In this dissertation, I discuss bounds for the set of possible number of zeros of a homogeneous linear recurring sequence over a finite field of q elements, based on an irreducible minimal polynomials of degree d and order m as the characteristic polynomial. I prove upper and lower bounds on the cardinality of the set of number of zeros. The set is determined when t= (qd-1)/m has the form qa+1 or q2a-qa+1 where a is a positive integer. The connection with coding theory is a key ingredient. Also it is proved that the upper bound defined here is the best bound for the cardinality of the set of zeros, in the sense that it is reached infinitely often.