Covering numbers of some linear groups with respect to quadratic matrices

Let [Formula: see text] be a commutative local ring with residue field [Formula: see text]. Suppose [Formula: see text] is an even natural number, [Formula: see text] contains at least [Formula: see text] elements and [Formula: see text] is a quadratic polynomial in [Formula: see text]. The main goal of this paper is to prove that the covering number [Formula: see text] with respect to the set of [Formula: see text]-quadratic matrices in [Formula: see text] is equal to or less than [Formula: see text]. Some corollaries are also presented.