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B) Write down a maximal complete subgraph of each of the following: G(3, 3), G(4, 3), G(5, 3). 0. Prove it. g. Ψ(10111) = 01111). Then Ψ defines a graph homomorphism from G(n, k) to itself. Prove it. ) ANSWERS: (a) Try this yourself. Note that it says that A2 (n, k) is the size of a maximal complete subgraph in G(n, k). (b) We give our complete graph as a list of vertices in each case: G(3, 3): {000, 111} (equally good would be {001, 110}, but it will be clear than neither subgraph can be enlarged without losing the completeness property); G(4, 3): {0000, 1110}; G(5, 3): {00000, 11100, 10011, 01111}.

That is, for (3,8,1): {000, 001, 010, 011, 100, 101, 110, 111} is the unique such code. 5. OPTIMISATION 47 For our fourth case it is no longer obvious how to construct a code. Under the circumstances it is prudent to check if such a code is impossible, by checking the BP and singleton bounds. In this case one finds that the BP bound fails, so there is no such code. An (undirected) graph G is a set VG of vertices together with a set EG of edges between them (for a more careful definition see Chapter 5).

CODING THEORY so for a linear code R = k/n Thus the bigger k is, the more information we transmit; the bigger n is, the longer it takes to transmit. But of course the bigger n−k is the more checking we are doing, so the better we can confirm or protect the information. Let us now examine some examples of linear codes. In particular, which of the codes we already looked at are linear? If S = F is a field then the repetition code Rn ⊂ F n is linear of dimension 1. 2. The parity-check code Pn ⊂ F n consists of all vectors u such that ui = 0 i We can consider the first n − 1 digits as information, and un as a check digit, simply defined as n−1 un = − ui .

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