Cyclic Codes For Correcting Burst Errors
From Hamming distance concept, a code with minimum distance can correct any errors. But in many channels error pattern is not very arbitrary, it occurs within very short segment of the message. Such kind of errors are called burst errors. So, for correcting such errors we will get a more efficient code of higher rate because of the less constraints. Cyclic codes are used for correcting burst error. In fact, cyclic codes can also correct cyclic burst errors along with burst errors. Cyclic burst errors are defined as
A cyclic burst of length is a vector whose nonzero components are among (cyclically) consecutive components, the first and the last of which are nonzero.
In polynomial form cyclic burst of length can be described as with as a polynomial of degree with nonzero coefficient . Here defines the pattern and defines the starting point of error. Length of the pattern is given by deg. Syndrome poynomial is unique for each pattern and is given by
A linear block code that corrects all burst errors of length or less must have at least check symbols. Proof: Because any linear code that can correct burst pattern of length or less cannot have a burst of length or less as a codeword because if it did then a burst of length could change the codeword to burst pattern of length, which also could be obtained by making a burst error of length in all zero codeword. Now, any two vectors that are non zero in the first components must be from different co-sets of an array to avoid their difference being a codeword of bursts of length . Therefore number of such co-sets are equal to number of such vectors which are . Hence at least co-sets and hence at least check symbol.
This property is also known as Rieger bound and it is similar to the singleton bound for random error correcting.
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