Distance matrices of subsets of the Hamming cube

Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of n+1 points {x0,x1,…,xn} in the Hamming cube Hn=({0,1}n,ℓ1). In this article we derive a formula for the determinant of the distance matrix D of an arbitrary set of m+1 points {x0,x1,…,xm} in...

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Bibliographic Details
Published in:Indagationes mathematicae 2021-05, Vol.32 (3), p.646-657
Main Authors: Doust, Ian, Robertson, Gavin, Stoneham, Alan, Weston, Anthony
Format: Article
Language:English
Subjects:
Apexes
Determinant
Distance matrix
Hamming cube
Matrices (mathematics)
Negative type
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Summary:Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of n+1 points {x0,x1,…,xn} in the Hamming cube Hn=({0,1}n,ℓ1). In this article we derive a formula for the determinant of the distance matrix D of an arbitrary set of m+1 points {x0,x1,…,xm} in Hn. It follows from this more general formula that det(D)≠0 if and only if the vectors x0,x1,…,xm are affinely independent. Specializing to the case m=n provides new insights into the original formula of Graham and Winkler. A significant difference that arises between the cases m
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2021.01.004