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Graham and Winkler derived a formula for the determinant of the distancematrix of a full-dimensional set of $n + 1$ points $\{ x_{0}, x_{1}, \ldots ,x_{n} \}$ in the Hamming cube $H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$. In thisarticle we derive a formula for the determinant of the distance matrix $D$ ofan arbitrary set of $m + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{m} \}$ in$H_{n}$. It follows from this more general formula that $\det (D) \not= 0$ ifand only if the vectors $x_{0}, x_{1}, \ldots , x_{m}$ are affinelyindependent. Specializing to the case $m = n$ provides new insights into theoriginal formula of Graham and Winkler. A significant difference that arisesbetween the cases $m < n$ and $m = n$ is noted. We also show that if $D$ is thedistance matrix of an unweighted tree on $n + 1$ vertices, then $\langle D^{-1}\mathbf{1}, \mathbf{1} \rangle = 2/n$ where $\mathbf{1}$ is the column vectorall of whose coordinates are $1$. Finally, we derive a new proof of Murugan'sclassification of the subsets of $H_{n}$ that have strict $1$-negative type.

Distance matrices of subsets of the Hamming cube