1 0 0 ]. Now, imagine that there are two bit-errors in the transmission, so the received codeword is [ 1 0 1 1 1 0 0 0 1 0 1 0 0 ]. In polynomial notation: In order to correct the errors, first calculate the syndromes. Taking we have and Next, apply the Peterson procedure by row-reducing the following augmented matrix. Due to the zero row, is singular, which is no surprise since only two errors were introduced into the codeword. However, the upper-left corner of the matrix is identical to , which gives rise to the solution The resulting error locator polynomial is which has zeros at and The exponents of correspond to the error locations. There is no need to calculate the error values in this example, as the only possible value is 1. Decoding with unreadable characters Suppose the same scenario, but the received word has two unreadable characters [ 1 0 ? 1 1 ? 0 0 1 0 1 0 0 ]. We replace the unreadable characters by zeros while creating the polynomial reflecting their positions We compute the syndromes and (Using log notation which is independent on GF(24) isomorphisms. For computation checking we can use the same representation for addition as was used in previous example. Hexadecimal description of the powers of are consecutively 1,2,4,8,3,6,C,B,5,A,7,E,F,D,9 with the addition based on bitwise xor.) Let us make syndrome polynomial compute Run the extended Euclidean algorithm: We have reached polynomial of degree at most 3, and as we get Therefore, Let Don't worry that Find by brute force a root of The roots are and (after finding for example we can divide by corresponding monom and the root of resulting monom could be found easily). Let Let us look for error values using formula where are roots of We get Fact, that should not be surprising. Corrected code is therefore [ 1 0 1 1 0 0 1 0 1 0 0]. Decoding with unreadable characters with a small number of errors Let us show the algorithm behaviour for the case with small number of errors. Let the received word is [ 1 0 ? 1 1 ? 0 0 0 1 0 1 0 0 ]. Again, replace the unreadable characters by zeros while creating the polynomial reflecting their positions Compute the syndromes and Create syndrome polynomial Let us run the extended Euclidean algorithm: We