A Study on Universal Codes With Finite Block Lengths

Abstract

Abstract-Based on random codes and typical set decoding, an alternative proof of Root and Varaiya's compound channel coding theorem for linear Gaussian channels is presented. The performance limit of codes with finite block length under a compound channel is studied through error bounds and simulation. Although the theorem promises uniform convergence of the probability of error as the block length approaches infinity, with short block lengths the performance can differ considerably for individual channels. Simulation results show that universal performance can be a practical goal as the block lengths become large.Index Terms-Compound channel, random coding bound, sphere-packing bound (SPB), universal code.

On Universal Properties of Capacity-Approaching LDPC Code Ensembles

“…where the last equality follows from (47). This lower bound on g 1 is attained for a BEC (since for a BEC whose erasure probability is p, (20) implies that the sequence {g k } is constant and…”

“…We refer the reader to recent studies on universal LDPC codes (see, e.g., [13], [30], [38] and [47]). A simple modification of the bounds derived in this paper makes them universal in the sense that they hold for the set of MBIOS channels which exhibit a given channel capacity.…”

Algebraic lattice codes achieve the capacity of the compound block-fading channel

“…The proof uses the techniques of [9], [10,Appendix], and consists of two parts: (i) an universal code for a finite set of channel matrices and (ii) fine quantization of the channel coefficients Proof. (i) Suppose that we have H 1 , .…”

“…The authors provide a technique to convert traditional random codes into universal ones, under the additional assumption that the norm of H is bounded (see also [10]). In other words, it is shown how to achieve the compound capacity for H ∩ S, where S is a compact set.…”

“…This represents an advantage of ideal lattices over the classic Gaussian random codes [9], [10] and standard Construction A [5]. This is made possible by exploiting the multiplicative structure of number fields and their group of units.…”