On More General Distributions of Random Binning for Slepian-Wolf Encoding

Abstract

Traditionally, ensembles of Slepian-Wolf (S-W) codes are defined such that every bin of each <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> -vector of each source is randomly drawn under the uniform distribution across the sets <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\{0,1,\ldots, 2^{nR_{X}}-1\}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\{0,1,\ldots, 2^{nR_{Y}}-1\}$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{X}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{Y}$</tex> are the coding rates of the two sources, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Y$</tex> , respectively. In a few recent works, where only one source is compressed and the other one serves as side information at the decoder, the scope is extended to variable–rate S-W (VRSW) codes, where the rate may depend on the type class of the source string, but still, the random–binning distribution is assumed uniform within the type–dependent, bin index set. In this expository work, we investigate the role of the uniformity of the random binning distribution from the perspective of the trade-off between the error exponent and the source coding exponent. To this end, we study a much wider class of random–binning distributions, which includes VRSW codes as a special case, but goes considerably beyond. We first show that, except for some pathological cases, the sub-ensemble of VRSW codes is as good as the large ensemble in terms the trade–off between the error exponent and the source coding exponent. Nonetheless, the wider class of ensembles is motivated in two ways. The first is that it outperforms VRSW codes in the above–mentioned pathological cases, and the second is that it allows robustness: in the event of unavailability of the compressed bit–stream from one of the sources, it still allows reconstruction of the other source within some controllable distortion.