Configurations of Extremal Even Unimodular Lattices

Scott Duke Kominers

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Article | International Journal of Number Theory | May 2009

Configurations of Extremal Even Unimodular Lattices

by Scott Duke Kominers

Abstract

We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors.

Keywords: Mathematical Methods;

Format: Print

8 pages

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Citation:

Kominers, Scott Duke. "Configurations of Extremal Even Unimodular Lattices." International Journal of Number Theory 5, no. 3 (May 2009): 457–464.

About the Author

Scott Duke Kominers

MBA Class of 1960 Associate Professor of Business Administration

Entrepreneurial Management

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Article | Bloomberg Opinion

How to End the Plasma Shortage for Coronavirus Patients

Scott Duke Kominers

Those who have recovered from the virus will donate more blood if given the right incentives.

Citation:
Kominers, Scott Duke. "How to End the Plasma Shortage for Coronavirus Patients." Bloomberg Opinion (May 11, 2020). View Details

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Working Paper | HBS Working Paper Series | 2020

Paying It Backward and Forward: Expanding Access to Convalescent Plasma Therapy Through Market Design

Scott Duke Kominers, Parag A. Pathak, Tayfun Sönmez and M. Utku Ünver

COVID-19 convalescent plasma (CCP) therapy is currently a leading treatment for COVID-19. At present, there is a shortage of CCP relative to demand. We develop and analyze a model of centralized CCP allocation that incorporates both donation and distribution. In order to increase CCP supply, we introduce a mechanism that utilizes two incentive schemes, respectively based on principles of “paying it backward" and “paying it forward." Under the first scheme, CCP donors obtain treatment vouchers that can be transferred to patients of their choosing. Under the latter scheme, patients obtain priority for CCP therapy in exchange for a future pledge to donate CCP if possible. We show that in steady-state, both principles generally increase overall treatment rates for all patients—not just those who are voucher-prioritized or pledged to donate. Our results also hold under certain conditions if a fraction of CCP is reserved for patients who participate in clinical trials. Finally, we examine the implications of pooling blood types on the efficiency and equity of CCP distribution.

Citation:
Kominers, Scott Duke, Parag A. Pathak, Tayfun Sönmez, and M. Utku Ünver. "Paying It Backward and Forward: Expanding Access to Convalescent Plasma Therapy Through Market Design." Harvard Business School Working Paper, No. 20-116, May 2020. View Details

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Article | Journal de Théorie des Nombres de Bordeaux | 2019

Configurations of Extremal Type II Codes via Harmonic Weight Enumerators

Noam D. Elkies and Scott Duke Kominers

We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for n ∈{8,24,32,48,56,72,96} every extremal Type II code of length n is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along the way we introduce "t12-designs" as a discrete analog of Venkov's spherical designs of the same name.

Keywords: Mathematical Methods;

Citation:
Elkies, Noam D., and Scott Duke Kominers. "Configurations of Extremal Type II Codes via Harmonic Weight Enumerators." Journal de Théorie des Nombres de Bordeaux 31, no. 3 (2019): 679–688. View Details

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Configurations of Extremal Even Unimodular Lattices

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