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We derive new approximate upper bounds on the error probability of linear codes with maximum likelihood decoding by applying an appropriate transformation of decision variables. The new bounds are simple to calculate and based on the standard union bound but give tight results in some range below the channel cutoff rate R0. In applying the bound to iterative decodable codes, such as turbo codes, we use the concept of uniform interleavers and determine the average distance spectrum. A comparison of our bounds to simulated error rates shows a significantly improved tightness similar to the tangential sphere bound of Poltyrev (1994), which is the best rigorous upper bound known up to now.