44,177 research outputs found

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    The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure

    A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

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    Using Singer polygons, we construct locally finite affine buildings of types ~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the ~C_2-buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type ~C_2. Integral and rational group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged

    Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes

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    An optimal constant-composition or constant-weight code of weight ww has linear size if and only if its distance dd is at least 2w−12w-1. When d≥2wd\geq 2w, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of d=2w−1d=2w-1 has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight ww and distance 2w−12w-1 based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight ww and distance 2w−12w-1 are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight ww and distance 2w−12w-1 are also determined for all w≤6w\leq 6, except in two cases.Comment: 12 page
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