Cyclic error correction codes.

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.