The Postage Stamp Problem: An algorithm to determine the $h$-range on the $h$-range formula on the extremal basis problem for $k=4$

Abstract:

Given an integral “stamp" basis $A_k$ with $1=a_1 < a_2 <\ldots < a_k$ and a positive integer $h$, we define the $h$-range $n(h,A_k)$ as \[ n(h,A_k)=\max \{N\in {\mathbf N} \mid n \leq N \Longrightarrow n= \sum _{1}^{k}x_{i}a_{i}, \sum _{1}^{k}x_{i}\leq h,\;\;n,\;x_{i} \in {\mathbf N}_{0}\}.\] ${\mathbf N}_{0}={\mathbf N}\cup \{0\}$. For given $h$ and $k$, the extremal basis $A_{k}^{*}$ has the largest possible extremal $h$-range \[ n(h,k)=n(h,A_{k}^{*})=\max _{A_k} n(h,A_k).\] We give an algorithm to determine the $h$-range. We prove some properties of the $h$-range formula, and we conjecture its form for the extremal $h$-range. We consider parameter bases $A_k=A_k(h)$, where the basis elements $a_i$ are given functions of $h$. For $k=4$ we conjecture the extremal parameter bases for $h\geq 11385$.