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For $n \ge 3$, we demonstrate the existence of quantum codes which are local in dimension $n$ with $V$ qubits, distance $V^{\frac{n-1}{n}}$, and dimension $V^{\frac{n-2}{n}}$, up to a $polylog(V)$ factor.
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Panteleev, P., Kalachev, G. (2022, June). Asymptotically good quantum and locally testable classical LDPC codes. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (pp. 375-388)
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