a) Let the parity check matrix of a C linear code be ' H ' . Show that the necessary and sufficient condition for finding a vector with weight ' t ' in the coset of C with syndrome ' v ' is that the vector ' v ' is written as the linear component of ' t ' columns in ' H ' .

b) Suppose that the minimum weight of a linear code C is ' d ' , where ' d ' is even-number. Show that at least two of the vectors with weight ' e + 1 ' are in the same coset, with ' e = [(d 1) / 2 ] ' .