Notes on classification of toric surface codes of dimension 5

This is an addendum to the beautiful paper by Little and Schwarz (Appl Algebra Eng Commun Comput 18:349–367, 2007) in which one case of toric surface codes of dimension 5 was missing in their classification result of toric surface codes of dimension less than 6. Our main purpose is to fill the gap of this paper. We find that our new code $${C_{P_5^{(7)}}}$$ enjoys more symmetry, and it has more codewords of minimum distance in general. However, over some special fields $${\mathbb{F}_{2^m}}$$, $${C_{P_5^{(5)}}}$$ and $${C_{P_5^{(7)}}}$$ have the same number of the codewords of minimum distance.

[1]  Diego Ruano,et al.  On the parameters of r-dimensional toric codes , 2005, Finite Fields Their Appl..

[2]  Ruud Pellikaan,et al.  The Newton Polygon of Plane Curves with Many Rational Points , 2000, Des. Codes Cryptogr..

[3]  David Joyner,et al.  Toric Codes over Finite Fields , 2002, Applicable Algebra in Engineering, Communication and Computing.

[4]  Johan P. Hansen,et al.  Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes , 2002, Applicable Algebra in Engineering, Communication and Computing.

[5]  Hal Schenck,et al.  Toric Surface Codes and Minkowski Sums , 2006, SIAM J. Discret. Math..

[6]  Johan P. Hansen,et al.  Toric Surfaces and Error-correcting Codes , 2000 .

[7]  John Little,et al.  On toric codes and multivariate Vandermonde matrices , 2007, Applicable Algebra in Engineering, Communication and Computing.

[8]  Yves Aubry,et al.  A Weil theorem for singular curves , 1996 .