Reed-Solomon Codes and the Deep Hole Problem

of the Dissertation Reed-Solomon Codes and the Deep Hole Problem (DRAFT) By Matt Keti Doctor of Philosophy in Mathematics University of California, Irvine, 2015 Professor Professor Daqing Wan, Chair In many types of modern communication, a message is transmitted over a noisy medium. When this is done, there is a chance that the message will be corrupted. An error-correcting code adds redundant information to the message which allows the receiver to detect and correct errors accrued during the transmission. We will study the famous Reed-Solomon code (found in QR codes, compact discs, deep space probes,. . . ) and investigate the limits of its error-correcting capacity. It can be shown that understanding this is related to understanding the “deep hole” problem, which is a question of determining when a received message has, in a sense, incurred the worst possible corruption. We partially resolve this in its traditional context, when the code is based on the finite field Fq or Fq, as well as new contexts, when it is based on a subgroup of Fq or the image of a Dickson polynomial. This is a new and important problem that could give insight on the true error-correcting potential of the Reed-Solomon code.