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

TL;DR

The paper resolves the asymptotic counting problem for inequivalent $q$-ary linear codes of length $n$ with dimension $k(n)$ under permutation, monomial, and semilinear equivalence. It leverages Burnside's lemma and sharp asymptotics for $q$-binomial coefficients to express the counts in terms of $\binom{n}{k(n)}_q$ and the total $S(n)$, with a key condition $(\star)$ on $k(n)$ ensuring the dominant term is captured. A central contribution is the precise asymptotics of $\binom{n}{k(n)}_q$, the connection to the discrete Gaussian theta distributions, and the resulting description of the proportion of codes with given dimension through Gaussian limits tied to Brownian motion. The work also resolves an open question by Wild by providing exact constants for the asymptotics of $S(n)$ in even and odd length cases, linking combinatorics of Grassmannians to probabilistic limit laws. Overall, the results bridge coding theory, $q$-combinatorics, and probabilistic limit theory, with explicit asymptotic formulas suitable for SEO and context-rich embeddings.

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.

Theorems & Definitions (30)

• Definition 2.1
• Proposition 3.1
• proof
• Example 3.1
• Example 3.2
• Proposition 3.2
• proof
• Remark 3.1
• Lemma 4.1
• proof