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In the modern systems of transfer and storage of information for correction of the arising mistakes noiseproof codes of Read-Solomon widely are used. With use of soft decisions apply decoding of these codes on the generalized minimum distance which advantage is simplicity of realization to correction of mistakes. In work the algorithm of decoding of codes of Read-Solomon on the generalized minimum distance which feature is use of the algebraic decoder correcting errors abroad a half of the minimum code distance with use of soft decisions is offered. The algebraic decoder realizes syndromic decoding and is based on application of analytical continuation of an algorithm of Berlekempa-Messi for 2τ iterations (τ-number of in addition corrected wrong symbols). He provides search of positions of tC+τ of wrong symbols in a code word (tC - number of the wrong symbols which are guaranteed corrected by a code) which locators would be the return to roots of a possible polynom of locators of errors of degree tC + τ. Search of positions of mistakes is carried out in ascending order of nadezhnost of symbols of the accepted code word. The efficiency of correction of mistakes was investigated by the offered algorithm in the channel with additive white Gaussian noise by imitating modeling on the COMPUTER. Researches were conducted for Read-Solomon's codes defined over the field of GF(28). The additional code prize provided with an algorithm at correction on iteration of three additional mistakes in relation to Read-Solomon (255,239,17) code reaches 0,26 dB. The additional code prize for Read-Solomon (255,127,129) code at correction on iteration of two additional mistakes has made about 0,1 dB. The additional code prize for Read-Solomon (255,41,215) code at correction on iteration of three additional mistakes has made about 0,17 dB.