Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding

2025/06/26 by Ilaria Cardinali, Cardinali, Ilaria, Luca Giuzzi +1 · 2 citations
Computer Science · Engineering · #14M12 #51E22 #94B05 #Advanced Wireless Communication Techniques #Coding theory and cryptography #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2506.21309

openalex publication_date 2025/06/26 · openalex created_date 2025/10/15 · openalex updated_date 2026/08/03

Abstract

Let V be a vector space over the finite field \mathbbF_q with q elements and Λ be the image of the Segre geometry PG(V)\otimesPG(V^*) in PG(V⊗ V^*). Consider the subvariety Λ₁ of Λ represented by the pure tensors x⊗ ξ with x∈ V and ξ∈ V^* such that ξ(x)=0. Regarding Λ₁ as a projective system of PG(V⊗ V^*), we study the linear code C(Λ₁) arising from it. The code C(Λ₁) is minimal code and we determine its basic parameters, itsfull weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.

Citations

• On minimal codes arising from projective embeddings of point-line geometries
• Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding
• Strong blocking sets and minimal codes from expander graphs
• Three Combinatorial Perspectives on Minimal Codes
• A geometric characterization of minimal codes and their asymptotic\n performance
• Full Characterization of Minimal Linear Codes as Cutting Blocking Sets
• Minimal linear codes arising from blocking sets
• Line Hermitian Grassmann Codes and their Parameters
• Line Polar Grassmann Codes of Orthogonal Type
• Automorphism groups of Grassmann codes
• Linear nonbinary covering codes and saturating sets in projective spaces

Cited by

• On minimal codes arising from projective embeddings of point-line geometries
• Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding

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