A note on matrix rigidity

Abstract

In this paper we give an explicit construction ofn×n matrices over finite fields which are somewhat rigid, in that if we change at mostk entries in each row, its rank remains at leastCn(log q k)/k, whereq is the size of the field andC is an absolute constant. Our matrices satisfy a somewhat stronger property, we will explain and call “strong rigidity”. We introduce and briefly discuss strong rigidity, because it is in a sense a simpler property and may be easier to use in giving explicit construction.

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References

  1. [1]

    D. Y. Grigorjev:Notes of the Leningrad Branch of the Steklov Mathematical Institute of the Academy of Science of the USSSR,60 (1976), 38–48.

    Google Scholar 

  2. [2]

    Pudlak, Savitzky, andA. Razborov: Observations on rigidity of Hadamard matrices, Personal Communication.

  3. [3]

    A. Razborov: On rigid matrices,Problems of Pure and Applied Mathematics (Literal Translation from Russian), to appear.

  4. [4]

    L. G. Valiant: Graph-theoretic arguments in low-level complexity. Technical Report, University of Edinburgh, 1977. Computer Science Report 13-77. Also in Proc. 6th Symp. on Mathematical Foundations of Computer Science, Tatranska Lomnica, Czechoslovakia 1978.

  5. [5]

    G. van der Geer, andJ. van Lint:Introduction to Coding Theory and Algebraic Geometry, Birkhäuser Verlag, Boston, 1988.

    Google Scholar 

  6. [6]

    J. van Lint:Introduction to Coding Theory, Springer-Verlag, New York, 1982.

    Google Scholar 

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This paper was written while on leave from Princeton, at the Hebrew University. The author wishes to acknowledge the National Science Foundation for supporting this research in part under PYI grant CCR-8858788, and a grant from the program of Medium and Long Term Research at Foreign Centers of Excellence.

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Friedman, J. A note on matrix rigidity. Combinatorica 13, 235–239 (1993). https://doi.org/10.1007/BF01303207

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AMS subject classification code (1991)

  • 05 B
  • 05 C
  • 68 Q
  • 94 B