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Bounds for the size of a minimal 1-perfect bitrade in a Hamming graph

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Abstract

We improve the available upper and lower bounds for the minimal size of the support of an eigenfunction of the Hamming graph H(n, q), where q > 2. In particular, the size of a minimal 1-perfect bitrade in H(n, q) is estimated. We show that the size of such a bitrade is at least 2n−(n−1)/q(q − 2)(n−1)/q for q ≥ 4 and 3n/2(1 − O(1/n)) for q = 3. Moreover, for n ≡ 1 mod q, where q is a prime power, we propose a construction of bitrades of size q (q−2)(n−1)/q2(n−1)/q+1.

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Correspondence to K. V. Vorob’ev.

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Original Russian Text © K.V. Vorob’ev, D.S. Krotov, 2014, published in Diskretnyi Analiz i Issledovanie Operatsii, 2014, Vol. 21, No. 6, pp. 3–10.

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Vorob’ev, K.V., Krotov, D.S. Bounds for the size of a minimal 1-perfect bitrade in a Hamming graph. J. Appl. Ind. Math. 9, 141–146 (2015). https://doi.org/10.1134/S1990478915010159

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  • DOI: https://doi.org/10.1134/S1990478915010159

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