Maximal -distance sets containing the representation of the Hamming graph

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• 作者：Saori Adachi^a ; ^{k630945d@m2.aichi-c.ed.jp} ; Rina Hayashi^b ; Hiroshi Nozaki^c ; ^{hnozaki@auecc.aichi-edu.ac.jp} ; Chika Yamamoto^d
• 关键词：Hamming graph ; Few-distance set ; Euclidean representation ; Erdős&ndash ; Ko&ndash ; Rado theorem
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：340
• 期：3
• 页码：430-442
• 全文大小：489 K
• 卷排序：340

文摘

A set XX in the Euclidean space RdRd is an mm-distance set   if the set of Euclidean distances between two distinct points in XX has size mm. An mm-distance set XX in RdRd is maximal   if there does not exist a vector xx in RdRd such that the union of XX and {x}{x} still has only mm distances. Bannai et al. (2012) investigated maximal mm-distance sets that contain the Euclidean representation of the Johnson graph J(n,m)J(n,m). In this paper, we consider the same problem for the Hamming graph H(n,m)H(n,m). The Euclidean representation of H(n,m)H(n,m) is an mm-distance set in Rm(n−1)Rm(n−1). We prove that if the representation of H(n,m)H(n,m) is not maximal as an mm-distance set for some mm, then the maximum value of nn is m2+m−1m2+m−1. Moreover we classify the largest mm-distance sets that contain the representation of H(n,m)H(n,m) for n≥2n≥2 and m≤4m≤4. We also classify the maximal 22-distance sets that are in R2n−1R2n−1 and contain the representation of H(n,2)H(n,2) for n≥2n≥2.