A Generalization of Montucla's Rectangle-to-Rectangle Dissection to Higher Dimensions

Abstract : Dissections of polytopes are a well-studied subject by ge-ometers as well as recreational mathematicians. A recent application in coding theory arises from the problem of parameterizing binary vectors of constant Hamming weight [TVS09], [SV09], which is shown to be equivalent to the problem of dissecting a tetrahedron to a brick. An application of dissections to a problem related to the construction of analog codes arises in [CVC13]. Here we consider the rectangle-to-rectangle dissection due to Montu-cla [Fre03]. Montucla's dissection is first reinterpreted in terms of the Two Tile Theorem [SV09]. Based on this, a cube-to-brick dissection is developed in R^n. We present a linear time algorithm (in n) that computes the dissection, i.e. determines a point in the cube given a point in a specific realization of the brick. An application of this algorithm to a previously reported analog coding scheme [CVC13] is also discussed.
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Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2016, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉
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Antonio Campello, Vinay Vaishampayan. A Generalization of Montucla's Rectangle-to-Rectangle Dissection to Higher Dimensions. Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2016, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉. 〈hal-01276485〉

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