{"paper":{"title":"On the structure of cube tiling codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej P. Kisielewicz","submitted_at":"2018-05-20T17:53:30Z","abstract_excerpt":"Let $S$ be a set of arbitrary objects, and let $S^d=\\{v_1...v_d\\colon v_i\\in S\\}$. A polybox code is a set $V\\subset S^d$ with the property that for every two words $v,w\\in V$ there is $i\\in [d]$ with $v_i'=w_i$, where a permutation $s\\mapsto s'$ of $S$ is such that $s''=(s')'=s$ and $s'\\neq s$. If $|V|=2^d$, then $V$ is called a cube tiling code. Cube tiling codes determine $2$-periodic cube tilings of $\\mathbb{R}^d$ or, equivalently, tilings of the flat torus $\\mathbb{T}^d=\\{(x_1,\\ldots ,x_d)({\\rm mod} 2):(x_1,\\ldots ,x_d)\\in \\mathbb{R}^d\\}$ by translates of the unit cube as well as $r$-perf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.07806","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}