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We prove a lower bound of the form N Ω(1) on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Z m , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N Ω(1) design for the Count q principle over Z m . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 N Ω(1) for the number of formulas in a constant-depth Frege proof of the modular counting principle Count N q from instances of thedoi:10.1007/bf01294258 fatcat:pa354ea7sbatdc4vfzudhizm6i