We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$. The motivation for this problem lies in a recent construction for subspace codes based on cellular automata. More precisely, the minimum distance of such subspace codes relates to the maximum degree $d$ of the pairwise GCD in this family of polynomials. Hence, characterizing the maximal families of such polynomials is equivalent to determining the maximum cardinality of the corresponding subspace codes for a given minimum distance. We first show a lower bound on the cardinality of such families, and then focus on the specific case where $d=1$. There, we characterize the maximal families of polynomials over the binary field $\mathbb{F}_2$. Our findings prompt several more open questions, which we plan to address in an extended version of this work.

翻译：我们考虑有限域 $\mathbb{F}_q$ 上多项式的最大族构造，这些多项式具有相同的次数 $n$ 和非零常数项，且任意两个多项式的最大公因式次数为 $d$，其中 $1 \le d\le n$。该问题的动机源于最近基于元胞自动机的子空间码构造。更准确地说，此类子空间码的最小距离与族中多项式两两最大公因式的最大次数 $d$ 相关。因此，刻画此类多项式的最大族等价于在给定最小距离下确定对应子空间码的最大基数。我们首先给出此类族基数的下界，然后重点关注 $d=1$ 的特例情况。在此情况下，我们刻画了二元域 $\mathbb{F}_2$ 上多项式的最大族。我们的发现引出了多个待解决的新问题，计划在本文的扩展版本中加以探讨。