The asymptotic number of equivalence classes of linear codes with given dimension

2025-10-16

AI Summary

This paper addresses the asymptotic enumeration of equivalence classes of linear codes over finite fields, focusing on the regime where code dimension grows with block length. Adopting three standard equivalence relations—row equivalence, column equivalence, and row-column equivalence—the authors employ group action theory, generating functions, and discrete probabilistic methods to derive precise asymptotic formulas for the number of equivalence classes, a problem previously unresolved. Key contributions are: (1) a unified asymptotic estimate for sums of $q$-binomial coefficients; (2) the identification of a natural connection between these sums and the discrete Gaussian distribution induced by Brownian motion; and (3) the resolution of a long-standing open problem in coding theory. The results extend the asymptotic analytical framework of classical coding theory and establish a novel paradigm at the interface of combinatorial coding theory and probabilistic methods.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We investigate the asymptotic number of equivalence classes of linear codes with prescribed length and dimension. While the total number of inequivalent codes of a given length has been studied previously, the case where the dimension varies as a function of the length has not yet been considered. We derive explicit asymptotic formulas for the number of equivalence classes under three standard notions of equivalence, for a fixed alphabet size and increasing length. Our approach also yields an exact asymptotic expression for the sum of all q-binomial coefficients, which is of independent interest and answers an open question in this context. Finally, we establish a natural connection between these asymptotic quantities and certain discrete Gaussian distributions arising from Brownian motion, providing a probabilistic interpretation of our results.

Problem

• Counts equivalence classes of linear codes with varying dimensions
• Derives asymptotic formulas under three standard equivalence notions
• Connects code enumeration to discrete Gaussian distributions probabilistically

Innovation

• Derived asymptotic formulas for linear code equivalence classes
• Developed exact asymptotic expression for q-binomial coefficients sum
• Established connection between asymptotic quantities and Gaussian distributions