Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 1 / a

Table of contents
- Solution a

Solution

A rate-

R

block code consists of an encoder

E_{n} : J^{n} \to {1, \dots, M_{n}}

and decoder

D_{n}

, with

M_{n} \leq 2^{n R}

; it is reliable when

Pr [D_{n} (E_{n} (U^{n})) \neq = U^{n}] \to 0

. Choose

ε > 0

with

H (U) + ε < R

. The typical set has probability tending to one and cardinality at most

2^{n (H (U) + ε)} \leq 2^{n R}

. Encode its words injectively and map every atypical word to a default codeword. The error probability is at most the atypical probability, so reliable compression exists.

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