By Bull C.G.

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**Example text**

The decoder will decide that c has been the transmitted vector when receiving c if this received vector is inside the sphere of radius d and none of the remaining M − 1 codewords are inside that 33 Information and Coding Theory sphere. Incorrect decoding happens when the number of errors produced during transmission is such that the received vector is outside the sphere of radius d and lies in the sphere of another codeword different from c. Incorrect decoding also occurs even if the received vector lies in the sphere of radius d, but another codeword is also inside that sphere.

X M }. Each symbol xi has the probability Pi of being emitted and contains information Ii . The symbol probabilities must be in agreement with the fact that at least one of them will be emitted, so M Pi = 1 (3) i=1 The source symbol probability distribution is stationary, and the symbols are independent and transmitted at a rate of r symbols per second. 2. Each symbol contains the information Ii so that the set {I1 , I2 , . . , I M } can be seen as a discrete random variable with average information M Hb (X ) = M Pi Ii = i=1 Pi logb i=1 1 Pi (4) 5 Information and Coding Theory xi , x j , ...

This suggests that a quantitative measure of symbol information will be inversely proportional to the probability of occurrence. Assuming an arbitrary message xi which is one of the possible messages from a set a given discrete source can emit, and P(xi ) = Pi is the probability that this message is emitted, the output of this information source can be modelled as a random variable X that can adopt any of the possible values xi , so that P(X = xi ) = Pi . Shannon defined a measure of the information for the event xi by using a logarithmic measure operating over the base b: Ii ≡ − logb Pi = logb 1 Pi (1) The information of the event depends only on its probability of occurrence, and is not dependent on its content.