Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemer\'edi-type result: for all c∈(0,1] and all positive integers N, subsets of density at least c in \0,1,,N-1\ⁿ contain hypercubes of arbitrarily large dimension as n grows.
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Bell et al. (2024) studied this question.
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