Lecture
Cyclic error correction codes.
In general, any correction code correcting t errors corrects any configuration of t errors. However, if it is known in advance that errors are located in a package, then codes can be constructed more efficiently. The error packet is described as e (x) = x i * b (x) (mod x n -1), where b (x) is a polynomial whose degree is not higher than t -1, x i is the packet locator, i number of discharge.
Syndromic polynomials S (x) for a packet error correcting CC must be different for any packet of length not more than t.
Example: g (x) = x 6 + x 3 + x 2 + x + 1, n = 15 and corrects packets of three or less errors.
e (x) = x i , i = 0, ..., 14
e (x) = x i (1 + x) (mod x 15 -1)
e (x) = x i (1 + x 2 ) (mod x 15 -1)
e (x) = x i (1 + x + x 2 ) (mod x 15 -1)
Direct calculation shows that the syndromes for all 56 possible packages are different. Therefore, g (x) = x 6 + x 3 + x 2 + x + 1 generates a code that corrects all packets of length 3.
As a rule, error correction CKs are synthesized by computer.
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Information and Coding Theory
Terms: Information and Coding Theory