Improving the Trifference Lower Bound via Non-Perfect Hashing
Chen Yuan, Ruiqi Zhu
原文摘要
We strictly improve the Körner-Marton lower bound $\frac14\log \frac{9}{5}$ on the asymptotic rate of trifferent codes. We obtain this improvement by constructing non-perfect $(9,3)$-hash codes with rate above $\log(9/5)$ and then applying the Tetra-code construction, which gives $R_3\ge\frac14R_{9,3}$. More generally, for every $q\ge4$, we prove $R_{q,3}>B_{q,3}+\frac1{250000}$. We also extend the method to $(p,2,3)$-list recovery. For every $q\ge4$ and $0\le p<\frac{q-2}{12(q-1)}$, we prove $R_q^{\mathrm{LR}}(p)>B_q^{\mathrm{LR}}(p)$. Here $B_{q,3}$ and $B_q^{\mathrm{LR}}(p)$ denote the respective probabilistic lower bounds. For every fixed $0\le p<1/18$, the rate improvement is bounded below by a positive constant independent of $q\ge4$. For every fixed $0\le p<1/12$, we also obtain a positive rate improvement independent of $q$ for all sufficiently large $q$.
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