Sharp Analytical Capacity Upper Bounds for Sticky and Related Channels

Abstract

We study natural examples of binary channels with synchronization errors. These include the duplication channel, which independently outputs a given bit once or twice, and geometric channels that repeat a given bit according to a geometric rule, with or without the possibility of bit deletion. We apply the general framework of Cheraghchi (STOC 2018) to obtain sharp analytical upper bounds on the capacity of these channels. Previously, upper bounds were known via numerical computations involving the computation of finite approximations of the channels by a computer and then using the obtained numerical results to upper bound the actual capacity. While leading to sharp numerical results, further progress on the full understanding of the channel capacity inherently remains elusive using such methods. Our results can be regarded as a major step towards a complete understanding of the capacity curves. Quantitatively, our upper bounds sharply approach, and in some cases surpass, the bounds that were previously only known by purely numerical methods. Among our results, we notably give a completely analytical proof that, when the number of repetitions per bit is geometric (supported on {0, 1, 2, . . . }) with mean growing to infinity, the channel capacity remains substantially bounded away from 1.

An Overview of Capacity Results for Synchronization Channels

“…4 Thus far, this is the only non-trivial capacity upper bound for any repeat channel in the "many replications" regime. [75] beats the numerical upper bound from [60].…”

“…Cheraghchi and Ribeiro [75], building on techniques from [68], derived tight analytical capacity upper bounds for the duplication and geometric sticky channels. As before, these bounds are maximums of concave smooth functions over [0, 1], and hence can be computed efficiently to any desired accuracy.…”

“…Techniques developed in [68] were later adapted to yield a better understanding of the capacity of the discrete-time Poisson channel [76], which is an important channel in optical communications. Finally, Cheraghchi and Ribeiro [75] studied the geometric deletion channel, which independently replicates bits according to a Geom p distribution. They obtain analytical capacity upper bounds for this channel, and notably give a fully analytical proof that the capacity of this channel is at most 0.73 bits/channel use (hence bounded away from 1) in the limit p → 1 (which corresponds to the expected number of replications growing to infinity).…”

“…Figure 9: Tight capacity upper bounds for the geometric sticky channel. The circle indicates where the analytical upper bound from[75] beats the numerical upper bound from[60].…”

Zero-Error Capacity of Duplication Channels

“…which is seen by considering which of the blocks from B q,ℓ,r is the first block in each particular codeword of C q,ℓ,r (n). The theorem is then obtained by a standard application of 2 The function v 2,1,1 (x) (in fact, the related function…”