Abstract
We introduce the notion of an extension set for an affine plane of order q to study affine designs \({\mathcal {D}}'\) with the same parameters as, but not isomorphic to, the classical affine design \({\mathcal {D}} = \mathrm {AG}_2(3,q)\) formed by the points and planes of the affine space \(\mathrm {AG}(3,q)\) which are very close to this geometric example in the following sense: there are blocks \(B'\) and B of \({\mathcal {D}'}\) and \({\mathcal {D}}\), respectively, such that the residual structures \({\mathcal {D}}'_{B'}\) and \({\mathcal {D}}_B\) induced on the points not in \(B'\) and B, respectively, agree. Moreover, the structure \({\mathcal {D}}'(B')\) induced on \(B'\) is the q-fold multiple of an affine plane \({\mathcal {A}}'\) which is determined by an extension set for the affine plane \(B \cong AG(2,q)\). In particular, this new approach will result in a purely theoretical construction of the two known counterexamples to Hamada’s conjecture for the case \(\mathrm {AG}_2(3,4)\), which were discovered by Harada et al. [7] as the result of a computer search; a recent alternative construction, again via a computer search, is in [23]. On the other hand, we also prove that extension sets cannot possibly give any further counterexamples to Hamada’s conjecture for the case of affine designs with the parameters of some \(\mathrm {AG}_2(3,q)\); thus the two counterexamples for \(q=4\) might be truly sporadic. This seems to be the first result which establishes the validity of Hamada’s conjecture for some infinite class of affine designs of a special type. Nevertheless, affine designs which are that close to the classical geometric examples are of interest in themselves, and we provide both theoretical and computational results for some particular types of extension sets. Specifically, we obtain a theoretical construction for one of the two affine designs with the parameters of \(\mathrm {AG}_2(3,3)\) and 3-rank 11 and for an affine design with the parameters of \(\mathrm {AG}_2(3,4)\) and 2-rank 17 (in both cases, just one more than the rank of the classical example).
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Notes
Not even a single counterexample is known, and the only proven instances are the few cases in which actually the conjecture in its strong version has been established, which all require \(q=2\) or \(q=3\).
See [2, §XII.5] for results and references concerning good blocks in general incidence structures. Other authors use the term prime block instead.
As the discussion of the special case \({\mathcal {D}}=\mathrm {AG}_2(3,4)\) given in [23, §4] shows, \({\mathcal {E}}\) may have further resolutions. However, these are of no interest to us.
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Acknowledgements
The authors wish to thank the three anonymous reviewers for their careful reading of the manuscript and their helpful comments. In particular, thanks are due to one of the reviewers for pointing out an error in our original proof of Corollary 2.9 and for providing Example 2.4. Yue Zhou: Research supported by a grant of the Alexander von Humboldt Foundation. Vladimir D. Tonchev: Research partially supported by NSA Grant H98230-16-1-0011.
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Communicated by J. D. Key.
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Jungnickel, D., Zhou, Y. & Tonchev, V.D. Extension sets, affine designs, and Hamada’s conjecture. Des. Codes Cryptogr. 86, 587–610 (2018). https://doi.org/10.1007/s10623-017-0344-6
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DOI: https://doi.org/10.1007/s10623-017-0344-6