bibkey: basu2019latticesubspacecodes authors: Pranab Basu, Navin Kashyap year: 2019 title: The Lattice Structure of Linear Subspace Codes doi: null url: https://arxiv.org/abs/1911.00721v1 claim: “A linear code 𝒰 in ℙ_q(n) has a unique indecomposable basis if and only if 𝒰 is closed under intersection.” strata_touched:

• D5/S3/Combinatorics/SubspaceCodes/BasuKashyapUniqueIndecomposableBasisRefutation license: citation-only triage: anchor

Basu and Kashyap, linear subspace codes

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DOI: null

Source: https://arxiv.org/abs/1911.00721v1

The arXiv v1 PDF has 24 pages. It uses consecutive numbering for definitions and propositions; the TeX labels below identify the statements independently of numbering.

• Page 2: the subspace distance is d_S(X,Y) = dim X + dim Y − 2 dim(X ∩ Y).
• Pages 4–5, Definition 1, TeX label L: “A subset 𝒰 ⊆ ℙ_q(n), with {0} ∈ 𝒰, is a linear subspace code if there exists a function ⊞ : 𝒰 × 𝒰 → 𝒰 such that: (i) (𝒰, ⊞) is an abelian group; (ii) the identity element of (𝒰, ⊞) is {0}; (iii) X ⊞ X = {0} for every group element X ∈ 𝒰; (iv) the addition operation ⊞ is isometric, i.e., d_S(X ⊞ Y₁, X ⊞ Y₂) = d_S(Y₁, Y₂) for all X, Y₁, Y₂ ∈ 𝒰.”
• Page 15, Definition 9, TeX label 8: “A linear code 𝒰 ⊆ ℙ_q(n) with the property that X ∩ Y ∈ 𝒰 whenever X, Y ∈ 𝒰 is said to be a linear code closed under intersection.”
• Page 19, Definition 11, TeX label 9: “A codeword Y ≠ {0} of a linear subspace code 𝒰 is said to be indecomposable if Y cannot be expressed as Y = Y₁ ⊞ Y₂ for any Y₁,Y₂ ∈ 𝒰 with dim Y₁, dim Y₂ < dim Y.”
• Page 21, Remark 5, TeX label R: “The set of indecomposable codewords in a linear subspace code closed under intersection is linearly independent with respect to the linear addition over 𝔽₂. This together with Proposition 17 imply that the indecomposable codewords are a basis for the vector space over 𝔽₂ formed by the linear code.”
• Page 22, Section 6: “We observed earlier that the indecomposable codewords in a linear subspace code constitute a basis for the vector space over 𝔽₂ formed by the code (Remark 5). We refer to such a basis as an indecomposable basis. A linear code closed under intersection has a unique indecomposable basis.”
• Page 23, Section 6, Conjecture 6.1, TeX label UIB: “A linear code 𝒰 in ℙ_q(n) has a unique indecomposable basis if and only if 𝒰 is closed under intersection.”

Encoding and scope

The field is any finite F : Type; the ambient space is Fin n → F. Codewords are Submodule F (Fin n → F). Intersection is ⊓, the zero subspace is ⊥, and dimension is Module.finrank F. The distance uses the displayed integer expression literally. A code records its subset and operation on that subset, together with all four axioms. The basis is an unordered finite subset of codewords: each codeword has exactly one representation as a finite subset sum under the code operation. This is a basis of the code’s 𝔽₂-space, rather than a basis of the ambient F-space.

The converse fails for every finite field. Split the coordinates into blocks I,P,Q of dimensions i,a,b, with 0 < i < a and i < b. Put A = I ⊕ P, B = I ⊕ Q, C = P ⊕ Q and use Klein addition on {⊥,A,B,C}. Exactly A,B are indecomposable, and they form its unique indecomposable basis; nevertheless A ⊓ B = I is outside the code. The settling module proves this family and negates the universal conjecture with F = ZMod 2 and (i,a,b) = (1,2,2).

The source’s intersection-closed direction and the qualified Remark 5 remain intact. Section 6’s recap omits Remark 5’s intersection-closed qualifier. The Braun–Etzion–Vardy cardinality conjecture and the source’s results proved under intersection closure are not refuted by this construction.