Some considerations for choosing a generating polynomial.

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.