arXiv subjects

Yuhao Zhao

Publications and source records attributed to Yuhao Zhao.

21 records · Page 2Linked to original sources

arXiv2024

Improved upper bounds for wide-sense frameproof codes

Frameproof codes have been extensively studied for many years due to their application in copyright protection and their connection to extremal set theory. In this paper, we investigate upper bounds on the cardinality of wide-sense $t$-frameproof codes. For $t=2$, we apply results from Sperner theory to give a better upper bound, which significantly improves a recent bound by Zhou and Zhou. For $t\geq 3$, we provide a general upper bound by establishing a relation between wide-sense frameproof codes and cover-free families. Finally, when the code length $n$ is at most $\frac{15+\sqrt{33}}{24}(t-1)^2$, we show that a wide-sense $t$-frameproof code has at most $n$ codewords, and the unique optimal code consists of all weight-one codewords. As byproducts, our results improve several best known results on binary $t$-frameproof codes.

math.CO↗

arXiv2023

On supersaturation for oddtown and eventown

We study the supersaturation problems of oddtown and eventown. Given a family $\mathcal A$ of subsets of an $n$ element set, let $op(\mathcal A)$ denote the number of distinct pairs $A,B\in \mathcal A$ for which $|A \cap B|$ is odd. We show that if $\mathcal A$ consists of $n+s$ odd-sized subsets, then $op(\mathcal A)\geq s+2$, which is tight when $s\le n-4$. This disproves a conjecture by O'Neill on the supersaturation problem of oddtown. For the supersaturation problem of eventown, we show that for large enough $n$, if $\mathcal A$ consists of $2^{\lfloor \frac n 2\rfloor}+s$ even-sized subsets, then $op(\mathcal A)\ge s\cdot2^{\lfloor \frac n 2\rfloor-1} $ for any positive integer $s\le \frac{2^{\lfloor\frac n 8\rfloor}} n$. This partially proves a conjecture by O'Neill on the supersaturation problem of eventown. Previously, the correctness of this conjecture was only verified for $s=1$ and $2$. We further provide a twice weaker lower bound in this conjecture for eventown, that is $op(\mathcal{A})\ge s\cdot 2^{\lfloor n/2\rfloor-2}$ for general $n$ and $s$ by using discrete Fourier analysis. Finally, some asymptotic results for the lower bounds of $op(\mathcal A)$ are given when $s$ is large for both problems.

math.CO↗

arXiv2019

Solvable lattice models for metals with Z2 topological order

We present quantum dimer models in two dimensions which realize metallic ground states with Z2 topological order. Our models are generalizations of a dimer model introduced in [PNAS 112,9552-9557 (2015)] to provide an effective description of unconventional metallic states in hole-doped Mott insulators. We construct exact ground state wave functions in a specific parameter regime and show that the ground state realizes a fractionalized Fermi liquid. Due to the presence of Z2 topological order the Luttinger count is modified and the volume enclosed by the Fermi surface is proportional to the density of doped holes away from half filling. We also comment on possible applications to magic-angle twisted bilayer graphene.

cond-mat.str-el↗