The authors consider a novel statistical measure of the typical size of words of low weight in a linear code over a finite field, and use it to obtain a nice characterization of dual toric codes coming from polytopes of degree one, among all dual toric codes. More specifically, let \(k\) be a finite field, \(V\subseteq k^t\) be a linear code. For a set \(S \subseteq \{1,\ldots,t\}\), let \(w_S\) be the weight of the lowest weight word of \(V\) whose support contains \(S\). For an integer \(1\leq s \leq t\) and a subset \(B \subseteq \{1,\ldots,t\}\), let the mode of size \(s\) of \(V\), denoted \(M^B(s,V )\), be the most frequent \(w_S\) as \(S\) ranges over all subsets of \(B\) of size \(s\). Then, the following characterization is obtained:
(Theorem 1.4, 2.7) Let \(P \subseteq \mathbb{R}^m\) be any full-dimensional lattice polytope with at least \(m + 2\) lattice points, and let \(c := codim(X_P)\). For all but finitely many finite fields \(\mathbb{F}_q\) there is a cofinal subset C of the poset of finite fields containing \(\mathbb{F}_q\) such that, for every \(\mathbb{F}_r \in C\), we have \[ M^{X_P(\mathbb{F}_q)^{\circ}} (c + 1,C^*_P(\mathbb{F}_r )) \in \{c + 2,c + 3\} \] Moreover, it equals \( c + 3\) if and only if \(P\) is a polytope of degree one.
In order to establish this result, the authors use a characterization of the minimum distance of \(C^*_P (\mathbb{F}_r)\) as the cardinality of smallest sets of \(\mathbb{F}_r\) points \(S \subseteq X_P^{\circ}\) which does not impose independent conditions on linear forms in \(\mathbb{P}(H^0 (P)^*)\). Along the way, they establish that the minimal distance of \(C^*_P (\mathbb{F}_r)\) is at least three, and at most \(c+3\) for sufficiently large \(r\). A second ingredient to the proof is to consider \(c+1\) tuples of points of \(X_P^{\circ}\) which are generic tuples. The weight of shortest words which contain a generic tuple is independent of the chosen tuple. By proving a Bertini type result that there are sufficiently many generic tuples for all but finitely many \(\mathbb{F}_q\), it is established that this common weight turns out to be the mode. In other words, the mode equals the cardinality of smallest sets that impose linearly dependent conditions on linear forms on \(\mathbb{P}(H^0 (P)^*)\) while including a generic tuple. For a cofinal set of fields containing \(\mathbb{F}_q\), the authors show that this number can only equal \(c+2\) or \(c+3\), and it equals \(c+3\) precisely when \(X_P\) is a variety of minimal degree. As established in \([4]\), toric varieties of minimal degree has a one-one correspondence with lattice polytopes of degree one.
In addition, using a recent combinatorial characterization of lattice polytopes of degree one [V. Batyrev and B. Nill, Mosc. Math. J. 7, No. 2, 195–207 (2007; Zbl 1134.52020)], the authors extend works in [J. Little and H. Schenck, SIAM J. Discrete Math. 20, No. 4, 999–1014 (2006; Zbl 1131.14026)] from dimension two to high dimensions and characterize the length, dimension and minimal distance of primal codes \(C_P(\mathbb{F}_q)\) associated with polytopes of degree one. Finally, the parameters of some examples are given when they come from Lawrence polytopes.