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\begin{center}
\vskip 1cm{\LARGE\bf Some Extremal Postage Stamp Bases
}
\vskip 1cm
\large
Michael F. Challis\\
Hempnall House\\
Lundy Green Hempnall\\
Norwich NR15 2NU \\
United Kingdom\\
\href{mailto:mike@hempnallhouse.co.uk}{\tt mike@hempnallhouse.co.uk}\\
\ \\
John P. Robinson \\
Center for Bioinformatics and Computational Biology \\
University of Iowa \\
Iowa City, IA  52242 \\
USA \\
\href{mailto:jprobin@engineering.uiowa.edu}{\tt john-robinson@uiowa.edu} \\
\end{center}

\vskip .2 in

\begin{abstract}
A set of $k$ positive integers is a postage stamp basis for $n$ if
every positive integer up to $n$ can be expressed as the sum of no more
than $h$ values from the set.  An extremal basis is one for which $n$
is as large as possible.  For the case $h=k=8$, the unique extremal
basis is $A=\{1,8,13,58,169,295,831,1036\}$, with $n=3485$.  Several
other new extremal bases are presented, along with corrections to 
a previous article.  
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{defin}[theorem]{Definition}
\newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}}
\newtheorem{examp}[theorem]{Example}
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\section{Introduction}

The global postage-stamp problem consists of determining, for given positive integers $h$ and $k$, a set of $k$ positive integers

$$A_k=\{a_1=1<a_2<\cdots<a_k\}$$

\noindent such that

\begin{enumerate}

\item[(a)] sums of $h$ or fewer of these $a_j$ can realize the numbers $1,2,\ldots,n$, and
\item[(b)] the value of $n$ is as large as possible.

\end{enumerate}

The $h$-range for a particular set $A_k$ is denoted by $n_h(A_k)$ and
the extremal value by $n_h(k)$.  Mossige \cite{Mos81} presented
efficient search algorithms for determining $n_h(k)$, which Shallit
\cite{Sha02} has shown to be NP hard in $k$.  Various techniques (e.g.,
Challis \cite{Cha93}) have been used to reduce the effort to compute
$n_h(k)$, and a further improvement is described below.

The most recent results are presented in an appendix.  Corrections and
improvements to Robinson \cite{Rob09} are also given.

\section{Tree representation}

A set $A_k$ is said to be admissible (strictly, $h$-admissible) if
$n_h(A_k) \geq a_k$; clearly inadmissible sets are of no interest when
searching for the extremal value $n_h(k)$.

The natural tree representation for all admissible postage stamp sets
associates $a_1=1$ with the root.  The root branches out to $h$
admissible nodes at the second level for the $a_2$ values
$2,3,\ldots,h+1$.  At level three and beyond, the number of successor
states for a particular state can vary.  For any given admissible
denomination vector $A$ there is a path.

For the diagonal case $h=k$, all possible admissible sets were examined
for the cases $h=6,7,8$ and about 100 million random admissible sets
for $h=9$.  Table \ref{pre} summarizes these experiments.  The average
fan-out is over all admissible sets.  Note that the average fan-out
increases down the tree and for the same level grows slowly with $h$.
The number of admissible sets for $h=9$ is an estimate based on the
experimental average values.

\begin{table}[!ht]
\centering
\begin{minipage}{\textwidth}
\centering
\bigskip

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
 & Admissible & \multicolumn{8}{c|}{Average fan-out by level} \\
$h$ & cases & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{2} & \multicolumn{1}{c}{3}  & \multicolumn{1}{c}{4} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{6} & \multicolumn{1}{c}{7} & \multicolumn{1}{c|}{8}\\  \hline
6 & $5.2\e{6}$ & 6 & 11.8 & 23.9 & 43.2 & 71.6 & & & \\  \hline
7 & $5.6\e{9}$ & 7 & 15.0 & 32.5 & 64.0 & 114 & 234 & & \\  \hline
8 & $2.5\e{13}$ & 8 & 18.5 & 42.8 & 90.8 & 174 & 307 & 525 & \\  \hline
9\footnotemark[1] & $3.7\e{17}$ & 9 & 22.3 & 54.7 & 125 & 256 & 472 & 860 & 1502 \\  \hline
\end{tabular}
\end{minipage}
\caption{Cases and fan-out}
\label{pre}
\end{table}

\footnotetext[1]{Estimates.}

\section{\texorpdfstring{Algorithm for case $h=k=8$}{Algorithm for case h=k=8}}

The number of admissible postage stamp sets grows rapidly for the diagonal case (Table \ref{pre}).  Level $k-1$ has the largest average fan-out; e.g., over 500 for $h=8$.  We extend the method of Challis \cite{Cha93} wherein a set $A$ can be rejected if it can be shown that a particular needed total $x$ cannot be represented by $A$.  Whereas Challis uses a software cache to speed up his \emph{generate}($x$) procedure, we use look-up tables.  We next outline this speed-up for $h=8$.

For each preamble of $a_1=1$ to $a_7$, eight intermediate look-up tables are calculated.  Each table lists all totals of $a_1$ to $a_7$ using $len$ denominations or less, for $1\leq len\leq 8$.  These calculations are amortized over all the admissible values of $a_8$.  Next, the target total $x$ is checked, in one look-up step in the table $len=8$, to determine if some combination of $a_1$ to $a_7$ totals to $x$.  If not, then the table $len=7$ is checked for the total $x-a_8$; if it exists, this yields a combination totaling $x$ using one copy of $x_8$.  If not, another $a_8$ is subtracted, and the $len=6$ table is checked for a combination using two copies of $x_8$.  Continue down to $len=1$, finally checking whether $8a_8=x$.  When a combination for $x$ is found, the scan down the eight tables is exited and the entire process is repeated with another target $x$.  If no combination equal to $x$ is found in these nine steps, $A$ with $a_8$ is rejected, and another value for $a_8$ is tested.

The set of target values for $x$ are the totals needed to equal the current best $A$.  If we generate all these totals, then $n_8(A)$ is computed, and the target set is updated.  This process yielded a net speed-up factor of more than 20 over the unoptimized procedure for $h=8$.

There are more than $10^{13}$ admissible sets in the $(8,8)$ case.  The unique extremal basis $A=\{1,8,13,58,169,295,831,1036\}$ with $n=3485$ was determined in 2002 by the first author and independently in 2009 by the second using the program described above.  Challis \cite{Cha93} describes two different algorithms (the \emph{K-program} and \emph{H-program}) and the 2002 result was obtained using the \emph{K-program}, whereas the 2009 result was obtained using an independent algorithm based on the \emph{H-program}.

\section{New extremal sets}

Many other new results have been obtained by Challis, and a complete table of extremal bases known to him at the time of writing (September 2009) is included as Appendix A to this paper.

These results have been obtained using the algorithms described in Challis \cite{Cha93}, but with one further improvement made to the $H$-program which enables a number of candidate sets $A_k$ to be rejected as a group.

Suppose that we have already determined a reasonably good lower bound $T$ for $n_h(k)$. In \cite{Cha93} we show that the following value $X < T$ is a ``difficult" target for $A_k$: that is, in almost all cases (provided $h \gg k$) there is no generation of $X$ as the sum of no more than $h$ values from $A_k$:
\begin{equation}
\label{eq:1}
X = (C_k - 1)a_k + (C_{k-1} - 1)a_{k-1} + \ldots + (C_2 - 1)a_2 + (a_2 - 1)
\end{equation}
where
$$T = C_ka_k + R_k\mbox{ for } 0 \leq R_k < a_k,$$
and
$$a_i = C_{i-1}a_{i-1} + R_{i-1}\mbox{ for }0 \leq R_{i-1} < a_{i-1}, \, 3 \leq i < k .$$

We now suppose that we can find a value $x$ such that
	$$x \leq X \mbox{ and } x  > (C_k - 2)a_k + (h - C_k + 2)a_{k-1}, $$
where the second condition above means that if $x$ is to have a valid
generation as the sum of no more than $h$ values from $A_k$, then it
will require at least $(C_k - 1)$ values $a_k$. Of course there is no
guarantee that we can find such values for given $T$ and $A_k$, but in
practice it turns out often to be the case.

Suppose further that $x$ cannot be generated by $A_k$ at all. We now show how under certain conditions it is possible to derive a value $x'$ which cannot be generated by $A_{k}'$ where
 $a_{k}' = a_k - r$, $r > 0$, $a_{i}' = a_i$, $1 \leq i < k$. As $a_k$ decreases, so $C_k$ may increase and $C_{k-1}$ may decrease; other values $C_i$ will remain unchanged. One extra condition that we require is that $C_{k-1}$ also remains unchanged. So we have:
$$a_{k}' = a_k - r,\,  C_{k}' = C_{k} + j,\,  r > 0,\,  j \geq 0$$
with
\begin{equation}
C_{(k-1)}' = C_{k-1}
\label{eq:2}
\end{equation}							

Now consider
$$x' = x + j(a_k - r) - r(C_k - 1)$$
and suppose further (our second condition) that $x'$, in analogy with $x$, satisfies
\begin{equation}
x'  > (C_{k}' - 2)a_{k}' + (h - C_{k}' + 2)a_{k-1}.
\label{eq:3}
\end{equation}					

We show that $A_{k}'$ cannot generate $x'$ by contradiction. Suppose that such a generation exists.  Then it must include exactly $(C_{k}' - 1)$ values $a_{k}'$ and so will be of the form:
$$x' = (C_{k}' - 1)a_{k}' + f_{k-1}a_{k-1} + \ldots + f_2a_2 + f_1$$  
with  
$$(C_{k}' - 1) + f_{k-1} + \ldots + f_1 \leq h$$

Substituting for $x'$, $C_{k}'$ and $a_{k}'$ we have
$$x + j(a_k - r) - r(C_k - 1) = (C_{k} + j - 1)(a_k - r) + f_{k-1}a_{k-1} + \ldots + f_2a_2 + f_1$$
or
$$x = (C_k - 1)a_k + f_{k-1}a_{k-1} + \ldots + f_2a_2 + f_1  \mbox{ with }  (C_{k}' - 1) + f_{k-1} + \ldots + f_1 \leq h .$$
But since $C_{k}' \leq C_k$ this means that $x$ can be generated by $A_k$, which is contrary to hypothesis.

We now know that $x'$ cannot be generated by $A_{k}'$, but in order to reject $A_{k}'$ we must also show that $x'$ is less than $T$. If we write
$$x = (C_k - 1)a_k + F,$$
we have
$$x' = (C_k - 1)a_k + F + j(a_k - r) - r(C_k - 1) = (C_{k}' - 1)a_{k}' + F .$$

From (\ref{eq:1}) we know that $F \leq (C_{k-1} - 1)a_{k-1} + \ldots + (C_2 - 1)a_2 + (a_2 - 1)$, and so
$$x' \leq (C_{k}' - 1)a_{k}' + (C_{k-1} - 1)a_{k-1} + \ldots + (C_2 - 1)a_2 + (a_2 - 1) = X'$$
where $X'$ is the difficult target for $A_{k}'$, so $x' < T$.

Once we have discovered a suitable value $x$ we can reject all sets $A_{k}'$ for $r=0,1,\dots$
without further ado so long as the two conditions (\ref{eq:2}) and (\ref{eq:3}) are met. It is not difficult to calculate when these conditions will fail, and as the average length of the ``chain" of rejected sets is quite substantial for large $h$, the extra cost of the requisite book-keeping code is more than offset by the savings made. This is illustrated in the following table:

\bigskip
\begin{center}
\begin{tabular}{cccc}
$k$&$h$&Average chain length&Speed increase\\
4&150&1088&7\%\\
5&35&395&11\%\\
6&14&124&9\%
 \end{tabular}
 \end{center}
 \bigskip

\section{Corrections to ``Some extremal postage stamp 2-bases"}

Of minor importance are the following corrections to Table 1 of the original article \cite{Rob09}:

\begin{center}
\begin{tabular}{cll}
Stride $s$&Original value&Correct value\\
13&1,654&1,658\\
17&246,196&246,169\\
21&69,076,273&69,075,740\\
\end{tabular}
\end{center}

A counting bug was found in the original program which was most serious for $s=21$.

The type of 2-bases investigated by Robinson \cite{Rob09} were also investigated by Mossige \cite{Mos91}, who first describes preambles $\mathrm{PA}(s,a_s)$ and then describes a property \emph{extensibility}.  Consider the following family of bases derived from $\mathrm{PA}(s,a_s)$:

$$A_{p+s}=\{1,a_2,\ldots,a_s=b_0,a_s+s=b_1,a_s+2s=b_2,\ldots,a_s+ps=b_p\}$$

\bigskip

We say $\mathrm{PA}(s,a_s)$ is \emph{extensible} if $A_{p+s}$ is admissible for all $p$.

Next we say that an extensible preamble $\mathrm{PA}(s,a_s)$ is
\emph{symmetricizable} if the family of the symmetric bases $A_k$
derived from it in the obvious way (see Robinson \cite{Rob09}) are
admissible for all $k\geq k_0$ for some $k_0$.  It is straightforward
to show that $\mathrm{PA}(s, a_s)$ is extensible if $A_{p+s}$ is
admissible for some $p$ such that $b_{p-1} \geq 2b_0$, and that it is
symmetricizable if the derived symmetric basis is admissible for some
$p$ such that $b_p \geq 2b_0$. These facts were used by Challis when
coding his algorithm to determine optimal preambles $\mathrm{PA}(s,
a_s)$.

What is perhaps surprising is that an extensible preamble is not
necessarily symmetricizable, and this is the reason for the two errors
in Table 2 of the original article:  the preambles for $s=16$ and
$s=24$, although extensible, are not symmetricizable.  As an example,
consider the 2-basis for $k=52$ derived from $\mathrm{PA}(16,61)$; it
is easy to show that the value 415 cannot be generated by this basis.

Let us call a preamble that is extensible but not symmetricizable \emph{anomalous}; then there are no anomalous preambles for $s<14$.  Even afterwards, there are very few, although the proportion increases slightly as $s$ increases.  For $s=21$, just 1.3\% of extensible preambles are anomalous.

Here are corrected versions of Table 2 and Table 3 from Robinson \cite{Rob09}:

\begin{table}[ht]
\caption{Most efficient PAs for $s=11,\ldots,26$}
\begin{center}
\begin{tabular}{cl}
$s$&$A$\\
11&$\{1,3,4,7,8,9,16,17,21,24,35\}$\\
13&$\{1,2,5,7,10,11,19,21,22,25,29,30,43\}$\\
15&$\{1,2,5,6,8,9,13,19,22,27,29,33,40,41,56\}$\\
16&$\{1,2,3,7,8,9,12,15,22,26,30,36,37,43,45,61\}$\\
17&$\{1,2,5,6,7,12,13,16,26,28,31,37,38,42,44,49,66\}$\\
19&$\{1,2,3,6,9,11,12,15,16,27,32,37,45,48,52,55,61,62,80\}$\\
20&$\{1,2,4,5,11,13,14,19,29,35,37,43,46,47,50,52,56,58,68,88\}$\\
21&$\{1,2,3,6,10,14,17,19,26,29,36,41,49,51,54,55,58,60,67,74,95\}$\\
22&$\{1,3,5,7,8,12,14,18,26,32,33,42,43,50,60,63,68,79,81,83,97,105\}$\\
24&$\{1,3,5,6,13,15,16,18,22,38,41,44,47,52,55,58,59,60,74,80,81,91,93,117\}$\\
%25&$\{1,3,5,6,9,12,15,16,20,23,33,36,39,44,57,60,67,71,72,79,88,93,99,102,124\}$\\
%25&$\{1,3,4,6,9,11,12,14,15,20,32,33,41,43,47,48,60,67,71,80,88,94,99,102,124\}$\\
%26&$\{1,3,5,6,9,13,15,17,20,21,30,33,44,49,50,63,64,74,77,86,92,94,97,106,114,132\}$\\
26&$\{1,3,4,6,7,14,16,19,20,28,36,38,39,48,49,60,61,70,76,77,89,93,95,99,109,135\}$
\end{tabular}
\end{center}
\end{table}

The changes are an alternative $\mathrm{PA}(16,61)$ that is symmetricizable, a new $\mathrm{PA}(17,66)$ that gives rise to an optimal basis for $k=57$, a replacement for $\mathrm{PA}(24,118)$, and the addition of $\mathrm{PA}(26,135)$.

\begin{table}[h!]
\caption{Best $n_2(A_k)$ for the symmetric bases}
\begin{center}
\begin{tabular}{lllllll}
$s$&&$a_s$&&$k$-range&&$n_2(A_k)$\\
11&&35&&30-40&&$22k-344$\\
13&&43&&40-43&&$26k-504$\\
15&&56&&43-52&&$30k-676$\\
16&&61&&52-56&&$32k-780$\\
17&&66&&56-58&&$34k-892$\\
19&&80&&58-62&&$38k-1124$\\
20&&88&&62-67&&$40k-1248$\\
22&&105&&67-80&&$44k-1516$\\
24&&117&&80-82&&$48k-1836$\\
%25&&124&&84-86&&$50k-2004$\\
%26&&132&&86-88&&$52k-2176$\\
26&&135&&82-?&&$52k-2164$
\end{tabular}
\end{center}
\end{table}

Interestingly, Mossige missed $\mathrm{PA}(16,61)$ and noted that for $9\leq s\leq 19$ all of the best bases were derived from preambles with odd $s$, so although he calculated as far as $s=23$ he omitted $s=20$ and $s=22$.

Recently (see section \ref{sec:AppA}), an exhaustive search for optimal 2-bases for $k=21$ found the following to be extremal:

\begin{center}
\begin{tabular}{lll}
$k$&$n(2,k)$&$A$\\
21&164&$\{1,3,4,6,10,13,15,21,29,37,45,53,61,69,73,75,78,79,82,84,88\}$\\
&&$\{1,3,4,6,10,13,15,21,29,37,45,53,61,69,73,75,78,79,80,82,84\}$\\
&&$\{1,3,4,6,10,13,15,21,29,37,45,53,61,67,69,72,76,78,79,81,82\}$\\
&&$\{1,3,4,5,8,14,20,26,32,38,44,50,56,62,68,74,77,78,79,81,82\}$\\
\end{tabular}
\end{center}

This is the first example for $k>14$ where there are non-symmetric extremal solutions in addition to the expected symmetric bases.


\section{Discussion}

It was found that $n(8)=3485$ is 40 larger than the next best $(8,8)$
set with $n_8(A)=3445$.  The improved rejection factor technique
presented in this note, when applied to the $h=9$ case, appears to be
about 50.  Our estimate of more than $10^{17}$ admissible cases
indicates that this approach is insufficient for $h=9$ with the current
computer technology.  In the random experiments it was found that
$n(9)  \geq9338$, which is more than twice the Fibonacci lower bound of
4180 \cite{Alt77}.

\section{\texorpdfstring{Appendix A:  Table of extremal ranges $n(h,k)$ with corresponding $h$-bases $A_k$}{Appendix A:  Table of extremal ranges n(h,k) with corresponding h-bases Ak}}
\label{sec:AppA}

Note that the smaller values in the following table were previously published by the first author \cite{Cha93}.

%Put the full text of results from Dr. challis' email in the Appendix as Table 4, or a several tables, in whatever format takes the least time for you to prepare.  Keep the current version in case the reviewer(s)/editor require just the new results.

%It would be really nice to include all the results as an Appendix (ie after the references); new results since 1993 could always be identified by an asterisk. But if the referees really don't want this, could I ask for a layout for Table 4 that is more akin to my 1993 appendix, with separate sections for h =2, h =3, k= 4, k =5 etc, and right-aligned columns of numbers?  The attached PostScript files show what I mean. We should also include the new results for k = 4: in 1993 I had confirmed that the sets defined by the formulae (A) and (B) were extremal as far as h=157,  but can now confirm for all h <= 241.

\bigskip

$\mathbf{h=2}$

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrrrrrrrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(2,k)$}&&$a_i$\\
3 & 8 && 1&3&4\\
4 & 12 && 1&3&5&6\\
5 & 16 && 1&3&5&7&8\\
6 & 20 && 1&2&5&8&9&10\\
6 & 20 && 1&3&4&8&9&11\\
6 & 20 && 1&3&4&9&11&16\\
6 & 20 && 1&3&5&6&13&14\\
6 & 20 && 1&3&5&7&9&10\\
7 & 26 && 1&2&5&8&11&12&13\\
7 & 26 && 1&3&4&9&10&12&13\\
7 & 26 && 1&3&5&7&8&17&18\\
\end{tabular}

\begin{sidewaystable}
$\mathbf{h=2}$ \textbf{(continued)}

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrrrrrrrrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(2,k)$}&&$a_i$\\
8 & 32 && 1&2&5&8&11&14&15&16\\
8 & 32 && 1&3&5&7&9&10&21&22\\
9 & 40 && 1&3&4&9&11&16&17&19&20\\
10 & 46 && 1&2&3&7&11&15&19&21&22&24\\
10 & 46 && 1&2&5&7&11&15&19&21&22&24\\
11 & 54 && 1&2&3&7&11&15&19&23&25&26&28\\
11 & 54 && 1&2&5&7&11&15&19&23&25&26&28\\
11 & 54 && 1&3&4&9&11&16&18&23&24&26&27\\
11 & 54 && 1&3&5&6&13&14&21&22&24&26&27\\
12 & 64 && 1&3&4&9&11&16&21&23&28&29&31&32\\
13 & 72 && 1&3&4&9&11&16&20&25&27&32&33&35&36\\
14 & 80 && 1&2&5&8&11&14&17&20&23&24&25&51&53&55\\
14 & 80 && 1&3&4&5&8&14&20&26&32&35&36&37&39&40\\
14 & 80 && 1&3&4&9&10&15&16&21&22&24&25&51&53&55\\
15 & 92 && 1&3&4&5&8&14&20&26&32&38&41&42&43&45&46\\
16 & 104 && 1&3&4&5&8&14&20&26&32&38&44&47&48&49&51&52\\
17 & 116 && 1&3&4&5&8&14&20&26&32&38&44&50&53&54&55&57&58\\
18 & 128 && 1&3&4&5&8&14&20&26&32&38&44&50&56&59&60&61&63&64\\
19 & 140 && 1&3&4&5&8&14&20&26&32&38&44&50&56&62&65&66&67&69&70\\
20 & 152 && 1&3&4&5&8&14&20&26&32&38&44&50&56&62&68&71&72&73&75&76\\
21 & 164 && 1&3&4&6&10&13&15&21&29&37&45&53&61&69&73&75&78&79&82&84&88\\
21 & 164 && 1&3&4&6&10&13&15&21&29&37&45&53&61&69&73&75&78&79&80&82&84\\
21 & 164 && 1&3&4&6&10&13&15&21&29&37&45&53&61&67&69&72&76&78&79&81&82\\
21 & 164 && 1&3&4&5&8&14&20&26&32&38&44&50&56&62&68&74&77&78&79&81&82\\
22 & 180 && 1&3&4&6&10&13&15&21&29&37&45&53&61&69&77&81&83&86&87&90&92&96\\
22 & 180 && 1&3&4&6&10&13&15&21&29&37&45&53&61&69&77&81&83&86&87&88&90&92\\
22 & 180 && 1&3&4&6&10&13&15&21&29&37&45&53&61&69&75&77&80&84&86&87&89&90\\
\end{tabular}
\end{sidewaystable}

\newpage
$\mathbf{h=3}$

\smallskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(3,k)$}&&$a_i$\\
3 & 15 && 1&4&5\\
4 & 24 && 1&4&7&8\\
5 & 36 && 1&4&6&14&15\\
6 & 52 && 1&3&7&9&19&24\\
6 & 52 && 1&4&6&14&17&29\\
7 & 70 && 1&4&5&15&18&27&34\\
8 & 93 && 1&3&6&10&24&26&39&41\\
9 & 121 && 1&3&8&9&14&32&36&51&53\\
10 & 154 && 1&2&6&8&19&28&40&43&91&103\\
11 & 186 && 1&2&3&8&11&26&38&56&69&85&89\\
11 & 186 && 1&4&6&13&16&27&44&49&73&81&91\\
12 & 225 && 1&3&8&13&15&16&49&53&84&88&108&114\\
13 & 271 && 1&4&6&14&16&20&39&56&79&100&113&122&131\\
14 & 323 && 1&2&4&9&15&27&38&43&46&97&107&127&147&157\\
\end{tabular}

\bigskip

$\mathbf{h=4}$

\smallskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(4,k)$}&&$a_i$\\
3 & 26 && 1&5&8\\
4 & 44 && 1&3&11&18\\
5 & 70 && 1&3&11&15&32\\
6 & 108 && 1&4&9&16&38&49\\
6 & 108 && 1&5&8&27&29&44\\
7 & 162 && 1&4&9&24&35&49&51\\
7 & 162 && 1&4&10&15&37&50&71\\
7 & 162 && 1&5&8&25&31&52&71\\
8 & 228 && 1&3&8&19&33&39&92&102\\
9 & 310 && 1&4&10&11&28&33&78&118&143\\
9 & 310 && 1&5&7&22&31&36&83&117&133\\
10 & 422 && 1&4&9&24&26&42&104&115&174&185\\
11 & 550 && 1&4&9&20&34&52&62&137&149&229&242\\
\end{tabular}

\bigskip

$\mathbf{h=5}$

\smallskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(5,k)$}&&$a_i$\\
3 & 35 && 1&6&7\\
4 & 71 && 1&4&12&21\\
4 & 71 && 1&5&12&28\\
5 & 126 && 1&4&9&31&51\\
6 & 211 && 1&4&13&24&56&61\\
6 & 211 && 1&5&8&33&54&67\\
7 & 336 && 1&4&13&24&30&87&106\\
8 & 524 && 1&6&8&33&48&77&183&236\\
9 & 726 && 1&4&13&18&51&92&163&208&223\\
10 & 1016 && 1&6&8&21&60&93&104&154&378&414\\
\end{tabular}

\newpage
$\mathbf{h=6}$

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(6,k)$}&&$a_i$\\
3 & 52 && 1&7&12\\
4 & 114 && 1&4&19&33\\
5 & 216 && 1&7&12&43&52\\
6 & 388 && 1&7&11&48&83&115\\
7 & 638 && 1&4&18&31&104&145&170\\
8 & 1007 && 1&5&18&29&97&170&219&308\\
9 & 1545 && 1&6&10&32&77&114&284&447&471\\
\end{tabular}

\bigskip

$\mathbf{k=2}$

\bigskip

Separate formulae are given for $h$ even and $h$ odd:

$$\begin{array}{rclll}
n(2t,2) & = & t(t+3) && \mbox{with } a_2=t+1 \mbox{ or } t+2\\
n(2t+1,2) & = & t(t+4)+2 && \mbox{with } a_2=t + 2
\end{array}$$

\bigskip

$\mathbf{k=3}$

\bigskip

\begin{minipage}{2.5in}
\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrr}
$h$&\multicolumn{1}{c}{$n(h,3)$}&&$a_1$&$a_2$&$a_3$\\
7 & 69 && 1&8&13\\
8 & 89 && 1&9&14\\
9 & 112 && 1&9&20\\
10 & 146 && 1&10&26\\
11 & 172 && 1&9&30\\
11 & 172 && 1&10&26\\
12 & 212 && 1&11&37\\
13 & 259 && 1&13&34\\
14 & 302 && 1&12&52\\
\end{tabular}
\end{minipage}
\begin{minipage}{2.5in}
\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrr}
$h$&\multicolumn{1}{c}{$n(h,3)$}&&$a_1$&$a_2$&$a_3$\\
15 & 354 && 1&12&52\\
16 & 418 && 1&15&54\\
17 & 476 && 1&14&61\\
18 & 548 && 1&15&80\\
19 & 633 && 1&18&65\\
20 & 714 && 1&17&91\\
21 & 805 && 1&17&91\\
22 & 902 && 1&19&102\\
22 & 902 && 1&20&92\\
\end{tabular}
\end{minipage}

\vspace{.2in}

\begin{minipage}{2.8in}

For $h \geq 23$, $n(h,3)$ and $a_i$ are given by the formulae:

\bigskip

$\begin{array}{rcl}
      a_2 & = & (6t + c_{21})\\
      a_3 & = & (2t + c_{31}) + (2t + c_{32})a_2\\
      n(h,3) & = & (4t + c_{41}) + (2t + c_{42})a_2\\
      && \hspace{.2in} +(3t + c_{43})a_3
\end{array}$
\end{minipage}
\hspace{.2in}
\begin{minipage}{2.6in}

where  $h = 9t + r$,  $0 \leq r \leq 8$,  and $c_{ij}$ are given by:

\bigskip
\centering
\begin{tabular}{rrrrrrrr}
$r$&&$c_{21}$&$c_{31}$&$c_{32}$&$c_{41}$&$c_{42}$&$c_{43}$\\
0&&3&1&1&0&0&0\\
1&&3&1&1&0&0&1\\
2&&5&2&1&1&0&1\\
3&&5&2&1&1&0&2\\
4&&7&3&1&2&0&2\\
5&&6&2&2&2&1&2\\
6&&8&3&2&3&1&2\\
7&&8&3&2&3&1&3\\
8&&10&4&2&4&1&3\\
\end{tabular}
\end{minipage}

\newpage

$\mathbf{k=4}$

\bigskip

\begin{minipage}{3.1in}            
\centering
\begin{tabular}{r@{\hspace{.1in}}r@{\hspace{.2in}}rrrrr}
$h$&\multicolumn{1}{c}{$n(h,4)$}&&$a_1$&$a_2$&$a_3$&$a_4$\\
7 & 165&&1&5&24&37\\
8 & 234 && 1&6&25&65\\
9 & 326 && 1&5&34&60\\
10 & 427 && 1&6&41&67\\
11 & 547 && 1&7&48&85\\
12 & 708 && 1&7&48&126\\
13 & 873 && 1&9&56&155\\
14 & 1094 && 1&8&61&164\\
15 & 1383 && 1&12&65&240\\
16 & 1650 && 1&11&78&216\\
17 & 1935 && 1&11&90&252\\
18 & 2304 && 1&16&73&338\\
19 & 2782 && 1&10&99&360\\
20 & 3324 && 1&16&103&488\\
21 & 3812 && 1&16&103&488\\
22 & 4368 && 1&12&121&561\\
23 & 5130 && 1&14&142&659\\
24 & 5892 && 1&16&163&757\\
25 & 6745 && 1&20&149&860\\
26 & 7880 && 1&16&194&734\\
27 & 8913 && 1&21&177&1006\\
28 & 9919 && 1&21&177&1006\\
29 & 11081 && 1&19&230&870\\
30 & 12376 && 1&18&254&969\\
31 & 13932 && 1&25&211&1410
\end{tabular}
\end{minipage}
\begin{minipage}{3.1in}
\begin{tabular}{r@{\hspace{.15in}}r@{\hspace{.2in}}rrrrr}
$h$&$n(h,4)$&&$a_1$&$a_2$&$a_3$&$a_4$\\
32 & 15657&& 1&25&236&1585\\
33 & 17242 && 1&25&236&1585\\
34 & 18892 && 1&24&225&1734\\
35 & 21061 && 1&28&264&1773\\
36 & 23445 && 1&22&355&1700\\
37 & 25553 && 1&29&303&2346\\
38 & 27978 && 1&22&355&2361\\
39 & 31347 && 1&30&343&2634\\
40 & 33981 && 1&30&343&2634\\
41 & 36806 && 1&31&353&3092\\
42 & 39914 && 1&27&465&2692\\
43 & 43592 && 1&34&389&3376\\
44 & 47536 && 1&34&423&3682\\
45 & 51218 && 1&34&423&3682\\
46 & 54900 && 1&28&564&3261\\
46 & 54900 && 1&34&423&3682\\
47 & 59702 && 1&37&460&4004\\
48 & 63891 && 1&38&473&4590\\
49 & 69362 && 1&38&509&4986\\
50 & 74348 && 1&38&509&4986\\
51 & 81303 && 1&39&563&5448\\
52 & 86751 && 1&39&563&5448\\
53 & 92199 && 1&39&563&5448\\
54 & 97836 && 1&41&630&6147
\end{tabular}
\end{minipage}

\bigskip

For $55 \leq h \leq 254$, $n(h,4)$ and $a_i$ are given by one of the following three sets of formulae:

\bigskip

$\begin{array}{rrcl}
        (A): & a_2 & = & (9t + c_{21})\\
        & a_3 & = & (4t + c_{31}) + (3t + c_{32})a_2\\
        & a_4 & = & (7t + c_{41}) + (2t + c_{42})a_2 + (2t + c_{43})a_3\\
        & n(h,4) & = & (2t + c_{51}) + ( t + c_{52})a_2 + (6t + c_{53})a_3 + (3t + c_{54})a_4
\end{array}$

\bigskip

$\begin{array}{rrcl}
        (B): & a_2 & = & (9t + c_{21})\\
        & a_3 & = & (2t + c_{31}) + (3t + c_{32})a_2\\
        & a_4 & = & (7t + c_{41}) + (2t + c_{42})a_2 + (2t + c_{43})a_3\\
        & n(h,4) & = & (4t + c_{51}) + (3t + c_{52})a_2 + (2t + c_{53})a_3 + (3t + c_{54})a_4
\end{array}$

\bigskip

$\begin{array}{rrcl}
        (C): & a_2 & = & (9t + c_{21})\\
        & a_3 & = & (4t + c_{31}) + (3t + c_{32})a_2\\
        & a_4 & = & (7t + c_{41}) + (2t + c_{42})a_2 + (2t + c_{43})a_3\\
        & n(h,4) & = & (t + c_{51}) + (4t + c_{52})a_2 + (6t + c_{53})a_3 + (3t + c_{54})a_4
\end{array}$

\bigskip

where  $h = 12t + r$,  $0 \leq r \leq 11$,  and $c_{ij}$ are given in the following table:

\newpage

$\mathbf{k=4}$ \textbf{(continued)}
\vspace{-.1in}
\begin{center}
\begin{tabular}{rrrrrrrrrrrrrr}
$r$&&&$c_{21}$&$c_{31}$&$c_{32}$&$c_{41}$&$c_{42}$&$c_{43}$&$c_{51}$&$c_{52}$&$c_{53}$&$c_{54}$&Valid for:\\
0&A&&2&1&0&1&0&1&$-3$&0&4&$-1$&$ 4 \leq t \leq  5$\\
0&A&&1&0&0&0&0&0&$-2$&0&1&1&$ 6 \leq t \leq 11$\\
0&B&&2&2&$-1$&3&$-1$&0&$-1$&$-2$&$-1$&4&$12 \leq t \leq 21$\\
1&A&&1&0&2&1&1&0&0&0&1&0&$ 5 \leq t \leq 21$\\
2&A&&2&1&1&1&1&1&$-3$&1&4&0&$ 5 \leq t \leq  6$\\
2&A&&1&0&2&1&1&0&0&0&1&1&$ 7 \leq t \leq 20$\\
2&B&&5&3&$-1$&6&$-1$&0&0&$-2$&$-1$&5&$21\leq t\leq21$\\
3&A&&3&1&2&2&1&1&$-1$&0&4&0&$ 1 \leq t \leq 20$\\
4&A&&3&1&2&2&1&1&$-1$&0&4&1&$ 2 \leq t \leq 20$\\
5&A&&3&1&2&2&1&1&$-1$&0&4&2&$ 4 \leq t \leq 20$\\
6&A&&3&1&2&2&1&1&$-1$&0&4&3&$ 5 \leq t \leq 20$\\
7&A&&7&3&2&5&1&2&$-1$&0&7&1&$ 2 \leq t \leq 11$\\
7&A&&8&4&1&7&1&0&0&1&1&5&$12 \leq t \leq 20$\\
8&A&&7&3&3&5&2&2&$-1$&1&7&1&$1 \leq t \leq 16$\\
8&A&&8&4&1&7&1&0&0&1&1&6&$	17 \leq t \leq 20$\\
9&A&&7&3&3&5&2&2&$-1$&1&7&2&$	1 \leq t \leq 20$\\   
10&A&&7&3&3&5&2&2&$-1$&1&7&3&$	4 \leq t \leq 19$\\
10&C&&11&6&1&10&1&0&0&3&0&7&$20\leq t\leq20$\\   
11&A&&10&4&3&7&2&2&0&1&7&3&$	2 \leq t \leq  7$\\
11&B&&11&4&2&10&1&2&3&1&1&6&$ 8 \leq t \leq 20$\\
\end{tabular}
\end{center}

The extremal bases of type A were independently investigated by Mossige \cite{Mos87}, who developed a procedure for determining the corresponding $h$-range. Selmer \cite{Sel92} showed the others to be given by similar formulae of type B.  Type C is a variant of type A.

\bigskip

$\mathbf{k=5}$

\smallskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrr}
$h$&\multicolumn{1}{c}{$n(h,5)$}&&$a_1$&$a_2$&$a_3$&$a_4$&$a_5$\\
7 & 345 && 1&8&11&64&102\\
8 & 512 && 1&9&15&78&115\\
8 & 512 && 1&9&15&80&118\\
9 & 797 && 1&9&23&108&181\\
10 & 1055 && 1&8&27&119&194\\
11 & 1475 && 1&10&34&165&270\\
12 & 2047 && 1&10&26&195&320\\
13 & 2659  && 1&13&34&242&409\\
14 & 3403 && 1&11&48&278&720\\
15 & 4422 && 1&14&50&325&782\\
16 & 5629 && 1&14&61&381&984\\
17 & 6865 && 1&13&67&326&1191\\
18 & 8669 && 1&14&75&500&1306\\
19 & 10835 && 1&14&89&523&1892\\
20 & 12903 && 1&14&102&589&1912\\
21 & 15785 && 1&14&88&727&2060\\
22 & 18801 && 1&18&97&858&2156\\
\end{tabular}

\newpage
$\mathbf{k=5}$ \textbf{(continued)}

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrr}
$h$&\multicolumn{1}{c}{$n(h,5)$} && $a_1$&$a_2$&$a_3$&$a_4$&$a_5$\\
23 & 22456 && 1&20&91&894&3330\\
24 & 26469 && 1&16&148&843&3894\\
25 & 31108 && 1&16&148&975&4554\\
26 & 36949 && 1&22&136&1168&4227\\
27 & 42744 && 1&22&162&1372&4889\\
28 & 49436 && 1&25&139&1510&5657\\
29 & 57033 && 1&23&170&1610&5811\\
30 & 66771 && 1&24&201&1718&7596\\
31 & 75558 && 1&23&192&1976&7018\\
32 & 86303 && 1&25&180&1916&8793\\
33 & 96852 && 1&28&202&2150&9867\\
34 & 110253 && 1&29&209&2434&11256\\
35 & 123954 && 1&27&231&2495&11464\\
36 & 140688 && 1&30&227&2839&12993\\
37 & 158389 && 1&31&234&2926&13391\\
38 & 178811 && 1&30&275&2947&16472\\
39 & 197293 && 1&29&300&3671&16677\\
40 & 223580 && 1&29&266&3382&18856\\
41 & 247194 && 1&32&294&3739&20847\\
42 & 273443 && 1&34&325&4133&23063\\
43 & 300747 && 1&33&342&4560&25414\\ 
44 & 331461 && 1&32&393&4562&25751\\
45 & 368894 && 1&32&457&5137&28671\\
46 & 401350 && 1&37&421&5602&31205\\
47 & 443231 && 1&33&336&5224&34431\\
48 & 490325 && 1&36&515&6304&35400\\
49 & 536399 && 1&34&422&6065&38741\\
50 & 586322 && 1&42&444&6906&45542\\
51 & 634430 && 1&36&482&7132&40052\\
52 & 699698 && 1&35&570&6602&50446\\
53 & 754166 && 1&40&462&7666&50721\\
54 & 823136 && 1&39&474&7840&51893\\
55 & 892139 && 1&42&511&8494&56238\\
56 & 968914 && 1&39&489&8580&65052\\
57 & 1052562 && 1&43&617&10091&66380\\
58 & 1150377 && 1&46&606&9531&72397\\
59 & 1236682 && 1&44&552&10237&77846\\
60 & 1325927 && 1&41&631&11205&74232\\
61 & 1420882 && 1&44&623&10432&89278\\
62 & 1547688 && 1&49&646&12050&91649\\
63 & 1678695 && 1&49&664&12338&93848\\
64 & 1782370 && 1&52&705&13100&99644\\
\end{tabular}

$\mathbf{k=5}$ \textbf{(continued)}

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrr}
$h$&\multicolumn{1}{c}{$n(h,5)$} && $a_1$&$a_2$&$a_3$&$a_4$&$a_5$\\
65 & 1888725 && 1&48&698&12988&111755\\
66 & 2036874 && 1&48&746&13252&113747\\
67 & 2165553 && 1&51&793&14087&120914\\
\end{tabular}

\bigskip

$\mathbf{k=6}$

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrr}
$h$&\multicolumn{1}{c}{$n(h,6)$}&&$a_1$&$a_2$&$a_3$&$a_4$&$a_5$&$a_6$\\
7 & 664 && 1&7&12&64&113&193\\
8 & 1045 && 1&9&14&65&170&297\\
9 & 1617 && 1&6&31&48&256&373\\
10 & 2510 && 1&9&31&96&366&411\\
11 & 3607 && 1&7&41&105&490&815\\
12 & 5118 && 1&6&47&120&565&946\\
13 & 7066 && 1&10&35&133&759&1304\\
14 & 9748 && 1&11&49&188&810&2109\\
15 & 12793 && 1&8&71&192&1215&1993\\
16 & 17061 && 1&15&49&285&1292&3043\\
17 & 22342 && 1&13&82&387&1723&4789\\
18 & 28874 && 1&13&94&354&1968&5062\\
19 & 36560 && 1&16&87&408&2351&6452\\
20 & 45754 && 1&17&93&436&2898&6897\\
21 & 57814 && 1&14&129&469&3585&8757\\
22 & 72997 && 1&17&109&624&3998&9618\\
23 & 87555 && 1&12&117&541&4487&11496\\
24 & 106888 && 1&19&138&782&5346&13991\\
25 & 129783 && 1&19&157&896&5656&19313\\
\end{tabular}

\bigskip

$\mathbf{k=7}$

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrr}
$h$&\multicolumn{1}{c}{$n(h,7)$}&&$a_1$&$a_2$&$a_3$&$a_4$&$a_5$&$a_6$&$a_7$\\
7 & 1137 && 1&7&18&62&104&244&259\\
7 & 1137 && 1&8&13&66&115&254&415\\
8 & 2001 && 1&6&28&47&127&412&602\\
9 & 3191 && 1&7&30&86&189&607&920\\
10 & 5047 && 1&6&29&96&246&857&1179\\
11 & 7820 && 1&10&34&153&380&1342&1487\\
12 & 11568 && 1&8&49&127&419&1566&2604\\
13 & 17178 && 1&12&40&223&544&2479&3253\\
\end{tabular}

\bigskip

$\mathbf{k=8}$

\bigskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrrr}
$h$&\multicolumn{1}{c}{$n(h,8)$}&&$a_1$&$a_2$&$a_3$&$a_4$&$a_5$&$a_6$&$a_7$&$a_8$\\
7 & 1911 && 1&4&17&31&117&209&513&550\\
7 & 1911 && 1&6&20&41&109&228&509&580\\
8 & 3485 && 1&8&13&58&169&295&831&1036\\
\end{tabular}



\pagebreak
\begin{thebibliography}{5}

\bibitem{Alt77}
R. Alter, J. Barnett, Remarks on the postage stamp problem with
applications to computers, {\em Congr. Numer.\/} {\bf 19} (1977),
43--59.

\bibitem{Cha93}
M. F. Challis, Two new techniques for computing extremal $h$-bases
$A_k$, {\em Computer J.} {\bf 36} (1993), 117--126.

\bibitem{Mos81}
S. Mossige, Algorithms for computing the $h$-range of the Postage stamp
problem, {\em Math. Comp.\/} {\bf 36} (1981), 575--582.

\bibitem{Mos87}
S. Mossige, On extremal $h$-bases $A_4$, {\em Math. Scand.\/} {\bf 61}
(1987), 5--16.

\bibitem{Mos91}
S. Mossige, Symmetric bases with large 2-range for $k\leq 75$, preprint,
1991.

\bibitem{Rob09}
J. P. Robinson, Some postage stamp 2-bases, \emph{J. Integer Seq.}
\textbf{12} (2009), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Robinson/robinson4.html"}{Article 09.1.1}.

\bibitem{Sel92}
E. S. Selmer, Associate bases in the postage stamp problem, {\em J.
Number Theory\/} {\bf 42} (3) (1992), 320--336.

\bibitem{Sha02}
J. Shallit, The computational complexity of the local postage stamp
problem, {\em ACM SIGACT News\/} {\bf 33} (2002), 90--94.

\bibitem{Slo09}
N. J. A. Sloane,
\href{http://www.research.att.com/~njas/sequences/}{The On-Line
Encylopedia of Integer Sequences}, 2009.

\end{thebibliography}

\bigskip
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\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B13.

\noindent \emph{Keywords: } $h$-basis, extremal $h$-basis.

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\noindent (Concerned with sequences
\seqnum{A001210}, \seqnum{A001211}, \seqnum{A001216}, \seqnum{A001215}, \seqnum{A001216}, \seqnum{A053346}, and \seqnum{A053348}.)

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\noindent
Received September 23 2009;
revised version received  November 18 2009; January 27 2010.
Published in {\it Journal of Integer Sequences}, January 30 2010.

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Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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