# Combinatorics Seminar 2018

Date: Sun, Jun 17, 2018

Time: 10:00 - 12:00

Venue: Middle Meeting Room

Title: On the equitable partitions of $H(12,2)$ with quotient matrix $[[3,9],[7,5]]$

Abstract:

In 2007, D.G. Fon-Der-Flaass presented a construction of perfect colorings (equitable partitions) of the $12$-cube with parameters $[[3,9],[7,5]]$ and showed that these colorings lie on a correlation-immunity bound. However, the number of the equivalence classes of the constructed colorings, as well as the existence of other colorings with the same parameters, remained unknown. We finish the characterization of this class of coloring by analyzing the Fourier transform of a hypothetical coloring. The Fourier transform of such coloring is shown to correspond to a system of $63$ $4$-subsets of the $12$-set $\{1,...,12\}$ that cover all $3$-subsets. Such covering system is redundant (a minimum covering is known to have 57 quadruples); however, additional properties allow to characterize all possibilities.

This is based on joint work with K. Vorob’ev.

Slides: View slides

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