A coding algorithm for constant weight vectors: a geometric approach based on dissections
IEEE Transactions on Information Theory
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Let v 1,…,v n be unit vectors in ℝn such that v i ⋅v j =−w for i≠j, where $-1 i=1 n λ i v i (1≥λ 1≥⋅⋅⋅≥λ n ≥0) form a “Hill-simplex of the first type,” denoted by $\mathcal {Q}_{n}(w)$. It was shown by Hadwiger in 1951 that $\mathcal {Q}_{n}(w)$is equidissectable with a cube. In 1985, Schöbi gave a three-piece dissection of $\mathcal {Q}_{3}(w)$into a triangular prism $c\mathcal {Q}_{2}(\frac{1}{2})\times I$, where I denotes an interval and $c=\sqrt{2(w+1)/3}$. In this paper, we generalize Schöbi’s dissection to an n-piece dissection of $\mathcal {Q}_{n}(w)$into a prism $c\mathcal {Q}_{n-1}(\frac{1}{n-1})\times I$, where $c=\sqrt{(n-1)(w+1)/n}$. Iterating this process leads to a dissection of $\mathcal {Q}_{n}(w)$into an n-dimensional rectangular parallelepiped (or “brick”) using at most n! pieces. The complexity of computing the map from $\mathcal {Q}_{n}(w)$to the brick is O(n 2). A second generalization of Schöbi’s dissection is given which applies specifically in ℝ4. The results have applications to source coding and to constant-weight binary codes.