AI 中文总结
该研究针对trifferent码，改进经典Körner--Marton构造，通过局部稀疏3-均匀超图方法将其下界提升至多项式因子√n，结合Tetra码级联得到更优下界。
AI 中文摘要
三元码若每三个不同码字都存在一个坐标使得它们的符号两两不同，则称为trifferent码。设T(n)为长度n的trifferent码的最大规模，经典Körner--Marton构造给出T(n)≥c₀(9/5)^(n/4)（c₀>0为绝对常数）。我们证明多项式改进的下界T(n)≥c√n(9/5)^(n/4)（c>0为绝对常数）。我们的证明改进了Körner--Marton级联的外码步骤：将不可区分三元组编码为3-均匀超图的边，随机稀疏其顶点集，删除高度顶点及所有剩余的长度为2和3的Berge环；所得局部稀疏超图可通过Verstraete和Wilson的定理得到大独立集，从而产生额外因子√n，再与长度为4的Tetra码级联即得到所述下界。
英文摘要
A ternary code is \emph{trifferent} if every three distinct codewords have a coordinate in which their symbols are pairwise distinct. Let $T(n)$ be the maximum size of a trifferent code of length $n$. The classical Körner--Marton construction gives $T(n)\ge c_0(9/5)^{n/4}$ for an absolute constant $c_0>0$. We prove the polynomial strengthening $T(n)\ge c\sqrt{n}(9/5)^{n/4}$ for an absolute constant $c>0$. Our proof refines the outer-code step in the Körner--Marton concatenation. We encode non separating triples as edges of a $3$-uniform hypergraph, randomly thin its vertex set, and remove high-degree vertices together with all remaining Berge cycles of lengths two and three. The resulting locally sparse hypergraph admits a large independent set by a theorem of Verstraete and Wilson, producing the additional factor $\sqrt n$. Concatenation with the length-four Tetra code then yields the stated lower bound.
CommentsAdded a brief discussion outlining a viable extension of the framework to generalized m-trifferent codes
