On the number of nowhere zero points in linear mappings

Abstract

LetA be a nonsingularn byn matrix over the finite fieldGF q ,k=⌊n/2⌋,q=p a,a≥1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF q )n such that bothx andAx have no zero component. We prove that forn≥2, and\(q > 2\left( {\begin{array}{*{20}c} {2n} \\ 3 \\ \end{array} } \right)\),P(A,q)≥[(q−1)(q−3)]k(q−2)n−2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)≥1, is true for allq≥n+2≥3 orq≥n+1≥4.

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Baker, R.D., Bonin, J., Lazebnik, F. et al. On the number of nowhere zero points in linear mappings. Combinatorica 14, 149–157 (1994). https://doi.org/10.1007/BF01215347

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AMS subject classification code (1991)

  • 06 C 10
  • 15 A 06
  • 11 T 99