Session S29 - Theory and Applications of Coding Theory

Wednesday, July 14, 15:30 ~ 15:55 UTC-3

Toric 3-fold codes and Minkowski length of lattice polytopes in $\mathbb{R}^3$

Jenya Soprunova

Kent State University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Fix a convex lattice polytope $P$ in $\mathbb{R}^n$, and define ${\mathcal L}_P$ to be the $\mathbb{F}_q$-vector space spanned by the monomials whose exponent vectors lie in $P$. The codewords of a toric code are obtained by evaluating polynomials in ${\mathcal L}_P$ at the points of the torus $(\mathbb{F}_q\setminus\{0\})^n$, taken in some fixed order. The question of computing or giving bounds on the minimum distance of toric codes has been studied by Hansen, Joyner, Little and Schenck, and others.

Our goal is to provide an estimate on the minimum distance of a toric code in terms of geometric invariants of $P$. In our earlier work, focusing on the case $n=2$, we provided such estimates involving a geometric invariant $L(P)$, the Minkowski length of $P$. In this talk I will concentrate on the case $n=3$ which involves much more challenging combinatorics as well as phenomena that do not arise in the case of toric surface codes.

Joint work with Kyle Meyer (UC San Diego) and Ivan Soprunov (Cleveland State University).

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