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.

中文翻译：

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七维复曲面码的分类

在本文中，我们根据单项式等价性给出了维数小于或等于 7 的复曲面代码的几乎完整分类。这是我们之前工作 [YZ]、[LYZZ] 的自然延伸。发现了更多对由非等价晶格多面体构造的单项式等价复曲面代码。出现了一个新现象，即在 Fq 上可以看出两个复曲面代码 CP (10) 7 和 CP (19) 7 的单项不等价，对于所有 q ≥ 8，除了 q = 29。变得奇怪而有趣。此外，参数，例如不同权重的码字数量，很大程度上取决于q。已经进行了更细致的分析，以获得可约多项式的可能不同族。