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This paper starts by considering the minimization of the Re&#x0301;nyi divergence subject to a constraint on the total variation distance. Based on the solution of this optimization problem, the exact locus of the points (D(QIIP1), D(QIIP2)) is determined when P1, P2, and Q are arbitrary probability measures which are mutually absolutely continuous, and the total variation distance between P1 and P2 is not below a given value. It is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet. This characterization yields a geometric interpretation of the minimal Chernoff information subject to a constraint on the variational distance. This paper also derives an exponential upper bound on the performance of binary linear block codes (or code ensembles) under maximum-likelihood decoding. Its derivation relies on the Gallager bounding technique, and it reproduces the Shulman-Feder bound as a special case. The bound is expressed in terms of the Re&#x0301;nyi divergence from the normalized distance spectrum of the code (or the average distance spectrum of the ensemble) to the binomially distributed distance spectrum of the capacity-achieving ensemble of random block codes. This exponential bound provides a quantitative measure of the degradation in performance of binary linear block codes (or code ensembles) as a function of the deviation of their distance spectra from the binomial distribution. An efficient use of this bound is considered.

On the Rényi Divergence, Joint Range of Relative Entropies, and a Channel Coding Theorem