Zero-Error Capacity of Duplication Channels

Abstract

This paper is concerned with the problem of errorfree communication over the i.i.d. duplication channel which acts on a transmitted sequence x1 · · · xn by inserting a random number of copies of each symbol xi next to the original symbol. The random variables representing the numbers of inserted copies at each position i are independent and take values in {0, 1, . . . , r}, where r is a fixed parameter. A more general model in which blocks of ℓ consecutive symbols are being duplicated, and which is inspired by DNA-based data storage systems wherein the stored molecules are subject to tandemduplication mutations, is also analyzed. A construction of optimal codes correcting all patterns of errors of this type is described, and the zero-error capacity of the duplication channel-the largest rate at which information can be transmitted through it in an error-free manner-is determined for each ℓ and r.

Asymptotic Behavior and Typicality Properties of Runlength-Limited Sequences

“…Now just note that ( 16) is equivalent to (8) (use (15a) to express γ in terms of α and β), and (15b)-(15c) to (10).…”

“…Proof: (a) Since S L (n, w, r) = S L (n, n − w, r) (this is seen by replacing 0's with 1's, and vice versa, in all sequences), the function σ L (ω, ρ) is symmetric in the argument ω, meaning that σ L (ω, ρ) = σ L (1 − ω, ρ). Further, differentiating (9) with respect to ω, and using (10), we get…”

“…That the second derivative is negative, and hence that σ L (ω, ρ) is concave in ω, is shown by differentiating (10), upon which one obtains…”

“…Equating the first derivative to zero and using (10), we find that the maximizer ρ ⋆ ω is defined by the set of equations ( 54) and (55). That the expression in (61) is negative, and hence that σ L (ω, ρ) is concave in ρ, is shown by differentiating (10), upon which one obtains…”

On the Reverse-Complement String-Duplication System

“…A construction correcting any number of tandem duplications of a fixed length k was found in [16]. Other codes that correct a prescribed number of duplications were constructed in [17], [18], [20], [40], addressing several duplication types. Additional codes that are capable of correcting a mixture of tandem duplications and substitutions or edits were described in [32]- [34].…”