{"paper":{"title":"Endomorphisms of Cuboidal Hamming Graphs, Latin Hypercuboids of Class $r$, and Mixed MDS Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Artur Schaefer","submitted_at":"2016-02-17T18:06:03Z","abstract_excerpt":"In this paper we investigate the existence of singular endomorphisms of the cuboidal Hamming graph $H(n_1,...,n_d,S)$ over the set $\\left[ n_1\\right]\\times \\left[ n_2\\right]\\times \\cdots \\times \\left[ n_d\\right]$, where $\\left[ n\\right]=\\{1,...,n\\}$, which is a generalisation of the well-known (cubic) Hamming graph over $\\left[ n\\right]^{d}$. Two vertices in $H$ are adjacent, if their Hamming distance lies in the set $S$. In this paper $S=\\{1,...,r\\}$, for some integer $1\\leq r\\leq d-1$, and we first show that the singular endomorphisms of minimal rank ( which is the size of their image) of $H"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05515","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"}