Question

(a) Consider binary codes determined by a parity-check matrix H. Let r be a vector of received symbols. The syndrome of r happens to be a sum of some columns of H. What columns are these?
Hint: They are determined by the errors occurring in transmissions.

(b) Let G be a k × n generating matrix of a code C the form G = [I_k | B] , where Ik is the k × k identity matrix, and B is a matrix of suitable dimensions. We want to encode an input word of the form w = (a_1, . . . , a_k) by G to obtain a codeword c = (c_1, . . . , c_n) ∈ C. What will be the first k symbols of codeword c?

(c) Consider coding in the binary alphabet, which encodes each bit symbol by repeating it three times: symbol x is encoded as codeword [x, x, x]. Give a generating matrix for this linear code.
Hint: The matrix has only one row.

Answer 1