Abstract
In this work, the sphere-covering bound on covering codes in Rosenbloom–Tsfasman spaces (RT spaces) is improved by generalizing the excess counting method. The approach focuses on studying the parity of a Rosenbloom–Tsfasman sphere (RT sphere) and the parity of the intersection of two RT spheres. We connect the parity of an RT sphere with partial sums of binomial coefficients and p-adic valuation of binomial coefficients. The intersection number of RT spaces is introduced and we determinate its parity under some conditions. Numerical applications of the method are discussed.
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The authors would like to thank the anonymous referees for their suggestions that greatly improved this paper.
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Communicated by Masaaki Harada.
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A. G. Castoldi was partially supported by Capes and Fundação Araucária; E. L. Monte Carmelo was supported by CNPq Grant: 311793/2016-0.
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Castoldi, A.G., Carmelo, E.L.d.M. & da Silva, R. Partial sums of binomials, intersecting numbers, and the excess bound in Rosenbloom–Tsfasman space. Comp. Appl. Math. 38, 55 (2019). https://doi.org/10.1007/s40314-019-0828-2
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DOI: https://doi.org/10.1007/s40314-019-0828-2