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Minimum Sum and Difference Covers of Abelian Groups
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Harri Haanpää

Laboratory for Theoretical Computer Science

Department of Computer Science and Engineering

Helsinki University of Technology

P.O. Box 5400, 02015 HUT

Finland

**Abstract:**
A subset * S * of a finite Abelian group * G * is said to be a sum
cover of * G * if every element of * G * can be expressed as the
sum of two not necessarily distinct elements in * S *, a strict sum
cover of * G * if every element of * G * can be expressed as the
sum of two distinct elements in * S *, and a difference cover of * G
* if every element of * G * can be expressed as the difference of
two elements in * S *. For each type of cover, we determine for small
* k * the largest Abelian group for which a * k *-element cover
exists. For this purpose we compute a minimum sum cover, a minimum
strict sum cover, and a minimum difference cover for Abelian groups of
order up to 85, 90, and 127, respectively, by a backtrack search with
isomorph rejection.

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Received June 11 2003;
revised versions received July 2 2003; March 16 2004; June 2 2004.
Published in *Journal of Integer Sequences* June 10 2004.

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