On Coding for Partial Erasure Channels

Authors

Carolyn Mayer, University of Nebraska - LincolnFollow

First Advisor

Christine A. Kelley

Date of this Version

5-2018

Document Type

Article

Citation

Mayer, Carolyn. "On Coding for Partial Erasure Channels," PhD diss., University of Nebraska, 2018.

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Christine A. Kelley. Lincoln, Nebraska: May, 2018

Copyright (c) 2018 Carolyn Mayer

Abstract

Error correcting codes have been essential to the technology we use in everyday life in digital storage, wireless communication, barcodes, and much more. Different channel models are used for different types of communication (for example, if information is sent to one person or to many people) and different types of errors. Partial erasure channels were recently introduced to model applications in which some information remains after an erasure event. These remnants of information may be used to increase the chances of successful decoding. We introduce a new partial erasure channel in which partial erasures correspond to individual bit erasures in the binary expansion of a 2^k-ary symbol or p-ary symbols in the expansion of a p^k-ary symbol. We show how multilevel coding and multistage decoding may be used on partial erasure channels and investigate cases in which partial erasure channels may be decomposed into simpler channels. Further, we show that partial erasure channels do not always decompose into simple erasure channels, and that when they do, the erasure channels may not be independent. The rest of this work focuses on three areas: fountain codes on partial erasure channels, relay channels with partial erasures, and graph-based codes for distributed storage. We adapt a class of fountain codes for use on partial erasure channels and show an improvement in terms of the number of symbols that must be generated for the successful decoding of such codes. In a relay channel setting, we consider a simple three node system with a sender, receiver, and relay where at least one of the links is a partial erasure channel. When the sender-receiver link is a degraded version of the sender-relay link, we determine the capacity of the channel. We also introduce a biregular hypergraph construction and find locality properties of codes based on these hypergraphs.

Adviser: Christine A. Kelley