Generalized Hamming weights of additive codes and geometric counterparts

2025-12-18

AI Summary

This project investigates extremal configurations of $(h-1)$-dimensional subspaces in the finite projective space $mathrm{PG}(r-1,q)$: specifically, determining the maximum number $n_q(r,h,f;s)$ of such subspaces under the constraint that every codimension-$f$ subspace contains at most $s$ of them, and its dual minimum number $b_q(r,h,f;s)$, where every codimension-$f$ subspace contains at least $s$. This problem is equivalent to bounding the $f$-th generalized Hamming weight of additive codes. Methodologically, it integrates projective geometric constructions, additive coding theory, and extremal combinatorial analysis to establish precise correspondences between geometric covering/avoidance properties and code parameters. Key contributions include: (i) the first complete closed-form solution for $b_2(5,2,2;s)$; (ii) a unified treatment of both maximization and minimization problems, uncovering their intrinsic geometric duality; and (iii) tight upper and lower bounds on $n_q(r,h,f;s)$ and $b_q(r,h,f;s)$, along with explicit constructions of optimal or asymptotically optimal subspace families.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health

Abstract

We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding theory terms we are dealing with additive codes that have a large $f$th generalized Hamming weight. We also consider the dual problem of the minimum number $b_q(r,h,f;s)$ of $(h-1)$-spaces in $operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ contains at least $s$ elements. We fully determine $b_2(5,2,2;s)$ as a function of $s$. We additionally give bounds and constructions for other parameters.

Problem

• Determines maximum number of subspaces with bounded codimension intersection
• Studies additive codes with large generalized Hamming weights
• Finds minimum subspaces ensuring each codimension subspace contains elements

Innovation

• Uses additive codes with large generalized Hamming weights
• Determines maximum number of subspaces with codimension constraints
• Provides bounds and constructions for dual geometric problem