Motivated by applications to noncoherent network coding, we study subspace codes defined by sets of linear cellular automata (CA). As a first remark, we show that a family of linear CA where the local rules have the same diameter -- and thus the associated polynomials have the same degree -- induces a Grassmannian code. Then, we prove that the minimum distance of such a code is determined by the maximum degree occurring among the pairwise greatest common divisors (GCD) of the polynomials in the family. Finally, we consider the setting where all such polynomials have the same GCD, and determine the cardinality of the corresponding Grassmannian code. As a particular case, we show that if all polynomials in the family are pairwise coprime, the resulting Grassmannian code has the highest minimum distance possible.

翻译：受非相干网络编码应用的启发，我们研究由线性细胞自动机（CA）集合定义的子空间码。首先，我们指出，一组局部规则具有相同直径（从而关联多项式具有相同次数）的线性CA族会诱导出一个格拉斯曼码。接着，我们证明此类码的最小距离由该族多项式中两两最大公因数（GCD）中出现的最大次数决定。最后，我们考虑所有此类多项式具有相同GCD的情形，并确定相应格拉斯曼码的基数。作为特例，我们证明：若该族中所有多项式两两互质，则所得格拉斯曼码具有可能的最大最小距离。