Cyclic codes

1 Purpose Cyclic codes
2 Formation of control bits
3 control scheme

Codes have become widespread with the sequential transmission of information with a small redundant equipment, not only Single but also group errors.

Or a binary code can be provided in the form of a polynomial N-1 some variable X

1 * x6 + 0 * x5 + 1 * x4 + 1 * x3 + 1 * x2 + 1 * x1 + 1 * x0 = x6 + x3 + x2

Polynomial generating P (X) number of digits nmk

The cyclic code N of a significant number is constructed by adding to all binary information the discharge to the controlled bits
the last ... Junior digits of the number and, therefore, the transmission is followed by the last one after the information bits.
The code combination is constructed as follows.
the information polynomial G (X) the underlying coding is multiplied by x ^ k. The resulting polynomial G (X) k ^ x is divisible by the generator polynomial to determine the remainder R (x). Then the remainder is added modulo 2 with C (x) x ^ k

F (x) = C (x) x ^ k (+) R (x)

By dividing the polynomial C (x) x ^ k by the generating polynomial R (x), in addition to the remainder R (x), the quotient is obtained

G (x) = Q (x) P (x) (+) P (x)

Q (x) - private branch

F (x) = G (x) x ^ k (+) R (x) = Q (x) P (x)
Whence it follows that the codes and the polynomial f (x) are divisible by the value p (x) and, in the absence of an error, should be divisible by P (x) without a remainder.

In addition, G (x) x ^ k has zeros in K bits. Therefore, the higher m bits of the F (x) code pattern are binary bits.
combinations of G (x), that is, informational.

Minors to the bits of the code combination F (x) are allocated to the remainder R (X) and represent the control bits

Incorrectly accepted combination

H (x) = F (x) (+) E (x)

Where E (x) is a polynomial of errors having a nonzero term in each distorted bit
When using a cyclic code as a corrective error correction, the determination of the number of the distortion of the discharge is made according to the type of residual obtained, the analysis of the remainder and the correction of the distorted discharge is performed automatically.
It should be noted that the cyclic code with d = 3 is considered to be a cyclic version of the Hamming code in which, unlike the latter, all control bits are placed at the end of the information combination.