Admissible tracks in Lai-Ding's secret sharing scheme

We consider Lai-Ding's secret sharing schemes, with the secret placed as a coefficient ai in f(x)=a0xc0+⋯+ak−1xck−1, and private shares obtained by applying f to the sequence of public identities t=(t0,…,tn−1)∈Fqn. In this setting, we focus on such sequences t for which the scheme turns out to be k-threshold. We prove certain quantitative properties of such sequences which generalize some of the results obtained by Schinzel et al. for the particular case of Shamir's type schemes.