1.2.2 The relative information transfer rate of cyclic codes

To determine the optimal parameters of CC use the expression for the relative information transfer rate

R = C/B = k/n. (1.11)

where C – information transfer rate, (bps);

B – modulation rate, (baud).

The length of the code combinations n should be chosen to ensure maximum capacity of a discrete communication channel. Obviously, when the larger the ratio k/n approaches 1, the smaller C differs from B and the higher the relative information transfer rate. It is known that for cyclic codes with minimum code distance d₀ = 3 the inequality is correct:

r ≥ log₂(n + 1).

Evidently, that than more n, the nearer the relation k/n is approached by 1. This inequality also is correct for large d₀, although exact correlations between r and n there are not. There are only high and lower bounds estimations of minimum code distance (1.7) – (1.10). From the point of bringing of the permanent redundancy in the code combination, it is advantageous to choose long code combinations, because with the increase of n a relative bandwidth increases, tending to the limit = 1.

In the real communication channels interferences are present, resulting in appearance of errors in the code combinations. At detecting an error, a decoder in the systems with the automatic repeat request (ARQ) the request of the code combinations group is produced. During request useful information is not passed, information transfer rate decreases therefore. It is possible to show that

(1.12)

where P_det – probability of the detected errors by a decoder (probability of request);

P_cr – probability of correct reception of the code combination;

M – capacity of the transmitter buffer on the certain number of the codewords.

After simplifications and transformations of the formula (1.12), for the case of independent distribution of errors, get the information transfer rate is

, (1.13)

where p_er – error probability in the communication channel;

L – the distance between the transmitter and receiver, km;

V – the signal propagation velocity, km/sec.

Thus, the relative information transfer rate of the channel is

. (1.14)

So, at presence of errors with the independent distribution in a communication channel a value R is a function of the parameters p_er, n, k, L and V. Consequently, there is optimal n, at that a relative information transfer rate will be maximal.

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

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

; (1.15)

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

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

If the cyclic code is used in the mode of the errors detection by multiplicity t, then the formula (1.15) with account (1.5) will be:

. (1.16)

With some approximation it is possible to associate the probability of errors of multiplicity t [P( t, n)] with probability of the undetected errors P_un er and the number of check bits in a code combination as follows: