Admissible tracks in Shamir's scheme

We consider Shamir's secret sharing schemes, with the secret placed as a coefficient ai of the scheme polynomial f(x)=a0+⋯+ak−1xk−1, determined by a sequence pairwise different public identities, called a track. If t defines a k-out-of-n Shamir's scheme then the track t is called (k,i)-admissible. If t is not a (k,i)-admissible track, we obtain the scheme with some privileged coalitions of less than k shareholders who can reconstruct the secret by themselves. No (k,i)-admissible tracks contain privileged coalitions. In Spież et al. [11] it is proved that the coalitions are common zeros of some elementary symmetric polynomials.We obtain some quantitative results on the tracks. Given i≠0,k−1 we prove that the number of (k,i)-admissible tracks of length n is , where the constant in the O-symbol depends on n, k and i. We also estimate the number of tracks being (k,i)-admissible for every i. We prove the existence and extendability of all tracks for sufficiently large q, giving algorithms for their constructing and extending.Furthermore, we investigate (k,i)-privileged coalitions of length k−1, which can reconstruct the secret, placed as ai, by themselves. We prove that the number of such coalitions is qk−2+O(qk−3), where the constant in the O-symbol depends on k and i.