By P. P. Martin

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Our strategy will be to use some terms, where necessary, without an initial definition; but to try to come back to them later and find a way to check that we really do agree on their meaning. To begin with, then, we may consider communication as the process whereby some ‘data’ held in one ‘machine’ is passed so that a copy is held in some other machine. This is somewhat analogous to the process whereby an idea held in your mind might be communicated to me, so that I then hold that same idea. 5 The extent to which this analogy works (or, if failing, we 4 What does ‘certain’ mean here?!

Answer: A q-ary repetition code has M = q and d = n for any q, n. Our first case is an example of this: {000000, 111111} is a (6,2,6) code. As we have set things up, all codewords are necessarily distinct. This means that d is necessarily at least 1. To make a d = 1 code, then, all we have to do is make any code at all. The biggest q-ary length n code has M = q n (just include every possible codeword). For binary n = 3, therefore, this biggest code has M = 8. That is, for (3,8,1): {000, 001, 010, 011, 100, 101, 110, 111} is the unique such code.

For A2 (n, d) we can deduce the even d cases from the odd. 19. The weight of a string x ∈ S n is w(x) = #non-zero entries in x 7 and roadies 44 CHAPTER 3. g. w(011) = 2 = w(10010). 20. Suppose x, y ∈ S n both have even weight. Then d(x, y) is even. , n} and, fixing x, y, nij = nij (x, y) = {k ∈ n | xk = i and yk = j} For example if x = 01101, y = 10110 then n00 = ∅ and n01 = {1, 4}. Now w(x) = |n10 | + |n11 | = 2l for some l, since it is even; and w(y) = |n01 | + |n11 | = 2m for some m similarly.

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