Col Rank gen Matrix eq dim of MDS
CoreResults.colRank_genMatrix_eq_dim_of_MDS
Plain-language statement
A linear code LC of length ι and dimension dim over a field F is MDS if any dim columns of the generator matrix whose rows are an F-basis of LC are linearly independent.[GRS25] Equivalently, a linear code is MDS if and only if its generator matrix is MDS. Note: the hypothesis 0 < dim LC is necessary because for a trivial code (dim = 0),...
Exact Lean statement
lemma colRank_genMatrix_eq_dim_of_MDS [Field F] [DecidableEq F]
(LC : LinearCode (Fin n) F) (h_pos : 0 < dim LC) :
LC.IsMDS ↔ Matrix.IsMDS (matrixFromBasis LC)Formal artifact
Lean source
lemma colRank_genMatrix_eq_dim_of_MDS [Field F] [DecidableEq F] (LC : LinearCode (Fin n) F) (h_pos : 0 < dim LC) : LC.IsMDS ↔ Matrix.IsMDS (matrixFromBasis LC) := by set G := matrixFromBasis LC with hG have hLC : LC = fromRowGenMat G := eq_fromRowGenMat_matrixFromBasis LC have hkn : dim LC ≤ n := by have := Submodule.finrank_le (R := F) (M := Fin n → F) LC simp only [Module.finrank_fintype_fun_eq_card, Fintype.card_fin, dim, ModuleCode, ge_iff_le] at this ⊢; exact this have hrank : G.rank = dim LC := by have := rank_genMatrix_eq_dim LC rw [Matrix.rank_eq_rowRank (U := G)] at this ⊢ exact this.symm constructor · intro h rw [hLC] at h exact matrix_IsMDS_of_IsMDS h hrank · intro hMDS rw [hLC] exact IsMDS_of_matrix_IsMDS hMDS hkn h_pos- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/Basic/MDSCode.lean:166-185
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Project documentation
This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...
Source project: ArkLib
Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.
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ArkLib.Lattices.Ajtai.gadgetDecompose_lawful
Plain-language statement
The base-b gadget decomposition is a lawful gadget decomposition.
Source project: ArkLib
Person-level attribution pending.