Polynomially Improved Lower Bounds for Trifferent Codes via Locally Sparse $3$-Uniform Hypergraphs

2026-07-28

AI Summary

This work investigates the maximum size $T(n)$ of trifferent codes in ternary alphabets, aiming to improve its lower bound. By refining the outer code in the Körner–Marton concatenated construction and modeling non-separating triples as edges of a 3-uniform hypergraph, the authors employ a combination of random vertex sparsification, removal of high-degree vertices, and elimination of Berge cycles to extract a large independent set. Integrating the Verstraete–Wilson independence number theorem with a Tetra code concatenation scheme, they achieve—for the first time—a polynomial gain of $\sqrt{n}$ over the classical exponential lower bound, establishing $T(n) \geq c\sqrt{n}\,(9/5)^{n/4}$. This result significantly improves upon the previous best-known bound of $c_0(9/5)^{n/4}$.

Abstract

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.

Problem

• trifferent codes
• lower bounds
• 3-uniform hypergraphs
• polynomial improvement
• coding theory

Innovation

• trifferent codes
• 3-uniform hypergraphs
• locally sparse
• independent set
• concatenated coding