Vector Trifference
Siddharth Bhandari, Abhishek Khetan
TL;DR
This work introduces vector trifferent codes as a geometric analogue of trifferent codes, placing the alphabet on the sphere $S^2$ and requiring that three codewords have mutually orthogonal coordinates in some position. The authors prove an upper bound $|\mathcal{C}| \le (\sqrt{2}+o(1))(\tfrac{3}{2})^n$ via a tensor-product subspace pairing argument, and show that the method yields a polynomial improvement $|\mathcal{C}| \lesssim n^{-1/4}(\tfrac{3}{2})^n$ in the discrete setting. They extend the technique to vector $k$-separating codes, obtaining asymptotic bounds $\log_2 |\mathcal{C}|/n \le \frac{k!}{k^{k-1}}+o(1)$, and discuss implications for improving Elias-type bounds and for broader alphabet generalizations.
Abstract
We investigate a geometric generalization of trifference, a concept introduced by Elias in 1988 in the study of zero-error channel capacity. In the discrete setting, a code C \subseteq {0,1,2}^n is trifferent if for any three distinct codewords x, y, z in C, there exists a coordinate i in [n] where x_i, y_i, z_i are all distinct. Determining the maximum size of such codes remains a central open problem; the classical upper bound |C| \leq 2 * (3/2)^n, proved via a simple pruning argument, has resisted significant improvement. Motivated by the search for new techniques, and in line with vectorial extensions of other classical combinatorial notions, we introduce the concept of vector trifferent codes. Consider C \subseteq (S^2)^n, where the alphabet is the unit sphere S^2 = { v in R^3 : ||v|| = 1 }. We say C is vector trifferent if for any three distinct x, y, z in C, there is an index i where the vectors x_i, y_i, z_i are mutually orthogonal. A direct reduction of the vectorial problem to the discrete setting appears infeasible, making it difficult to replicate Elias's pruning argument. Nevertheless, we develop a new method to establish the upper bound |C| \leq (sqrt(2) + o(1)) * (3/2)^n. Interestingly, our approach, when adapted back to the discrete setting, yields a polynomial improvement to Elias's bound: |C| \lesssim n^(-1/4) * (3/2)^n. This improvement arises from a technique that parallels, but is not identical to, a recent method of the authors, though it still falls short of the sharper n^(-2/5) factor obtained there. We also generalize the concept of vector trifferent codes to richer alphabets and prove a vectorial version of the Fredman-Komlos theorem (1984) for general k-separating codes.