Counting generalized Reed-Solomon codes

In this article we count the number of $[n, k]$ generalized Reed-Solomon (GRS) codes, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of $[n, 3]$ MDS codes with $n=6, 7, 8, 9$.

[1]  Robert Rolland.  The number of MDS [7, 3] codes on finite fields of characteristic 2 , 2005, Applicable Algebra in Engineering, Communication and Computing.

[2]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[3]  Aart Blokhuis,et al.  Finite Geometries , 2018, Des. Codes Cryptogr..

[4]  Richard C. Singleton,et al.  Maximum distance q -nary codes , 1964, IEEE Trans. Inf. Theory.

[5]  David G. Glynn.  An Invariant for Hypersurfaces in Prime Characteristic , 2012, SIAM J. Discret. Math..

[6]  J. Hirschfeld.  Finite projective spaces of three dimensions , 1986 .

[7]  Beniamino Segre,et al.  Lectures on modern geometry , 1962 .

[8]  David G. Glynn.  Rings of geometries I , 1987, J. Comb. Theory, Ser. A.

[9]  Krishna V. Kaipa,et al.  An Asymptotic Formula in \(q\) for the Number of \([n, k]~q\) -Ary MDS Codes , 2013, IEEE Trans. Inf. Theory.

[10]  J. Thas,et al.  General Galois geometries , 1992 .

[11]  Wei-Liang Chow,et al.  On the Geometry of Algebraic Homogeneous Spaces , 1949 .

[12]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

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

[14]  Arne Dür,et al.  The automorphism groups of Reed-Solomon codes , 1987, J. Comb. Theory A.

[15]  David G. Glynn,et al.  A condition for arcs and MDS codes , 2011, Des. Codes Cryptogr..

[16]  J. Hirschfeld.  Projective Geometries Over Finite Fields , 1980 .

[17]  Simeon Ball,et al.  On sets of vectors of a finite vector space in which every subset of basis size is a basis II , 2012, Designs, Codes and Cryptography.

[18]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[19]  Jan De Beule,et al.  On sets of vectors of a finite vector space in which every subset of basis size is a basis II , 2012, Des. Codes Cryptogr..

[20]  Alexei N. Skorobogatov,et al.  Formula for the number of [9, 3] MDS codes , 1995, IEEE Trans. Inf. Theory.

[21]  David G. Glynn.  The non-classical 10-arc of PG(4, 9) , 1986, Discret. Math..

[22]  G. Lachaud,et al.  Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes , 2001 .