Communications in Analysis and Geometry

Volume 28 (2020)

Number 2

On classification of toric surface codes of dimension seven

Pages: 263 – 319

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n2.a3

Authors

Naveed Hussain (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and School of Data Sciences, Huashang College Guangdong University of Finance and Economics, Guangzhou, Guangdong, China)

Xue Luo (School of Mathematical Sciences, Beihang University, Beijing, China)

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Mingyi Zhang (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)

Huaiqing Zuo (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

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. 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 $C_{P_7} {}_{\mathsf{(10)}}$ and $C_{P_7} {}_{\mathsf{(19)}}$ can be discerned on $\mathbb{F}_q$, for all $q \geq 8$, 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.

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This work is financially supported by National Natural Science Foundation of China (NSFC) (11961141005, 11531007). Zuo thanks the support of NSFC (11771231); Luo acknowledges the support of NSFC (11871003) and the Fundamental Research Funds for the Central Universities (YWF-20-BJ-J-644); and Yau is supported by Tsinghua University start-up fund and Tsinghua University education foundation fund (042202008).

Received 14 May 2017

Accepted 2 November 2017

Published 6 May 2020