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A new tight upper bound on the maximum-likelihood (ML) word and bit-error decoding probabilities for specific codes over discrete channels is presented. It constitutes an enhanced version of the Gallager upper bound and its variations resulting from the Duman-Salehi second bounding technique. An efficient technique is developed that, in the case of symmetric channels, overcomes the difficulties associated with the direct computation of the proposed bound. Surprisingly, apart from the distance and input-output weight enumerating functions (IOWEFs), the bound depends also on the coset weight distribution of the code.