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.