Modular periodicity of binomial coefficients

We prove that if the signed binomial coefficient viewed modulo p is a periodic function of i with period h in the range 0⩽i⩽k, then k+1 is a power of p, provided h is not too large compared to k. (In particular, 2h⩽k suffices). As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H