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The binary Golay code $C$ is defined as the 12-dimensional vector space over $Z_{2}$ spanned by the 759 octads interpreted as vectors with eight 1s and 16 0s. The MOG is constructed by considering two 3-dimensional spaces over $Z_{2}$ , the Point space and the Line space, whose codewords are of length 8, and gluing three copies together in such a way as to obtain a 12-dimensional subspace of the 24-dimensional space P(Ω), consisting of all subsets of Ω. The minimal weight codewords in this 24-dimensional space are shown to have weight 8 and to total 759. The construction thus proves that a Steiner system S(5, 8, 24) exists, and provides a unique label for each codeword in the binary Golay code. We exhibit a natural isomorphism between the 24-dimensional space P(Ω) factored by $C$ and the dual space $C^{⋆}$ , and identify its elements as 24 monads, 276 duads, 2024 triads and $(\binom{24}{4}) / 6 = 1771$ sextets; this last division by 6 occurs because two tetrads 4 whose union is an octad are congruent modulo $C$ .

This paper describes an efficient and fast algorithm for finding the minimum distance between two convex polyhedrons in a three dimensional space. To obtain the minimal distance, the proposed computational scheme is based on a direct approach to minimizing the distance function which produces a succession of optimal search directions along object boundaries. This algorithm combines the gradient projection method'; and an additional optimal search direction when the gradient projection method leads to a zigzagging phenomenon. In this case, the additional optimal search direction accelerates significantly the convergence of the process. Extensive numerical experiments with convex polyhedra show the performance of this algorithm when compared with that of previous approaches. The proposed algorithm may be very helpful in solving the computation of minimal distance between a pair of convex sets, the collision detection problem or to track the closest points of moving convex objects.

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4 - The Binary Golay Code

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A fast algorithm for distance calculation between convex objects using the optimization approach

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2 results

4 - The Binary Golay Code

Summary

A fast algorithm for distance calculation between convex objects using the optimization approach