For a given linear code, a lower bound on the error exponent under 
maximum likelihood decoding is investigated at low rates. In the 
analysis of the decoding error probability, it is important to 
characterize the error exponent since it indicates how fast the 
probability of decoding error converges to zero asymptotically. 
Over a symmetric memoryless channel, an error exponent is derived 
for a given linear code. A sufficient condition for achieving the 
expurgated exponent, which is the best among known error exponents 
at low rates, is given. Over a general discrete memoryless channel, 
the same analysis shows the expected error exponent of the ensemble 
of a linear code and its cosets.