Abstract

We introduce a new method for the study of linear codes, termed subspace decomposition, which provides new insights into code structure and performance. Briefly, subspace decomposition comprises the evaluation of complex properties of a linear code by evaluating simple properties of its constituent subspaces. We find that this technique has application to a number of problems in coding theory. When applied to the problem of evaluating the performance of secrecy codes, subspace decomposition provides an efficient method for computing the equivocation of coset codes over the binary erasure wiretap channel. It is also especially efficient in computing the chi squared divergence, a metric which relates to an eavesdropper's maximum likelihood decoding performance. We use subspace decomposition to prove theoretical results about certain classes of codes, including a new code construction termed subspace exclusion codes, which are optimal in a number of situations. We also employ subspace decomposition to the problem of finding good coset codes of any size via gradient descent. As part of this effort, we develop specialized methods to produce a discrete code from a continuous code definition vector. These methods result in secrecy codes superior to those produced by other known methods. The results of these efforts also indicate remarkably poor performance for secrecy codes based on duals of low-density parity check codes. We also present a method for efficiently determining whether different codes are equivalent based on the properties of their subspaces. A wide variety of applications for subspace decomposition are also identified for future research.

Degree

PhD

College and Department

Ira A. Fulton College of Engineering; Electrical and Computer Engineering

Rights

https://lib.byu.edu/about/copyright/