Feedback Insertion-Deletion Codes

TL;DR

This paper introduces a new feedback-based insertion-deletion coding problem, reduces it to substitution channels, and establishes the maximal asymptotic rate, revisiting and elaborating on Zigangirov's partial results.

Contribution

It formulates the feedback insertion-deletion coding problem, reduces it to substitution channels, and fully determines the asymptotic rate, improving on Zigangirov's analysis.

Findings

Maximal asymptotic rate for feedback insertion-deletion codes established.

Reduction of insertion-deletion problem to substitution channel problem.

Revised and detailed proof of Zigangirov's lower bound.

Abstract

In this paper, a new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Suppose that the encoder transmits $n$ binary symbols one-by-one over a channel, in which some symbols can be deleted and some additional symbols can be inserted. After each transmission, the encoder is notified about the insertions or deletions that have occurred within the previous transmission and the encoding strategy can be adapted accordingly. The goal is to design an encoder that is able to transmit error-free as much information as possible under the assumption that the total number of deletions and insertions is limited by $\tau n$, $0<\tau<1$. We show how this problem can be reduced to the problem of transmitting messages over the substitution channel. Thereby, the maximal asymptotic rate of feedback insertion-deletion codes is completely established. The maximal asymptotic rate for the adversarial substitution channel has been partially determined by Berlekamp and later finished by Zigangirov. However, the analysis of the lower bound by Zigangirov is quite complicated. We revisit Zigangirov's result and present a more elaborate version of his proof.