On classification of toric surface codes of dimension seven

Abstract

In this paper, we give an almost complete classification of toric surface codes of dimension less than or equal to 7, according to monomially equivalence. This is a natural extension of our previous work [YZ], [LYZZ]. More pairs of monomially equivalent toric codes constructed from non-equivalent lattice polytopes are discovered. A new phenomenon appears, that is, the monomially non-equivalence of two toric codes C P (10) 7 and C P (19) 7 can be discerned on Fq , for all q ≥ 8, except q = 29. This sudden break seems to be strange and interesting. Moreover, the parameters, such as the numbers of codewords with different weights, depends on q heavily. More meticulous analyses have been made to have the possible distinct families of reducible polynomials.