多线性形式凯莱图与张量码的特征值方法
The multilinear forms Cayley graph and the eigenvalue method for tensor codes
AI总结：
研究将图论与编码理论的框架推广到有限域上张量空间，以张量秩为度量。通过分析秩一张量生成的凯莱图的谱，得到递归表达式并应用于推导张量码维度的界。
AI中文摘要：
Delsarte为汉明度量和秩度量码建立了图论（更一般地，关联方案）与编码理论之间的联系。汉明度量码和秩度量码的环境度量空间可视为由权重为1的字生成的凯莱图。我们专注于将此框架推广到有限域上的张量空间，以张量秩为度量。该空间对应于由秩一张量生成的凯莱图，对于至少3阶的张量，它不是距离正则的。我们表明该图的谱有一个递归表达式，并且取决于足够大维度的张量子空间与塞格雷簇之间的可能交集。3阶张量的该图的谱可以用由这些张量生成的秩度量码的秩分布来表示。特别地，我们得到了任意有限域上2x3x3张量的图的完整谱。我们应用此结果使用特征值方法推导张量秩度量中张量码维度的界，特别是比率型界。
英文摘要：
The connections between graph theory, and more generally association schemes, and coding theory were established by Delsarte for the Hamming metric and rank-metric codes. The ambient metric space of Hamming-metric codes and rank-metric codes can be seen as Cayley graphs generated by words of weight one. The metrics considered then coincide with the geodesic distances of these distance-regular graphs. We focus on a generalisation of this framework to the space of tensors over a finite field, endowed with the tensor-rank as a metric. This space corresponds to the Cayley graph generated by rank-one tensors, which is not distance-regular for tensors of order at least 3. We show that the spectrum of this graph has a recursive expression and depends on the possible intersections between tensor subspaces of large enough dimension and the Segre variety. The spectrum of this graph for 3-order tensors can be expressed with the rank distribution of the rank-metric codes generated by these tensors. In particular, we obtain the complete spectrum of the graph for 2x3x3 tensors over any finite field. We apply this result to derive bounds on the dimension of tensor codes in the tensor-rank metric using the eigenvalue method, and in particular the ratio-type bound.
