Abstract
The binary k-dimensional simplex code is known to be a \(2^{k-1}\)-batch code and is conjectured to be a \(2^{k-1}\)-functional batch code. Here, we offer a simple, constructive proof of a result that is “in between” these two properties. Our approach is to relate these properties to certain (old and new) additive problems in finite abelian groups. We also formulate a conjecture for finite abelian groups that generalizes the above-mentioned conjecture.
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Notes
Note added in proof: Very recently, we learned that Conjecture 5.2 is not true. For more details, see https://arxiv.org/pdf/2110.07421.pdf.
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Acknowledgements
We thank Lev Yohananov and Eitan Yaakobi for sharing their (at that time) unpublished work [15] with us. This research was supported by the Estonian Research Council Grants PRG49 and PSG114, and by the European Regional Development Fund via CoE Project EXCITE.
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Communicated by T. Etzion.
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Part of this work has been presented at WCC 2022 (the Twelfth International Workshop on Coding and Cryptography). See [8] for the extended abstract.
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Hollmann, H.D.L., Khathuria, K., Riet, AE. et al. On some batch code properties of the simplex code. Des. Codes Cryptogr. 91, 1595–1605 (2023). https://doi.org/10.1007/s10623-022-01173-6
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DOI: https://doi.org/10.1007/s10623-022-01173-6