On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors
<|reference_start|>On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors: 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.<|reference_end|>
@article{gadouleau2024on,
    title={On Maximal Families of Binary Polynomials with Pairwise Linear Common
  Factors},
    author={Maximilien Gadouleau, Luca Mariot, Federico Mazzone},
    journal={arXiv preprint arXiv:2405.08741},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.08741},
    primaryClass={cs.DM cs.CR math.CO}
}