Binary matroid representation of cyclic families

Definitions and mathematical statement

Fix integers $n\ge1$ and $1\le k\le n$, and a $k$-element subset $B\subseteq\mathbb Z/n\mathbb Z$. For $i\in\mathbb Z/n\mathbb Z$, put $B+i=\{b+i:b\in B\}$, with addition modulo $n$. A binary matroid on this labelled ground set is specified by a matrix over $\mathbb F_2$: a subset is independent precisely when its columns are linearly independent. A basis is a maximal independent set.

The conjecture asks whether, for every choice of $n,k,B$, there exists a binary matroid of rank $k$ for which all the translates $B+i$ are bases. Equivalently, can one choose column vectors $a_0,\ldots,a_{n-1}\in\mathbb F_2^k$ so that \[ \det[a_j]_{j\in B+i}=1 \quad\text{for every }i\in\mathbb Z/n\mathbb Z? \] Column order is immaterial to nonsingularity; over $\mathbb F_2$ every nonzero determinant equals one.

The representing matrix may depend on $B$. The requirement does not say that these translates are the only bases, or that cyclic translation must preserve every independent set of the resulting matroid. Different indices may give the same translate, imposing the same constraint twice. The field is fixed to two elements: a construction over a sufficiently large field does not establish the asserted binary representation.

Short English statement

Can every cyclic family of translates of one fixed subset be simultaneously realized as bases of a binary matroid?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000007 (AMR-029-0007), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Egres Open - List of open problems, Open-problem page: Binary matroid representation of cyclic families. Original source indexed by AMR-029-0007. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Binary matroid representation of cyclic families. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Binary_matroid_representation_of_cyclic_families.