With 1 parity bit we can detect all 1-bit errors. Show that at least one generalization fails, as follows:
(a) Show that if messages m are 8 bits long, then there is no error detection code e = e(m) of size 2 bits that can detect all 2-bit errors. Hint: Consider the set M of all 8-bit messages with a single 1 bit; note that any message from M can be transmuted into any other with a 2-bit error, and show that some pair of messages m1 and m2 in M must have the same error code e.
(b) Find an N (not necessarily minimal) such that no 32-bit error detection code applied to N-bit blocks can detect all errors altering up to 8 bits.