Randomly chosen index assignments are asymptotically bad for uniform sources

Abstract

It is known that among all redundancy-free codes (or index assignments), the natural binary code minimizes the mean-squared error (MSE) of the uniform source and uniform quantizer on a binary symmetric channel. We derive a code which maximizes the MSE and demonstrate that the code is linear and its distortion is asymptotically equivalent, as the blocklength grows, to the expected distortion of an index assignment chosen uniformly at random.

Channel-optimized scalar quantizers with erasure correcting codes

“…The usual goal to design an index *Correspondence: ye.li@siat.ac.cn 1 Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China Full list of author information is available at the end of the article assignment for noisy channels is to minimize the endto-end MSD over all possible index assignments. Some famous index assignments, such as natural binary code (NBC), Gray code and randomly chosen index assignments, are studied on a binary symmetric channel (BSC) in [16][17][18][19][20][21]. In [16], it is proved that NBC is optimal for uniform scalar quantizers and uniform source.…”

Quantizers with uniform encoders and channel optimized decoders

“…Then we extend a result in [4] by showing that most index assignments are asymptotically bad (Theorem 7.1), and we extend results in [3], [6], and [7] by computing the mean squared error resulting from the Natural Binary Code (Theorem 7.3), the Folded Binary Code (Theorem 7.5), the Gray Code (Theorem 7.7), and a randomly chosen index assignment (Theorem 7.9). As comparisons, we state previously known mean squared error formulas for channel unoptimized decoders (i.e.…”

“…Theorem 7.8 was stated in [6], and [4] contains a concise proof. Let £ ± # ö 7 ÷ be a random variable denoting the MSE of a channel unoptimized uniform quantizer with a randomly chosen index assignment.…”

“…In [4] it was shown that randomly chosen index assignments for a channel unoptimized uniform quantizer are asymptotically bad in the sense that their MSE approaches that of the worst possible index assignment in the limit as ¹ B D…”

“…In [3] explicit mean squared error formulas were computed for uniform sources on binary asymmetric channels with various structured classes of index assignments. In [4] it was shown that for the uniform source and uniform quantizer the mean squared error resulting from a randomly chosen index assignment was, on average, equal in the limit of large to that of the worst possible index assignment. In this sense the result showed that randomly chosen index assignments are asymptotically bad.…”