In this talk we will discuss lower and upper bounds for the size of k-AP-free subsets of Znm, that is, for rk(Znm, in certain cases. Specifically, we will discuss some lower bounds given by Elsholtz and myself. In the case m=4,k=3 we present a construction which gives the tight answer up to n≤5 and point out some connection with coding theory. We will also mention some open questions (and some partial answers) about related linear algebraic problems.