向量三差（Vector Trifference）
Vector Trifference
AI总结：
该研究提出向量三差码的几何推广概念，通过新方法证明其规模上界，且将方法适配到离散场景后得到Elias三差码上界的多项式改进，还推广了相关概念并证明向量版Fredman-Komlos定理。
摘要：
我们研究了三差（trifference）概念的几何推广，该概念由Elias于1988年在零错误信道容量研究中提出。在离散场景下，若码集C⊆{0,1,2}^n满足：对C中任意三个不同码字x、y、z，存在坐标i∈[n]使得x_i、y_i、z_i两两不同，则称C为三差码。确定这类码的最大规模仍是核心开放问题；通过简单剪枝论证证明的经典上界|C|≤2*(3/2)^n，长期以来未得到显著改进。受寻找新技术的动机驱动，同时契合其他经典组合概念的向量扩展思路，我们提出向量三差码的概念。考虑码集C⊆(S²)^n，其中字母表为单位球面S²={v∈ℝ³:||v||=1}。若对C中任意三个不同的x、y、z，存在索引i使得向量x_i、y_i、z_i两两正交，则称C为向量三差码。将向量问题直接约简到离散场景似乎不可行，导致难以复刻Elias的剪枝论证。尽管如此，我们提出了一种新方法，证明了上界|C|≤(√2+o(1))*(3/2)^n。有趣的是，将我们的方法适配回离散场景后，可得到Elias上界的多项式改进：|C|≲n^(-1/4)*(3/2)^n。这一改进源自一项与作者近期提出的方法相似但不完全相同的技术，不过该结果仍未达到此前得到的更优的n^(-2/5)因子。我们还将向量三差码的概念推广到更丰富的字母表，并证明了适用于一般k-分离码的Fredman-Komlos定理（1984）的向量版本。
英文摘要：
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.
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