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We present some upper bounds on the size of nonlinear codes and their restriction to systematic codes and linear codes. These bounds are independent of other known theoretical bounds, e.g., the Griesmer bound, the Johnson bound, the Plotkin bound, and of linear programming bounds. One of the new bound is actually an improvement of a bound by Zinoviev, Litsyn, and Laihonen. Our experiments show that in the linear case our bounds provide the best value in a wide range, compared with all other closed-formula upper bounds. In the nonlinear case, we also compare our bound with the linear programming bound and with some improvements on it, show that there are cases where we beat these bounds. In particular, we obtain a new bound in Brouwer's table for A3(16,3).