Every summer, many of our students participate in various summer research programs. Students work as full participants in the processes of creating a research plan, executing a research project, and preparing results for presentation in a public forum. Learn more about the research done by your fellow mathematics and statistics students.
This week's panelists include:
Khanh Mai ‘28 worked as part of the Kenyon Summer Science Scholars this year. As a generalization of cyclic and constacyclic codes, polycyclic codes offer a promising framework for constructing codes with good properties. Since the dual of a polycyclic code is not necessarily polycyclic, characterizing their dual presents theoretical challenges. In this research, he investigated polycyclic codes associated with trinomials and quadrinomials. He established that all dual-containing polycyclic codes are, in fact, constacyclic. Furthermore, he proved several duality results for polycyclic codes associated with trinomials and quadrinomials, and presented many examples of polycyclic codes with optimum or best-known parameters. Finally, he resolved an open conjecture from a recent manuscript regarding binary polycyclic codes associated with trinomials.
Lucas Waite ‘28 also worked as part of the Kenyon Summer Science Scholars over the summer. Recent progress has produced the first exponential improvement in nearly fifty years to the general asymptotic upper bounds for binary codes. By contrast, the corresponding lower-bound problem remains open. Lucas studied this problem, the Gilbert-Varshamov conjecture, through the lens of graph theory by defining the proximity graph of a metric space, in which codes correspond to independent sets and the Gilbert-Varshamov bound arises from the classical degree bound on independence. Lucas thus investigated whether fixed subgraph counts in the Hamming proximity graph can force independent sets exponentially larger than those guaranteed by Gilbert-Varshamov.
Join us on Tuesday, Sept. 22, at 11:10 am in Hayes Hall 109 (please note the date and time change) to hear these exciting presentations and perhaps learn how you too can get involved in summer research programs. We hope to see you there!