Chapter2
17. (a) M has eight elements: there are only four values for e, so there
must be m1 and m2 in M with e(m1)=e(m2). Now if m(1) is transmuted into
m(2) by a two bit error, then the error0code e cannot detect this.
(b)For a crude estimate, let M be the set of N-bit messages with four
1's and all the rest zeros. The size of M is (Nchose4)=N!/4!(n-4)!.
Any element of M can be transmuted into any other by an 8-bit error.
If we take N large enough that the size of M is bigger thean 2**32, then
as in part (a) there must be for any 32-bit error code function e(m)
be elements m(1) and m(2) of M with e(m(1))-e(m(2)). To find a
sufficiently large N, we note N!/4!(N-4)! > N-3)**$/24; it
thus suffices to find N so that (N-3)**4>24*2**32, or approximately
10**11. N approximately 600 flies.