Lower Asymptotic Bound on the Number of Linear Code Words in a Sphere of Given Radius in $(F_q)^n$

The random coding method is applied to determine the asymptotic lower bound on the number of words of a linear code from $(F_q)^n$, contained in a sphere of given radius. The bound is uniform relative to the center of the sphere. An upper bound is also derived on the covering radius of linear code. An upper estimate is obtained on the proportion of codes for which these bounds are valid.