Combinatorial channels from partially ordered sets

A central task of coding theory is the design of schemes to reliably transmit data though space, via communication systems, or through time, via storage systems. Our goal is to identify and exploit structural properties common to a wide variety of coding problems, classical and modern, using the framework of partially ordered sets. We represent adversarial error models as combinatorial channels, form combinatorial channels from posets, identify a structural property of posets that leads to families of channels with the same codes, and bound the size of codes by optimizing over a family of equivalent channels. A large number of previously studied coding problems that fit into this framework. This leads to a new upper bound on the size of s-deletion correcting codes. We use a linear programming framework to obtain sphere-packing upper bounds when there is little underlying symmetry in the coding problem. Finally, we introduce and investigate a strong notion of poset homomorphism: locally bijective cover preserving maps. We look for maps of this type to and from the subsequence partial order on q-ary strings.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Electrical & Computer Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Cullina, Daniel Francis

Contributors dc:contributor

• Kiyavash, Negar
• Barg, Alexander
• Hajek, Bruce
• Milenkovic, Olgica
• Srikant, R.

• coding theory
• combinatorics, partially ordered set
• deletion errors

Statement dc:rights

• Copyright 2016 Daniel Cullina

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/95346

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/95346