环几何、双权码与强正则图
Ring geometries, Two-Weight Codes and Strongly Regular Graphs
摘要：
已知有限域$\F_q$上的线性二权码$C$既对应于$\F_q$上的射影空间中的满足$a$或$b$点中的每个超平面的多集，对于某些整数$a<b$，又对应于强正则图，其顶点可以用$C$的码字标识。在这里，我们将这个经典的结果的情况下，一个环线性码正好有两个非零齐次重量和多个点集在一个相关的射影环几何。我们将证明有限弗罗贝尼乌斯环上的二权码产生强正则图，并利用环几何给出二权码的一些构造。这些例子都产生了具有非平凡参数的强正则图的无限族。
英文摘要：
It is known that a linear two-weight code $C$ over a finite field $\F_q$ corresponds both to a multiset in a projective space over $\F_q$ that meets every hyperplane in either $a$ or $b$ points for some integers $a<b$, and to a strongly regular graph whose vertices may be identified with the codewords of $C$. Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homogeneous weights and multisets of points in an associated projective ring geometry. We will show that a two-weight code over a finite Frobenius ring gives rise to a strongly regular graph, and we will give some constructions of two-weight codes using ring geometries. These examples all yield infinite families of strongly regular graphs with non-trivial parameters.
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