Lecture
Some considerations for choosing a generating polynomial.
Code words in the Central Committee can be systematic and non-systematic. If the information word i (x) is given, then when encoding this word with a systematic code: C (x) = i (x) * x r + R [(i (x) * x r ) / g (x)], if non-systematic , then C (x) = i (x) * g (x).
To obtain the generating polynomial g (x), code words are used; f 1 , f 2 , f s are simple polynomials into which the polynomial x n -1 is expanded. Then the polynomial g (x) can be constructed as an arbitrary number of products of the function f. 2 s - various combinations of these works (f 1 (x) * ... * f s (x)). (2 s -2) - various combinations of these simple polynomials.
Example : n = 15 = 2 4 -1 = 15, n = 2 m -1 (m = 3,4,5 ...) - if we choose the length of the code like that, then the code is called primitive.
X 15 -1 = (x + 1) (x 2 + x + 1) (x 4 + x + 1) (x 4 + x 2 +1) (x 4 + x 3 + x 2 + x + 1)
As can be seen from the decomposition, the generating polynomial, for example, can be obtained in three ways:
q 1 (x 8 ) = a 1 * a 2 , q 2 (x 8 ) = a 1 * a 3 , q 3 (x 8 ) = a 2 * a 3
r = 8, k = nr = 7 => (n, k) = (15, 7)
Conclusion: from q 1 , q 2 , q 3 it is necessary to choose such generating polynomials that would provide the maximum corrective ability of the code. At first glance, the degrees of these polynomials are equal to 8 and the correcting ability of the code should be the same. However, when checking, it turns out that q 1 does not provide t = 2, and q 2 , q 3 provide. If there is a choice of generating polynomials of the same degree, then one can relatively prefer one of them only after verification.
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Information and Coding Theory
Terms: Information and Coding Theory