By Garrett P.

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1 + P log p + q log q is nearly 1. The requirement in the experiment was that the rate should be at least t. We see that for e > 0 and n sufficiently large there is a code C of length n, with rate nearly 1 and such that Pc < e. 3 we treat some technical details to be used later. e. it depends on wonly. The number of errors in a received word is a random variable with expected value np and variance np(l - pl. 4) Since p < t, the number p:= lnp + bJ is less than tn for sufficiently large n. Let Bp(x) be the set of words y with d(x, y) :::;; p.

The code is a singleerror-correcting code. Our explanation of decoding rules was based on two assumptions. First of all we assumed that during communication all codewords are equally likely. Furthermore we used the fact that if n l > n2 then an error pattern with n 1 errors is less likely than one with n 2 errors. This means that ify is received we try to find a codeword x such that d(x, y) is minimal. This principle is called maximum-likelihood-decoding. 2. 2. 1. Let us state the problem. We have a binary symmetric channel with probability p that a symbol is received in error (again we write q := 1 - pl.

The most common of these is given in the following definition. 7) Definition. If C is a code of length n over the alphabet IFq we define the extended code C by If C is a linear code with generator matrix G and parity check matrix H then C has generator matrix G and parity check matrix Ii, where G is obtained by adding a column to G in such a way that the sum of the columns of G is 0 and where 1... - H 1 o o o If C is a binary code with an odd minimum distance d, then C has minimum distance d + 1 since all weights and distances for C are even.

### A rationality principle by Garrett P.

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