Third-order Analysis of Channel Coding in the Moderate Deviations Regime

The channel coding problem in the moderate deviations regime is studied; here, the error probability sub-exponentially decays to zero, and the rate approaches the capacity slower than $O(1/\sqrt n )$. The main result refines Altuğ and Wagner’s moderate deviations result by deriving lower and upper bounds on the third-order term in the asymptotic expansion of the maximum achievable message set size. The third-order term of the expansion employs a new quantity called the channel skewness. For the binary symmetric channel and most practically important (n,ϵ) pairs, including n ∈ [100, 500] and ϵ ∈ [10
⁻¹⁰
,10
⁻¹
], an approximation up to the channel skewness is the most accurate among several expansions in the literature.