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

Abstract

In this work we consider the communication of information in the presence of synchronization errors. Specifically, we consider permutation channels in which a transmitted codeword x = (x 1 ,. .. , x n) is corrupted by a permutation π ∈ S n to yield the received word y = (y 1 ,. .. , y n) where y i = x π(i). We initiate the study of worst case (or zero error) communication over permutation channels that distort the information by applying permutations π which are limited to displacing any symbol by at most r locations, i.e. permutations π with weight at most r in the ℓ ∞-metric. We present direct and recursive constructions, as well as bounds on the rate of such channels for binary and general alphabets. Specific attention is given to the case of r = 1.

Zero-Error Capacity of $ P $ -ary Shift Channels and FIFO Queues

“…One can imagine a queuing system with a finite buffer which drops an incoming packet whenever the buffer is full, or a molecular communication system in which some of the particles never arrive at the receiving side. Another extension are models in which reordering of particles/packets is allowed (see, e.g., [1], [9], [14]). For example, due to properties of most molecular communication systems, it is reasonable to assume that the order in which the particles arrive at the receiving side is not necessarily the same as the one in which they were transmitted.…”

On Coding Over Sliced Information

“…Finally, we also mention the permutation channel [12], [13], [20], which is similar to our setting, and yet it is farther away in spirit than the aforementioned works. In that channel, a vector over a certain alphabet is transmitted, and its symbols are received at the decoder under a certain permutation.…”

Codes in the Space of Multisets—Coding for Permutation Channels With Impairments

“…Namely, the number of deletions growing linearly with the block-length is another natural asymptotic regime. Note that the upper bound in (31) is valid in this regime and can be expressed in the corresponding asymptotic form via (34). As for the lower bound, one can apply the familiar Gilbert-Varshamov bound.…”

“…grows exponentially in n with exponent (1 + q)H 1 1+q (see (34)), and since it has linearly many summands, there must exist λ ∈ (0, 1) such that q λn n−1 λn grows exponentially in n with the same exponent. By using Stirling's approximation, one can find the exponent of q λn n−1 λn as a function of λ, and check by differentiation that it is maximized for unique λ = λ * = q 1+q .…”

“…To the best of our knowledge, apart from[28], there is only a handful of papers discussing coding and related problems for channels with random reordering of symbols, e.g.,[54],[40],[16],[27]. Channel models with restricted reordering errors have also been studied in the literature, e.g.,[2],[31],[34],[43]; in these and similar works it is assumed that only certain permutations are admissible during a given transmission and, consequently, they require quite different methods of analysis.…”