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arXiv · 2510.14424

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

Andrea Di Giusto·Alberto Ravagnani
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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.

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Andrea Di Giusto, Alberto Ravagnani. 2025-10-16. The asymptotic number of equivalence classes of linear codes with given dimension. https://arxiv.org/abs/2510.14424

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