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Unique indecomposable basis conjecture for linear subspace codes

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Let U\mathcal{U} be a linear code in the projective space Pq(n)\mathbb{P}_q(n). An indecomposable basis is a basis of the vector space over F2\mathbb{F}_2 formed by the code whose elements are indecomposable codewords. The code U\mathcal{U} is closed under intersection when U∩V∈UU\cap V\in\mathcal{U} for all U,V∈UU,V\in\mathcal{U}. Unique indecomposable basis conjecture. A linear code U\mathcal{U} in Pq(n)\mathbb{P}_q(n) has a unique indecomposable basis if and only if U\mathcal{U} is closed under intersection.

The conjecture concerns the relationship between the internal basis structure of a linear subspace code and closure under the meet operation of the projective lattice. The source presents it as an open question, motivated by examples of codes with more indecomposable codewords than their dimension.

References

Primary source

Pranab Basu and Navin Kashyap, “The Lattice Structure of Linear Subspace Codes”, arXiv:1911.00721 (2019).

coding-theoryfinite-geometrysubspace-codes
Posted

Progress summary

Refreshed
Open

No public source reports a proof or counterexample, so the conjecture remains open.

The conjecture, formulated in the November 2, 2019 paper The Lattice Structure of Linear Subspace Codes, asserts that uniqueness of an indecomposable basis is equivalent to closure under intersection. The paper presents this as an open question.

Known results

  • Intersection-closed codes have a unique decomposition into indecomposable codewords, which form a basis.
  • Non-intersection-closed examples can have up to 2n−12^n-1 indecomposable codewords and may lack uniqueness.
  • The 2020 dissertation On Linear Codes in Projective Spaces records the same conjecture without proving the converse or giving a counterexample.

Current status (as of September 2026): The intersection-closed case is settled, but the converse and hence the full unique indecomposable basis conjecture remain open; no public proof, counterexample, or claimed resolution was found.

Sources

Solutions 0

No solutions have been posted yet.