Classifying toric surface codes of dimension 7 (Q2238473)

For the classification up to monomial equivalence of toric surface codes of dimension \(7\), the authors independently recover the results of \textit{N. Hussain} et al. [Commun. Anal. Geom. 28, No. 2, 263--319 (2020; Zbl 1455.14055)]. Essentially, they follow the same strategy as in previous cases by explicitly constructing the \(22\) equivalence classes of polygons with \(7\) lattice points by finding all possible ways to add an extra point to the the lattice equivalence classes of polygons with \(6\) lattice points. The main result is that these \(22\) classes of polygons generate monomially equivalent toric surface codes of dimension \(7\), with some exceptions and two cases over \({\mathbb F}_8\) that remain open. The minimum distances of these codes is explicitly given or bounded, when the cardinality of the base finite field is large. Denoting these equivalence classes of polygons by \(P_7^{(i)}\), with \(1\leq i\leq 22\), in the paper under review the authors correct an error in [Hussain et al., loc. cit.] for the minimum distance of the codes associated to the polygons \(P_7^{(i)}\) for \(i=16,18,19\), and prove that the minimum distance for the code associated to \(P_7^{(22)}\) is \((q-2)(q-3)\) improving the result of [Hussain et al., loc. cit.] that obtained this number as an upper bound.