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For given DMC W , for type P and numbers 0 ≤ E ≤ E Rr (P, E, W ) = min |Rsp (P, E , W ) + E − E|+ . E :E ≤E Proof. 13) we see Rr (P, E, W ) = = = min V :D(V W |P )≤E min |IP,V (X ∧ Y ) + D(V W |P ) − E|+ min E :E ≤E V :D(V W |P )=E |IP,V (X ∧ Y ) + E − E|+ min |Rsp (P, E , W ) + E − E|+ . 17. Involving Ecr = Ecr (P, W ) = min E : ∂Rsp (P, E, W ) ≥ −1 . ∂E we can write for all E > 0 Rr (P, E, W ) = Rsp (P, E, W ), if E ≤ Ecr , |Rsp (P, Ecr , W ) + Ecr − E|+ , if E ≥ Ecr . Proof. Since the function Rsp (P, E, W ) is convex in E, then for the values of E less than Ecr (P, W ) the tangency of the tangent is less than −1, and for E greater than Ecr (P, W ), it is equal or greater than −1.

3 of the Gallager’s book [79]. 5 for E(R, W ) from the book by Csisz´ar and K¨orner [51]. Consider the function called random coding bound for C(E, W ) Rr (P, E, W ) = min V :D(V W |P )≤E |IP,V (X ∧ Y ) + D(V W |P ) − E|+ , Rr (E, W ) = max Rr (P, E, W ). 5. For DMC W , for all E > 0 the following bound of E-capacity holds Rr (E, W ) ≤ C(E, W ) ≤ C(E, W ). 1 from [51]. 6. 20) there exist M distinct vectors x(m) from such that for any m ∈ M, any conditional types V, V , and N large enough, the following inequality is valid TPN (X), N TP,V (Y |x(m)) N TP,V (Y |x(m )) m=m ≤ N |TP,V (Y + |x(m))| exp{−N E − D(V W |P ) }.

3 Sphere Packing Bound for E-capacity 17 and denoted by Rsp (E, W ), of the E-capacity C(E, W ) for the average error probability. 9) for E(R, W ). Let V : X → Y be a stochastic matrix. 15) Rsp (E, W ) = max Rsp (P, E, W ). 4. For DMC W , for E > 0 the following inequalities hold C(E, W ) ≤ C(E, W ) ≤ Rsp (E, W ). Proof. Let E and δ be given such that E > δ > 0. 4) is 1 M W N {Y N − g −1 (m)|f (m)} ≤ exp{−N (E − δ)}. 1)), then there exists a “major” type P ∗ such that f (M) TPN∗ (X) ≥ M (N + 1)−|X .

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Finite Fields by Tanja Lange


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