The asymptotic number of equivalence classes of linear codes with given dimension
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.
Keywords
Cite
@article{arxiv.2510.14424,
title = {The asymptotic number of equivalence classes of linear codes with given dimension},
author = {Andrea Di Giusto and Alberto Ravagnani},
journal= {arXiv preprint arXiv:2510.14424},
year = {2025}
}