2 Main positions

2.1 Correcting ability of error-control codes. Correcting ability of error-control code is expressed by the guaranteed correctable errors value q_cor  (d_min – 1)/2 and by the guaranteed detectable errors value q_det  d_min – 1, where d_min – code distance.

Noise immunity of transmission system with a error-control code is described by the р_d = f (E_b/N₀), where р_d is bit error probability at the output of the decoder, E_b/N₀ is the ratio of signal energy expended on the transfer of one bit to the power density of noise (SNR) at the input of the demodulator.

Frequently the noise immunity of the transmission system with a error-control code it is convenient to describe by the gain coding (CG) value (see below).

2.2 Syndromes decoding. For (n, k) block error-control codes today the syndromes decoding is basic for a discovery and errors correction.

Syndromes method of errors correction is based on a simple rule: on the syndrome of code word the errors location is determined. Therefore under the syndrome of code word understand the result of decoder calculation on the set rules of number s = (s₁, s₂, …, s_r), r = n – k, which testifies to the errors detected and determines their placing (configuration) in code word. In binary codes a syndrome is written down in the binary number system, that its digits s₁, s₂, …, s_r take on a value 0 and 1.

A zero syndrome specifies on that the received code word is permitted, that the detected errors are not present. Nonzero syndrome is corresponding to some errors configurations. Thus possible such situations:

• the syndrome received by calculation corresponds to some configuration of errors which meets at q ≤ q_cor, in this situation the decoder "corrects" errors, but correction not true, and, most likely, the quantity of erroneous symbols in a code word grows;

• the syndrome received by calculation does not correspond to any of configurations of errors which can be at q ≤ q_cor, in this situation the decoder "refuses" decoding and passes such code word on an output with errors.

Thus, the syndromes decoding of binary codes is taken to the calculation of syndrome and search that or by other method errors configurations which errors are corrected on. Correction of an error in some binary symbol is reduced to its inversion and carried out by addition of an erroneous symbol with 1 on the module 2.The generalized block diagram of syndromes decoder is resulted in figure 1.

In the scheme of figure 1 “Syndrome analyzer” is the most complex device. In it on the calculated syndrome errors configuration, on which then the errors corrected with corrector, is set. As a syndrome can be calculated only after reception of all code word (block), for the errors correction in a code word needed to delay on n symbols, which is carried out n-digit shift register. After establishment of errors configuration an analyzer symbols of the decoding word are consistently "pushed" from a shift register, and from an analyzer symbols "1" act in those moments of time, when it is necessary to invert erroneous symbols.