Problem 1:
Let C be the nullspace of the 1 x 9 matrix
H = [1...1]
(6.2.6)
In other words, let C be the parity check code of length 9 (8 data bits, 1 parity check bit).
(a) Use our standard methods to find a basis B for C.
(b) What is the dimension of C? In general, what will the dimension of the parity check code of length n be?
(c) Let G be the matrix whose columns are the vectors in the basis B from part (a). For a given message bitstring m, explain why the encoding procedure of Example 6.2.8 is equivalent to transmitting x = Cm.
Let C be the repetition code of length 5.
(a) What is the dimension of C? Explain your answer.
(b) Let C' be the code of length 5 consisting of all x such that x1 = x2 (i.e., the first two coordinates of x are the same). Find a parity check matrix H2 for C'. Suggestion: What system of linear equations defines C'?)
(c) Now suppose H is a k x n parity check matrix for C (still the repetition code of length 5). What is the value of n? What is the smallest possible value of k? (Suggestion: Rank-nullity.)
(d) Find a parity check matrix for C of smallest possible size, as found in part (c).