论文标题
有限字段上的帧：正交几何形状中的等缘线
Frames over finite fields: Equiangular lines in orthogonal geometry
论文作者
论文摘要
我们研究了有限正交几何形状中的等缘线，专门针对等缘紧密框架（ETF）。与满足某些参数约束的强大ETF和强级规则图（SRG）之间的已知对应关系，我们证明有限正交几何形状中的ETF与SRG的模块化概括密切相同。我们有限字段设置中的约束较弱，除$ v \ leq 1300 $ Vertices上的所有已知的SRG参数外，所有限制都至少满足其中一个。将我们的结果应用于三角形图，我们推断出Gerzon的结合是在无限多个维度的有限正交几何形状中获得的。我们还展示了与实际ETF的连接，并得出了有限正交几何形状中ETF的必要条件。作为一种应用，我们表明Gerzon的界限无法在有限的正交几何形状〜5中获得。
We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5.
