Variations on the Gallager bounds with some applications

By generalizing the framework of the second version of the Duman–Salehi bound encompassing also deterministic and random codes, we demonstrate its rather broad features and show that this variation provides the natural bridge between the 1961 and 1965 Gallager bounds. This approach entails a natural geometric interpretation, encompassing also a large class of efficient recent bounds (or their Chernoff versions), which are demonstrated to be special cases of the generalized second version of the Duman–Salehi bound. Implications and applications of these observations are pointed out, referring to known bounds as well as a novel extended version of the Shulman–Feder bound.