The class of q-ary list permutation invariant linear codes is introduced in this paper along with probabilistic arguments that validate their existence when certain conditions are met. The specific class of codes is characterized by an upper bound that is tighter than the generalized Shulman-Feder bound and relies on the distance of the codes' weight distribution to the binomial (multinomial, respectively) one. The bound applies to cases where a code from the proposed class is transmitted over a q-ary output symmetric discrete memoryless channel and list decoding with fixed list size is performed at the output. In the binary case, the new upper bounding technique allows the discovery of list permutation invariant codes whose upper bound coincides with sphere-packing exponent. Furthermore, the proposed technique motivates the introduction of a new class of upper bounds for general q-ary linear codes whose members are at least as tight as the DS2 bound as well as all its variations for the discrete channels treated in this paper.

This paper establishes new achievable rates for nonlinear Volterra communication channels using refined versions of the Azuma-Hoeffding inequality. The characteristics of these rates are illuminated in special cases of interest that include time invariant linear channels with memory, memoryless non-linear channels, and Volterra channel models.

Random coding theorems and achievable rates for nonlinear additive noise channels are presented. Modeling the channel's nonlinear behavior as a causal, stationary Volterra system, upper bounds on the average error probability are obtained for maximum likelihood and weakly typical set decoding. The proposed bounds are deduced by treating correct decoding regions as subspaces of high concentration measure and deploying exponential martingale inequalities. Due to the union bound effect and the i.i.d. assumption imposed on the codewords components, the deduced exponents constitute only lower bounds on the true random coding exponents of nonlinear channels. Cubic and fourth-order nonlinearities are used as examples to illustrate the relation of the random coding exponents and achievable rates with respect to the channel's parameters.

Random coding performance bounds for$L$-list permutation invariant binary linear block codes transmitted over output symmetric channels are presented. Under list decoding, double and single exponential bounds are deduced by considering permutation ensembles of the above codes and exploiting the concavity of the double exponential function over the region of erroneous received vectors. The proposed technique specifies fixed list sizes$L$for specific codes under which the corresponding list decoding error probability approaches zero in a double exponential manner. The single exponential bound constitutes a generalization of Shulman-Feder bound and allows the treatment of codes with rates below the cutoff limit. Numerical examples of the new bounds for the specific category of codes are presented.

A random coding theorem for nonlinear additive Gaussian channels is presented. Modeling the channel's nonlinear behavior as a causal, stationary Volterra system and under maximum likelihood decoding, an upper bound on the average error probability is obtained. The proposed bound is deduced by deploying exponential martingale inequalities. Cubic nonlinearities are used as example to illustrate the validity of the random coding exponent and the usefulness of the proposed technique in the treatment of nonlinear channels.

A new tight upper bound on the maximum-likelihood (ML) word and bit-error decoding probabilities for specific codes over discrete channels is presented. It constitutes an enhanced version of the Gallager upper bound and its variations resulting from the Duman-Salehi second bounding technique. An efficient technique is developed that, in the case of symmetric channels, overcomes the difficulties associated with the direct computation of the proposed bound. Surprisingly, apart from the distance and input-output weight enumerating functions (IOWEFs), the bound depends also on the coset weight distribution of the code.