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%           FPSAC/SFCA '94, May 23-27, 1994   at DIMACS
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%   LaTeX file of problem submitted by Laurent Habsieger (May 24, 1994)
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\begin{document}
\begin{center}
{FPSAC/SFCA '94}\\
{PROBLEM SESSION}\\
{Submission received May 24, 1994}\\[.2in]
{\bf AN EXTREMAL PROBLEM}\\[.1in]
Laurent Habsieger\\[.2in]
\end{center}
Let $V$ be a finite vector space and $S$ an $s$-element multiset of
elements of $V$. Let $m\in {\Bbb N}$, and define
  $$ {\cal F}_m = \{f:V\rightarrow {\Bbb N}\ | \ \sum_{y\in S}f(x+y)
      \geq m \mbox{ for all }x\in V\} $$
  $$ g(m) = \min_{f\in {\cal F}_m}\left( \sum_{x\in V}f(x)\right). $$
\noindent When is it true that
  $$ g(m+s) = |V|+g(m)? $$
One can easily prove that $g(m+s)\le |V|+g(m)$. The equality holds
when ${\rm dim}V=1$ or $V=\Bbb F_2^2$. One can expect to find a
smallest value of $m$ from which the equality is satisfied. If this
smallest value happens to be $0$, numerous applications could be
found.

\noindent
This problem deals with linear-programming in integer points
and arises in the theory
of covering codes (e.g.\ the football pool problem).

\end{document}


