Summary: Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds we derive are applied to a code with an average spectrum, and the result is compared to the random coding exponent. We show that the bound considered here for BSC case coincides asymptotically with the random coding bound. For the case of AWGN channel, we show that \textit{E. R. Berlekamp}'s tangential bound [Proc. IEEE 68, 564-593 (1980)] can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it.