Coding Theory

(14 points) Let n be a positive integer and let C En, ie., let C be the binary code consisting of all length n binary vectors of even weight: (a) Find a parity check check matrix H for C. (Hint: If x = x1x2 . .-Tn, then w(a) T1 +x2+...+xn (mod 2) (b) Find a generator matrix for C. (c) Compute the minimum distance of both C and its dual code C. Justify your answers. (d) How many elements does C contain? How many elements does C1 contain? Justify your answers (e) Suppose that n is odd. Is C1 perfect? Why or why not? (f) Suppose that n is odd and 2 3. Is CCC? Is C C C? (g) Is C is a cyclic code? Justify your answer. (14 points) Let n be a positive integer and let C En, ie., let C be the binary code consisting of all length n binary vectors of even weight: (a) Find a parity check check matrix H for C. (Hint: If x = x1x2 . .-Tn, then w(a) T1 +x2+...+xn (mod 2) (b) Find a generator matrix for C. (c) Compute the minimum distance of both C and its dual code C. Justify your answers. (d) How many elements does C contain? How many elements does C1 contain? Justify your answers (e) Suppose that n is odd. Is C1 perfect? Why or why not? (f) Suppose that n is odd and 2 3. Is CCC? Is C C C? (g) Is C is a cyclic code? Justify your