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By Li G., Datta S.

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9. e. the syndrome of the received word. If at least one coefficient of the syndrome is not equal to zero, then the signal of error detection appears at the output of the OR gate. 9 The error detector for the (15,9) code CYCLIC CODES 39 The correction of the error pattern also can be done with the help of the syndrome. In most cases, to solve this problem it is necessary to keep in memory the table, in which every syndrome corresponds to the error pattern. For cyclic codes the solution is not so complex, because it is possible to keep in memory only those syndromes that correspond to error patterns containing a nonzero symbol in the first position.

AnÀ2 Þ is the codeword of the same code. Let each n-dimension vector f ¼ ðf0 ; f1 ; . . ; fnÀ1 Þ; fi 2 K correspond to the polynomial f ðxÞ ¼ f0 þ f1 x þ . . þ fnÀ1 xnÀ1 2 K½xŠ. Then each n-tuple corresponds to the polynomial of degree of no more than n À 1. Hereafter we will not distinguish between vector and the corresponding polynomial. 26 BLOCK CODES Let aðxÞ be the codeword of the cyclic code of length n. Consider the polynomial xaðxÞ mod ðxn À 1Þ: xaðxÞ ¼ anÀ1 xn þ anÀ2 xnÀ1 þ . . þ a1 x2 þ a0 x; and the residue of xaðxÞ on modulo ðxn À 1Þ is equal to xaðxÞ mod ðxn À 1Þ ¼ anÀ2 xnÀ1 þ .

E1 þ v2k À1 : ð2:44Þ ... : ... e2r À1 þ v2k À1 It is obvious that different rows of this array do not contain the same elements. Therefore, the number of rows is equal to 2r . The syndromes of all vectors in the same row are identical: H Á ðei þ vj1 ÞT ¼ H Á ðei þ vj2 ÞT ¼ H Á eTi ; and the syndromes of the elements from the different rows are different. The standard array is the method of writing the whole n-dimension vector space. There can occur any error vector in the channel, but the code can correct only one received vector from the row of the standard array, because the vectors, placed in the same row, have the identical syndromes.

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A bootastrap approach to non-parametric regression for right censored data by Li G., Datta S.

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