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  "path": "/abs/2604.13953v1",
  "publishedAt": "2026-04-16T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Michael Levet"
  ],
  "textContent": "**Authors:** Michael Levet\n\nIn this paper, we exhibit $\\textsf{AC}^{3}$ isomorphism tests for coprime extensions $H \\ltimes N$ where $H$ is elementary Abelian and $N$ is Abelian; and groups where $\\text{Rad}(G) = Z(G)$ is elementary Abelian and $G = \\text{Soc}^{*}(G)$. The fact that isomorphism testing for these families is in $\\textsf{P}$ was established respectively by Qiao, Sarma, and Tang (STACS 2011), and Grochow and Qiao (CCC 2014, SIAM J. Comput. 2017). The polynomial-time isomorphism tests for both of these families crucially leveraged small (size $O(\\log |G|)$) instances of Linear Code Equivalence (Babai, SODA 2011). Here, we combine Luks' group-theoretic method for Graph Isomorphism (FOCS 1980, J. Comput. Syst. Sci. 1982) with the fact that $G$ is given by its multiplication table, to implement the corresponding instances of Linear Code Equivalence in $\\textsf{AC}^{3}$. As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using $\\textsf{AC}$ circuits of depth $O(\\log^3 n)$ and size $n^{O(\\log \\log n)}$. This improves upon the previous bound of $n^{O(\\log \\log n)}$-time due to Grochow and Qiao (ibid.).",
  "title": "Parallel Algorithms for Group Isomorphism via Code Equivalence"
}