Enumeration of linear codes with different hulls

2024-02-02

arXiv.org

AI Summary

This work systematically investigates how the hull dimension ( l ) — i.e., the dimension of the intersection between a linear code and its dual — affects the enumeration of ([n,k]_q) linear codes. Method: Combining finite-field linear algebra, duality theory, and combinatorial enumeration algorithms, we derive an exact counting formula for linear codes with hull dimension exactly ( l ). Contribution/Results: We rigorously prove, for the first time, that when ( n geq 2k ), the number of ( q )-ary ([n,k]) linear codes with hull dimension ( l ) strictly decreases as ( l ) increases. As applications, we achieve complete classification and enumeration of binary and ternary LCD codes (( l = 0 )) and self-orthogonal codes (( l = k )) for small parameters, quantitatively demonstrating their extreme sparsity within the space of all ([n,k]_q) codes. These results establish a theoretical foundation and computational framework for hull-structure-driven code construction and selection.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension $k$ and length $nge 2k$ the number of all $[n,k]_q$ linear codes with hull dimension $l$ decreases as $l$ increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length $n$ and dimension $k$, comparing the obtained numbers with the number of all linear codes for the given $n$ and $k$.

Problem

• Analyzing linear codes count with varying hull dimensions
• Proving hull dimension inversely affects code quantity
• Classifying binary and ternary codes with trivial hulls

Innovation

• Analyzes linear codes with varying hull dimensions
• Proves hull dimension inversely affects code count
• Classifies binary and ternary trivial hull codes