On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors

We consider the construction of maximal families of polynomials over the finite field Fq, 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 d 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 F₂. Our findings prompt several more open questions, which we plan to address in an extended version of this work.