Feng-Rao distance of an element
Definition
Let \(S\) be a numerical semigroup and let \(s \in S\). It is defined the Feng-Rao distance of \(s\) in \(S\), denoted by \(\delta_{FR}(s)\), as
\[ \delta_{FR}(s) = \min \{ |D(s')| ~ : ~ s \le s' \text{ and } s' \in S\}, \]
where \(D(s)\) denotes the set of divisors of \(s\) in \(S\).
It can be proven that for any numerical semigroup \(S\) with conductor \(C(S)\) and genus \(g(S)\), if \(s \in S\) is such that \(s \ge 2C(S) - 1\), then
\[ \delta_{FR}(s) = s + 1 - 2g(S). \]
Examples
\(\circ\) Let \(S = \langle 4, 6, 11 \rangle\). It is obtained that \(C(S) = 14\) and \(g(S) = 7\). Therefore, if \(s \in S\) is such that \(s \ge 2C(S) - 1 = 27\), then \(\delta_{FR}(s) = s - 13\). For example, for \(s = 35 = 4 \cdot 6 + 11 \in S\), it holds \(\delta_{FR}(35) = 22\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 17, 35, 53, 78, 90 \rangle\), in GAP:
gap> S := NumericalSemigroup(17, 35, 53, 78, 90);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), an element \(s\) of it and \(r = 1\), the function FengRaoDistance computes the Feng-Rao distance of \(n\) in \(S\).
gap> FengRaoDistance(S, 1, 231);
26If we consider \(r \in \mathbb{N} \setminus \{0, 1\}\), the function computes the r-th Feng Rao distance of \(n\) in \(S\).
References
https://gap-packages.github.io/ numericalsgps.