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Improved lower bounds on the rigidity of Hadamard matrices

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Abstract

We writef=Ω(g) iff(x)≥cg(x) with some positive constantc for allx from the domain of functionsf andg. We show that at least Ω(n 2/r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank belowr. This improves the previously known bound Ω(n 2/r 2). If we additionally know that the changes are bounded above in absolute value by some numberθ≥n/r, then the number of these entries is bounded below by Ω(n 3/(rθ 2)), which improves upon the previously known bound Ω(n 2/θ 2).

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Authors and Affiliations

  1. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, USSR

    B. S. Kashin & A. A. Razborov

Authors
  1. B. S. Kashin
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  2. A. A. Razborov
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Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 535–540, April, 1998.

The research of the first author was supported by the Russian Foundation for Basic Research under grants No. 96-01-00094 and No. 96-15-96102 and by the INTAS Foundation under grant No. 93-1376. The research of the second author was supported by the Russian Foundation for Basic Research under grants No. 96-01-01222 and No. 96-15-96090.

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Kashin, B.S., Razborov, A.A. Improved lower bounds on the rigidity of Hadamard matrices. Math Notes 63, 471–475 (1998). https://doi.org/10.1007/BF02311250

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  • DOI: https://doi.org/10.1007/BF02311250

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