Abstract
A Heffter array \(H(m,n;s,t)\) is an \(m \times n\) matrix with nonzero entries from \(\mathbb {Z}_{2ms+1}\) such that (i) each row contains \(s\) filled cells and each column contains \(t\) filled cells, (ii) every row and column sum to 0, and (iii) no element from \(\{x,-x\}\) appears twice. Heffter arrays are useful in embedding the complete graph \(K_{2ms+1}\) on an orientable surface where the embedding has the property that each edge borders exactly one s-cycle and one t-cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be constructed in the case when \(s=m\), i.e every cell is filled. In this paper we concentrate on square arrays with empty cells where every row sum and every column sum is \(0\) in \({\mathbb {Z}}\). We solve most of the instances of this case.
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Acknowledgments
Much of this research was done at the University of Queensland while the second and fourth authors were visiting (at separate times). We thank the Ethel Raybould Visiting Fellowship and the Turkish Science Foundation 2219 Program for providing funds for these visits.
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Dedicated to the memory of our friend and colleague Scott Vanstone.
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Cryptography, Codes, Designs and Finite Fields: In Memory of Scott A. Vanstone”.
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Archdeacon, D.S., Dinitz, J.H., Donovan, D.M. et al. Square integer Heffter arrays with empty cells. Des. Codes Cryptogr. 77, 409–426 (2015). https://doi.org/10.1007/s10623-015-0076-4
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DOI: https://doi.org/10.1007/s10623-015-0076-4