We present a general theory to obtain good linear network codes utilizing the osculating nature of algebraic varieties. In particular, we obtain from the osculating spaces of Veronese varieties explicit families of equidimensional vector spaces, in which any pair of distinct vector spaces intersect in the same dimension. Linear network coding transmits information in terms of a basis of a vector space and the information is received as a basis of a possible altered vector space. Ralf Koetter and Frank R. Kschischang [KK08] introduced a metric on the set of vector spaces and showed that a minimal distance decoder for this metric achieves correct decoding if the dimension of the intersection of the transmitted and received vector space is sufficiently large. The proposed osculating spaces of Veronese varieties are equidistant in the above metric. The parameters of the resulting linear network codes are determined.
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机译：我们提出了一个一般的理论，以利用代数变种的密合性来获得良好的线性网络代码。特别是，我们从Veronese变种的密合空间中获得了等维矢量空间的显式族，其中任意一对不同的矢量空间在同一维上相交。线性网络编码根据向量空间来传输信息，并且该信息被接收为可能的改变后的向量空间的基础。 Ralf Koetter和Frank R. Kschischang [KK08]在向量空间集合上引入了一个度量，并表明，如果所发送和接收的向量空间的交点的维数足够大，则用于该度量的最小距离解码器可以实现正确的解码。拟议的Veronese变种的密合空间在上述度量标准上是等距的。确定所得线性网络代码的参数。
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