On the Capacity of the Precision-Resolution System

Abstract

Abstract-Arguably, the most prominent constrained system in storage applications is the (d; k)-run-length limited (RLL) system, where every binary sequence obeys the constraint that every two adjacent 1's are separated by at least d consecutive 0's and at most k consecutive 0's, namely, runs of 0's are length limited. The motivation for the RLL constraint arises mainly from the physical limitations of the read and write technologies in magnetic and optical storage systems. We revisit the rationale for the RLL system, reevaluate its relationship to the constraints of the physical media and propose a new framework that we call the Precision-Resolution (PR) system. Specifically, in the PR system there is a separation between the encoder constraints (which relate to the precision of writing information into the physical media) and the decoder constraints (which relate to its resolution, namely, the ability to distinguish between two different signals received by reading the physical media). We compute the capacity of a general PR system and compare it to the traditional RLL system. Index Terms-Run-length limited (RLL), constrained coding, capacity of constrained channels.

Coding for Optimized Writing Rate in DNA Storage

“…The PR framework was introduced in [17] as a way to improve RLL constraints. In essence, the RLL constraint was originally devised to solve the problem of clocking differences between the transmitter and receiver.…”

“…Thus, the constrained code employed should make sure run lengths may be unambiguously recoverable at the receiver side. It was shown in [17] that the choice of allowable run lengths in the RLL constraint is sub-optimal, and that the PR framework provides an optimal set of runlengths, thus increasing the system's capacity.…”

“…This does not seem to be the case described in [10], and we must assume the quantizer at the receiver may err. 3) Multiple copies -The setting described in [17] allows the encoder to create a single output that is transmitted over the channel, and decoded at the receiver. Here, however, multiple copies of the DNA string may be created simultaneously.…”

“…We may use these extra copies to improve our overall scheme. We therefore describe a new framework that builds upon the work of [17] and generalizes it.…”

Coding for the $\boldsymbol \ell _\infty $ -Limited Permutation Channel

“…The proof of Corollary 13 follows standard techniques (see, e.g., [14]). We also mention that in the other extreme case in which q = |Σ| is large (w.r.t.…”