Volume 55, Nº 1 (2019)

This work is a survey on completely regular codes. Known properties, relations with other combinatorial structures, and construction methods are considered. The existence problem is also discussed, and known results for some particular cases are established. In addition, we present several new results on completely regular codes with covering radius ρ = 2 and on extended completely regular codes.

We consider the problem of determining extreme values of the Rényi entropy for a discrete random variable provided that the value of the α-coupling for this random variable and another one with a given probability distribution is fixed.

An exact expression for the probability of inversion of a large spin is established in the form of an asymptotic expansion in the series of Bessel functions with orders belonging to an arithmetic progression. Based on the new asymptotic expansion, a formula for the inversion time of the spin is derived.

This paper considers a multimessage network where each node may send a message to any other node in the network. Under the discrete memoryless model, we prove the strong converse theorem for any network whose cut-set bound is tight, i.e., achievable. Our result implies that for any fixed rate vector that resides outside the capacity region, the average error probability of any sequence of length-n codes operated at the rate vector must tend to 1 as n approaches infinity. The proof is based on the method of types and is inspired by the work of Csiszár and Körner in 1982 which fully characterized the reliability function of any discrete memoryless channel with feedback for rates above capacity. In addition, we generalize the strong converse theorem to the Gaussian model where each node is subject to an almost-sure power constraint. Important consequences of our results are new strong converses for the Gaussian multiple access channel with feedback and the following relay channels under both models: the degraded relay channel (RC), the RC with orthogonal sender components, and the general RC with feedback.