All ( 2^r - 1 ) non-zero syndromes correspond exactly to single-error positions.
The core metric here is – the number of bit positions in which two codewords differ. coding and information theory hamming pdf
His book, Coding and Information Theory (first published in 1980 by Prentice-Hall), is the culmination of a lifetime of thinking about how to manage information in an imperfect world. All ( 2^r - 1 ) non-zero syndromes
Given ( k ) data bits ( \mathbfd ): [ \mathbfc = \mathbfd G ] coding and information theory hamming pdf