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.
My Research
Current Work
Dissertation link: Codes from norm-trace curves: local recovery and fractional decoding.
My current 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 norm-trace curves. These are defined over \(\mathbb{F}_{q^r}\) by
\[ x^{\frac{q^r-1}{q-1}} = y^{q^{r-1}} + … + y^q + y \]
we can see that the Hermitian curve is recovered when \(r=2\). The contexts where these curves appear are detailed below.
The Hermitian-lifted codes of López, Malmskog, Matthews, Piñero-González, and Wootters are based on the Hermitian curve over \(\mathbb{F}_{q^2}\).
The more general norm-trace curves can be used in this construction. Norm-trace-lifted codes over binary fields compare well to the Hermitian-lifted codes, having better asymptotic rate, and an explicit basis for the set of evaluation functions (ISIT 2022, and Ph.D. Dissertation, 2022)
The fractional decoding algorithm of Reed Solomon codes given by Santos draws connections between collaborative decoding of Reed Solomon codes and fractional decoding. In conjunction with Matthews and Santos, I look at how this connection yields a procedure for fractional decoding of codes from the Hermitian curve (ISIT, July 2021).
The decoding procedure given for codes from the Hermitian curve doesn't utilize the full structure of the Hermitian curve. The procedures can be greatly improved, and allow for extension to another class of codes, the Hermitian-lifted and norm-trace-lifted codes (Ph.D. Dissertation, 2022).
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?