66795

Dec od i nu cyclic codes

Following the Conclusion to check whether thc received sequcnce i<*) <=> v is a codę wordofan (/>,&) cyclic codę, v(jc) isdividedby g(x), a rcminder r(x) is dcsignatcd.

g(x) *    g(x)

This rcminder can be considcred as syndromc polynomial of i'(jc),

j>(.r) = r(x) <=> s = r

and it dctermincs a decodcr dccision. following the stratcgy of (5.2.25) for crror dctccting decoding or (5.2.26) for error correcting decoding.

For a cyclic codę, relationship between a codeword, error patiem and the received sequence (5.1.1), in polynomial notation (indexeshave been omitted):

v(x)= H<.t)©zU)

Then, from (7.1.22) and (7.1.20):

r{x)

g(x)

»*’(*)

S(v)

and taking into account

g(x)    g(x)

As can be seen, for each error polynomial z(ar), the corresponding syndrome polynomial s(.v)can be designated. Thus, for an (n,k) cyclic codę, error-syndrome look-up table can be constructed and used to error correction. The decoder decision (5.2.26), in polynomial notation: