Let 
be the transmitted
code vector and 
be the
corresponding received vector.
Then 
is the
error pattern added by the channel, where 
is a symbol
from GF 
.
Assume that the received vector has 
errors in positions 
. Then
.
Define 
, 
, the error
location numbers. Then, compute the 2t syndromes as follows:
.
Expanding we obtain a sequence of 2t algebraic syndrome equations:
The syndrome equations can be translated into a series of linear
equations by defining the error-locator polynomial
. has as its roots the inverses of the error locators
:
Assume that 
errors have occurred, where t is the error
correcting capability of the code. Then, we obtain the following
matrix equation:
