On Bounds for $E$-capacity of DMC
  Random coding, expurgated and sphere packing bounds are derived by method of types and method of graph decomposition for $E$-capacity of discrete memoryless channel (DMC). Three decoding rules are considered, the random coding bound is attainable by each of the three rules, but the expurgated bound is achievable only by maximum-likelihood decoding. Sphere packing bound is obtained by very simple combinatorial reasonings of the method of types. The paper joins and reviews the results of previous hard achievable publications. 
Evgueni A. Haroutunian (Associate Member, IEEE)