7.2 Choice of a generating polynomials of a cyclic code

Theoretical questions of a choice of optimum parameters and synthesis of code combinations of a cyclic code are considered in [1, 2, 4].

It is obvious, that introduction of necessary size of redundancy will be defined by length of an information part k, a preset value of admissible probability of error Р_{un er}, multiplicity of detected errors t_d and quality of the communication channel.

For engineering calculations wide application was found model of the errors stream, offered by L. P. Purtov, which with sufficient accuracy for practice describes characteristics of errors stream with packing.

Investigating statistics of errors in a communication channel, it has been noticed, that the probability of errors occurrence of multiplicity t in n is equal to a digit code combination:

; (1)

Where α ‑ factor of bursts of errors in the discrete channel.

For the channel without grouping (without memory) α = 0, and if α = 1 errors are concentrated in one package.

The cyclic code is necessary for detection of number of errors by multiplicity t with code distance not less then the formula 1 will become:

. (2)

With some approach it is possible to connect probability of errors occurrence of multiplicity t [P ( t, n)] with probability not found out errors P_{un er} and number of verifying categories in a code combination as follows:

(3)

Having substituted in the formula 3 value P ( t, n) and, having executed transformation, we will calculate r

(4)

At calculation on the personal computer it is more convenient to use decimal logarithms. After transformations:

(5)

As in this formula n = k + r, demanded value r can be defined by size selection r, satisfying to an inequality:

. (6)

Size selection r is necessary for beginning with 3 and to increase on 1 until the inequality will be satisfied.

Knowing size r, i.e. size of the higher degree of a generating polynomialsials, it is necessary to choose a corresponding polynomials from table 4.

For example, we will calculate quantity of verifying symbols and we will choose generating polynomialsials for following initial data:

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

• Probability of not detected errors decoder P_{un er} = 1,5*10^-6;

• The minimum code distance d₀ = 3;

• Factor of bursts α = 0,6.

Let's substitute in the formula (6) initial data, and also 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 not executed

r = 7: - the inequality is carried out. Therefore, value r = 7.

For a choice of a generating polynomials from table 4 it is possible to take advantage of any of three resulted polynomials for the quantity of verifying symbols equal 7. We will choose the second polynomials: x⁷ + x⁴ + x³ + 1.

Table 4

Degree of a generating polynomials	Kind of polynomials
1	x+1
2	x2+x+1
3	x3+x+1 x3+x2+1
4	x4+x+1 x4+x3+1 x4+x3+x2+x+1
5	x5+x3+1 x5+x3+x2+1 x5+x4+x2+x+1 x5+x4+x3+x2+1
7	x7+x3+1 x7+x4+x3+1 x7+x3+x2+x+1
8	x8+x4+x3+x+1 x8+x5+x4+x3+1 x8+x7+x5+x+1
9	x9+x4+x2+x+1 x9+x5+x3+x2+1 x9+x6+x3+x+1
10	x10+x3+1 x10+x4+x3+x+1 x10+x8+x3+x2+1
11	x11+x2+1 x11+x7+x3+x2+1 x11+x8+x5+x2+1
12	x12+x6+x4+x+1 x12+x9+x3+x2+1 x12+x11+x6+x4+x2+x+1
13	x13+x4+x3+1 x13+x10+x9+x+1 x13+x12+x11+x2+1
14	x14+x13+x11+x9+1 x14+x12+x10+x4+x2+x+1 x14+x12+x2+x+1
15	x15+x12+x3+x+1 x15+x13+x5+x+1 x15+x14+x13+x10+x2+x+1
16	x16+x15+x7+x2+1 x16+x14+x12+x3+x2+x+1 x16+x12+x5+x+1