Mathematical Notes

, Volume 63, Issue 4, pp 471–475 | Cite as

Improved lower bounds on the rigidity of Hadamard matrices

  • B. S. Kashin
  • A. A. Razborov
Article

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(n2/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(n2/r2). 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(n3/(2)), which improves upon the previously known bound Ω(n2/θ2).

Key words

rigidity of matrices Hadamard matrices spectral methods Hoffman-Wielandt inequality 

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Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • B. S. Kashin
    • 1
  • A. A. Razborov
    • 1
  1. 1.V. A. Steklov Mathematics InstituteRussian Academy of SciencesUSSR

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