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However, it can be proven that decoding up to minimum distance alone is not good enough to prove Shannon’s Theorem. ) Instead we note that a received block of large length n is most likely to contain sn symbol errors where s = p(m − 1) is the probability of symbol error. Therefore in proving Shannon’s theorem we look at large numbers of codes, each of which we decode using SSρ for some radius ρ a little larger than sn. A family C of codes over A is called a Shannon family if, for every > 0, there is a code C ∈ C with PC < .
5 we have |C| · |Ssn (∗)| ≤ mn , which, upon taking logarithms, is logm (|C|) + logm (|Ssn (∗)|) ≤ n . 3. SHANNON’S THEOREM AND THE CODE REGION 25 We divide by n and move the second term across the inequality to find κ(C) = n−1 logm (|C|) ≤ 1 − n−1 logm (|Ssn (∗)|) . The righthand side approaches 1 − Hm (s) = Cm (p) as n goes to infinity; so, for C to be a contributing member of a Shannon family, it should have rate at most capacity. 4) Proposition. If C is a Shannon family for mSC(p) with 0 ≤ p ≤ 1/m, then lim inf C∈C κ(C) ≤ Cm (p).
1) Theorem. ) Let F be a field with m elements, and consider a mSC(p) with p < 1/m. Set Lκ = { linear codes over F with rate at least κ }. Then Lκ is a Shannon family provided κ < Cm (p). 2 Hamming weight minimum weight Forney (1966) proved a strong version of this theorem which says that we need only consider those linear codes of length n with encoder/decoder complexity on the order of n4 (but at the expense of using very long codes). Thus there are Shannon families whose members have rate approaching capacity and are, in a theoretical sense, practical1 .