On Reliability Function Of BSC: Expanding The Region, Where It Is Known Exactly

In the paper notation from [1] is used. A binary symmetric channel (BSC) with crossover probability 0 < p < 1/2 and q = 1 − p is considered. Let F n denote the set of all 2 binary n-tuples, and d(x,y), x,y ∈ F n denote the Hamming distance between x and y. A subset C = C(M,n) = {x1, . . . ,xM} ⊆ F n is called a code of length n and cardinality M . The minimum distance of the code C is d(C) = min{d(xi,xj) : i 6= j}. Code rate is R(C) = n log2M . Everywhere below log z = log2 z. Cardinality of a set A is denoted by |A|. Code spectrum (distance distribution)B(C) = (B0, B1, . . . , Bn) is the (n+ 1)–tuple with components Bi = |C| |{(x,y) : x,y ∈ C, d(x,y) = i}| , i = 0, 1, . . . , n. (1)