Generalized Feng-Rao distance of an element
Definition
Let \(S\) be a numerical semigroup, let \(s \in S\) and \(r \in \mathbb{N} \setminus \{0\}\). It is defined the generalized Feng-Rao distance of \(s\) in \(S\), denoted by \(\delta_{FR}^r(s)\), as
\[ \delta_{FR}^r(s) = \{ |D(s_1, \ldots, s_r)| ~ : ~ s \le s_1 \le \cdots \le s_r \text{ and } s_1, \ldots, s_r \in S\}, \]
where \(D(s_1, \ldots, s_r) = \bigcup_{i = 1}^r D(s_i)\) and \(D(s_i)\) denotes the set of divisors of \(s_i \in S\). If \(r = 1\), we have the Feng-Rao distance of \(s\) in \(S\).
Given a numerical semigroup \(S\) with conductor \(C(S)\) and genus \(g(S)\), it can be proven that for all \(m \ge 2C(S) - 1\),
\[ \delta_{FR}^2(m) = m + 1 - 2g(S) + \min \{|Ap(S, n)| ~~ : ~ n \in \mathbb{N} \setminus \{0\}\}, \]
where \(Ap(S,n)\) is the Apéry set of the integer \(n\) in \(S\).
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle = \{0, 3, 5, \rightarrow \}\). We have \(C(S) = 5\) and \(g(S) = 3\). On the other hand, we know that for all \(n \in S\), it holds \(|Ap(S,n)| = n\), then it is sufficient to compute the Apéry set for \(n \in \{1, 2, 4\}\) and for \(n = m(S) = 3\), where \(m(S)\) denotes the multiplicity of \(S\). It is obtained that
\[ Ap(S, 1) = \{0, 3, 5\}, ~~ Ap(S,2) = \{0, 3, 6\}, \]
\[ |Ap(S, m(S))| = m(S) = 3, ~~ Ap(S,4) = \{0, 3, 5, 8\}. \]
Thus, if \(m \ge 2C(S) - 1 = 9\),
\[ \delta_{FR}^2(m) = m + 1 - 2g(S) + 3 = m -2. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 14, 15, 20, 24 \rangle\), in GAP:
gap> S := NumericalSemigroup(14, 15, 20, 24);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), an element \(n \in S\) and \(r \in \mathbb{N} \setminus \{0\}\), the function FengRaoDistance computes the \(r\) generalized Feng-Rao distance of \(n\) in \(S\).
gap> FengRaoDistance(S, 1, 62);
8
gap> FengRaoDistance(S, 2, 62);
12
gap> FengRaoDistance(S, 3, 62);
14References
https://gap-packages.github.io/ numericalsgps.