On the generalized Feng-Rao numbers of numerical semigroups generated by intervals (Q2840632)

Let \(S\) be a numerical semigroup, i.e., a subsemigroup of the additive semigroup of positive integers, containing all large integers, for \(\alpha\in S\) put \(D(\alpha)=\{\beta\in S:\;\alpha-\beta\in S\}\), and denote by \(\nu(\alpha)\) the cardinality of \(D(\alpha)\). One defines the \textit{Feng-Rao distance} \(\delta_{FR}(\alpha)\) of \(S\), by NEWLINE\[NEWLINE\delta_{FR}(\alpha)=\min\{\nu(\beta):\;\beta\in S, \beta\geq\alpha\}.NEWLINE\]NEWLINE This notion was generalized by \textit{P. Heijnen} and \textit{R. Pellikaan} [IEEE Trans. Inf. Theory 44, No. 1, 181--196 (1998; Zbl 1053.94581)] in the following way.NEWLINENEWLINELet \(m\in S\) and \(r\geq1\). Any \(r\)-element subset \(M\) of \(S\cap[m,\infty]\) is called a \((S,m,r)\) configuration, and denote by \(D(M)\) the union of the set of divisors of all elements of \(M\). The function NEWLINE\[NEWLINE\delta^r(m) = \min\{\#D(M):\;M\;\text{is\;a\;} (S,m,r)\;\text{configuration}\}NEWLINE\]NEWLINE is called the \textit{\(r\)th Feng-Rao distance} of \(S\). It is known \textit{J. I. Farrán} and \textit{C. Munuera} [Discrete Appl. Math. 128, No. 1, 145--156 (2003; Zbl 1037.94014)] that the number \(E(S,r) = \delta^r(m)-m-1+2g\) (where \(g\) denotes the number of positive integers \(n\notin S\)) does not depend on \(m\). It is called the \textit{\(r\)th Feng-Rao number} of \(S\).NEWLINENEWLINEThe authors present an algorithm for computing the \(r\)th Feng-Rao number for any numerical semigroup \(S\). They apply it in the special case when \(S\) is generated by an interval, and show that in this case one can obtain explicit formulas for the \(r\)-th Feng-Rao numbers.