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Build a Sporadic Group in Your Basement

by Paul E. Becker, Martin Derka, Sheridan Houghten, and Jennifer Ulrich

Year of Award: 2018

Publication Information: The American Mathematical Monthly, Volume 124, Number 4, April 2017, Pages 291-305.

Summary: Most finite simple groups fall into a few easy to understand categories, but there are a few sporadic (some might say "freakish") examples that defy easy classification. The Mathieu groups are the most accessible and applicable among these sporadic groups. With a lively and informative discussion of error-correcting codes, this article describes how the Mathieu groups connect to the extended Golay code. The authors give their readers a clear path of how different models of the Golay code connect together and give a way to use this to build the Mathieu groups in a simple and beautiful representation. (Note that no basements were harmed in the writing or reading of this paper.)

Response from the Authors:

It is a great honor to receive the MAA Paul R. Halmos-Lester R. Ford Award. We are very grateful for this recognition by the MAA. We are particularly pleased our paper was selected from the pages of American Mathematical Monthly, which consistently produces quality expository articles. We would like to thank the editor and referees, whose suggestions significantly improved the paper.

This article is a summary of a ten-year conversation between mathematicians and computer scientists. We set out to explore a specific question from coding theory. Our collaboration resulted in several narrowly-focused papers; this Monthly article is the other stuff. It is composed of ideas traded back-and-forth, translated through different viewpoints, and flavored by experiences along the way. Eventually, these different viewpoints became the most interesting aspect of our work.

We hope our paper encourages young mathematicians and computer scientists not only to develop a love for their respective fields, but also to pursue interdisciplinary problems. For inspiration, they could consider the question that led to our work: the (possible) existence of the length-72 extremal code. If such a code existed, it would be the third in an interesting sequence that starts with the extended Golay code.

About the Authors:

Paul E. Becker received his MS from Michigan State University and PhD in mathematics from Central Michigan University. He is an associate professor of mathematics at Penn State Behrend. In his spare time, he kayaks and operates a family blueberry farm.

Martin Derka likes combining academic research with industry-oriented projects. He is a graduate of Masaryk University in the Czech Republic, Brock University (MSc, under the supervision of Sheridan Houghten), and University of Waterloo (PhD in computer science, 2018). His industrial experience includes both the start-up scene and large corporations. He has interned as a software engineer at Google and is co-founder and CTO of Car Media 2.0 (formerly Car Pics 2.0). In his free time, Martin enjoys traveling, outdoors, all kinds of sports, and rock-metal music.

Sheridan Houghten received her PhD degree in computer science from Concordia University, Montreal. She is a professor of computer science at Brock University. Her research interests encompass bioinformatics, computational intelligence, coding theory, and combinatorial optimization.

Jennifer Ulrich received her MS degree in mathematics from Texas A&M University. She holds a BS degree from Penn State Behrend, where she is a lecturer in mathematics. In her free time, she enjoys knitting, reading, television, and being the chauffeur for her three children.