Choice of the generator polynomial of the cyclic code

Theoretical questions of optimum parameters choice of a cyclic code are considered in the item 1.2.2 of this teaching manual.

Knowing the minimum code distance d ₀ and the length of information part of the code combination k (for the task 2.1.1 k = 16), we can calculate the length of the check part of the code combination r. According to the minimum of redundancy of CC, we have to use formula (1.19). As in this formula n = k + r, demanded value r can be defined by size selection r, satisfying to an inequality:

. (2.1)

Size selection r it is necessary to begin from 3 and to increase on 1 until the inequality will be executed.

Knowing size r, i.e. size of the higher degree of a generator polynomial, it is necessary to choose a corresponding polynomial from table 1.1.

For example, let’s calculate the amount of check bits and choose generator polynomial for following initial data:

- Probability of an error in a communication channel р _er = 3*10^-5;

- Probability of undetected errors by the decoder P _{un er} = 1,5*10^-6;

- The minimum code distance d ₀= 3;

- Factor of burst α = 0,6.

Let's substitute to the formula (2.1) initial data, and the value r since 3:

r = 3: - the inequality is not executed

r = 4: - the inequality is not executed

r = 5: - the inequality is not executed

r = 6: - the inequality is executed. There is no generator polynomials with the maximum degree 6 in the table 1.1, therefore, value r = 7.

For a choice of a generator polynomial from table 1.1 it is possible to take advantage of any of three resulted polynomials for the amount of check symbols 7. We will choose the second polynomial: x ⁷ + x ⁴ + x ³ + 1.

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