Introduction to coding theoryCoding 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. ResearchRelevant links
My researchMy research primarily involves the Hermitian curve \[ x^{q+1} = y^q + y \] which is the unique maximal curve over \(\mathbb{F}_{q^2}\). A natural extension of the Hermitian curve are the normtrace curves. These are defined over \(\mathbb{F}_{q^r}\) by \[ x^{\frac{q^r1}{q1}} = y^{q^{r1}} + … + y^q + y \] we can see that the Hermitian curve is recovered when \(r=2\). The contexts where these curves appear are detailed below.
General ThoughtsAnother topic that catches my attention is the presence of the field trace \[ \text{tr}_{\mathbb{F}_{q^\ell} / \mathbb{F}_q}(\alpha) = \underset{j=1}{\overset{\ell1}{\sum}} \alpha^{q^j} \] 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 lessthanclassical amounts of information? Government contractingJohns Hopkins Applied Physics Laboratory
