Arithmetic progressions with constant weight

Let k≤n be two positive integers, and let F be a field with characteristic p. A sequence f : {1,...,n} → F is called k-constant, if the sum of the values of f is the same for every arithmetic progression of length k in {1,...,n}. Let V(n, k, F) be the vector space of all k-constant sequences. The constant sequence is, trivially, k-constant, and thus dim V(n, k, F)≥1. Let m(k, F)= min^∞_n=k dim V(n, k, F), and let c(k, F) be the smallest value of n for which dim V(n, k, F)=m(k, F). We compute m(k, F) for all k and F and show that the value only depends on k and p and not on the actual field. In particular, we show that if p#2224;k (in particular, if p = 0) then m(k, F) = 1 (namely, when n is large enough, only constant functions are k constant). Otherwise, if k = p^rt where r ≥ 1 is maximal, then m(k, F) = k - t. We also conjecture that c(k, F) = (k - 1)t + φ(t), unless p > t and p divides k, in which case c(k, F) = (k - 1)p + 1 (in case p#2224;A: we put t = k), where φ(t) is Euler's function. We prove this conjecture in case t is a multiple of at most two distinct prime powers. Thus, in particular, we get that whenever k = q^s1₁ q^s2₂ where q₁, q₂ are distinct primes and p ≠ q₁, q₂, then every k-constant sequence is constant if and only if n≥q^2s1₁ q^2s2₂ - q^s1-1₁ q^s2-1₂(q₁ + q₂ - 1). Finally, we establish an interesting connection between the conjecture regarding c(k, F) and a conjecture about the non-singularity of a certain (0, 1)-matrix over the integers.