179 research outputs found
Lattice polytopes in coding theory
In this paper we discuss combinatorial questions about lattice polytopes
motivated by recent results on minimum distance estimation for toric codes. We
also prove a new inductive bound for the minimum distance of generalized toric
codes. As an application, we give new formulas for the minimum distance of
generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure
On nondegeneracy of curves
A curve is called nondegenerate if it can be modeled by a Laurent polynomial
that is nondegenerate with respect to its Newton polytope. We show that up to
genus 4, every curve is nondegenerate. We also prove that the locus of
nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is
min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional
The Small Stellated Dodecahedron Code and Friends
We explore a distance-3 homological CSS quantum code, namely the small
stellated dodecahedron code, for dense storage of quantum information and we
compare its performance with the distance-3 surface code. The data and ancilla
qubits of the small stellated dodecahedron code can be located on the edges
resp. vertices of a small stellated dodecahedron, making this code suitable for
3D connectivity. This code encodes 8 logical qubits into 30 physical qubits
(plus 22 ancilla qubits for parity check measurements) as compared to 1 logical
qubit into 9 physical qubits (plus 8 ancilla qubits) for the surface code. We
develop fault-tolerant parity check circuits and a decoder for this code,
allowing us to numerically assess the circuit-based pseudo-threshold.Comment: 19 pages, 14 figures, comments welcome! v2 includes updates which
conforms with the journal versio
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