 2009, ArXiv.

[10]  Alberto Ravagnani,et al.  A G ] 9 A ug 2 01 3 ON THE GEOMETRY OF HERMITIAN ONE-POINT CODES EDOARDO BALLICO , 2014 .

[11]  Rudolf Lide,et al.  Finite fields , 1983 .

[12]  Nicola Durante,et al.  On the intersection pattern of a unital and an oval in PG(2, q2) , 2009, Finite Fields Their Appl..

[13]  Chiara Marcolla.  On structure and decoding of Hermitian codes , 2013 .

[14]  F. Torres,et al.  Algebraic Curves over Finite Fields , 1991 .

[15]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[16]  Massimiliano Sala,et al.  On the Hermitian curve, its intersections with some conics and their applications to affine-variety codes and Hermitian codes , 2012 .

[17]  Henning Stichtenoth,et al.  A note on Hermitian codes over GF(q2) , 1988, IEEE Trans. Inf. Theory.

[18]  Alberto Ravagnani,et al.  On the minimum distance and the minimum weight of Goppa codes from a quotient of the Hermitian curve , 2012, ArXiv.

[19]  Chiara Marcolla,et al.  On the geometry of small weight codewords of dual algebraic geometric codes , 2011, ArXiv.