Introduction to coding theory

Coding theory is the study of error correcting codes, the purpose of which are to add redundancy into data to increase robustness against errors. These codes are applied almost anywhere data is stored or transmitted, such as within CDs or on the servers used by Google / Facebook / Netflix; even QR codes are an example of an error correcting code.

Algebraic codes in particular oftentimes have good parameters. In the 1960s, Reed and Solomon proposed codes over general finite fields \(\mathbb{F}_q := \mathbb{F}_{p^n}\), where \(p\) is prime. Though this was thought at the time to be of little interest because the codes are nonbinary, they later found their first application on board the Voyagers 1 and 2, and were crucial to the formation of CD technology.

Research

Relevant links

• Research Statement

• Dissertation Link

My research

The information on my published works can be found in the papers themselves. The information on projects I would pursue while involving undergraduate students at a university can be found in my teaching statement. On the remainder of this page, I detail the work I am actively pursuing.

First, I am interested in the classification of basis monomials for Hermitian-lifted codes. These are codes which use the Hermitian curve

\[ x^{q+1} = y^q + y \]

which is the unique maximal curve over \(\mathbb{F}_{q^2}\), to construct codes with nonzero asymptotic rate. The computational examples in the original Hermitian-lifted codes paper suggest an internal structure that might be justified theoretically. This has the potential to not only provide a better lower bound on the asymptotic rate of these codes, but also to help prepare these codes for application by giving explicit formulation for the monomials.

Second, I am interested in properties of the asymptotic normalized rate function \(\alpha_q(\delta)\), where \(\delta\) is the relative minimum distance. Because the inquiry into this function has been primarily on determining upper and lower bounds, there are only a handful of properties about this function which are known, including continuity and the formulation of the tangent lines at the endpoints. However, there are many properties which are not known, such as differentiability, convexity, and even any nontrivial values of the function; it is these which I hope to uncover by reconsidering older work on this problem.

General Thoughts

Another topic that catches my attention is the presence of the field trace

\[ \text{tr}_{\mathbb{F}_{q^\ell} / \mathbb{F}_q}(\alpha) = \underset{j=1}{\overset{\ell-1}{\sum}} \alpha^{q^j} \]

both in the repair scheme of Guruswami and Wootters and fractional decoding algorithms, where each exploit the fact that \(\text{tr}_{\mathbb{F}_{q^\ell} / \mathbb{F}_q}(\alpha) \in \mathbb{F}_q\) for any \(\alpha \in \mathbb{F}_{q^\ell}\). In what other instances can either this property, or similar properties, be utilized to decode with less-than-classical amounts of information?