Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a sub-thermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. Here we quantify these notions by identifying a universal, subleading logarithmic contribution to the volume law entanglement entropy: $S^{(2)}(A)=kappa L_A+frac{3}{2}log L_A$ which bounds the mutual information between a qudit inside region $A$ and the rest of the system. Specifically, we find the power law decay of the mutual information $I({x}:bar{A})propto x^{-3/2}$ with distance $x$ from the regions boundary, which implies that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$. We obtain these results by mapping the entanglement dynamics to the imaginary time evolution of an Ising model, to which we can apply field-theoretic and matrix-product-state techniques. Finally, exploiting the error-correction viewpoint, we assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code to obtain a bound on the critical measurement strength $p_{c}$ as a function of the qudit dimension $d$: $p_{c}log[(d^{2}-1)({p_{c}^{-1}-1})]le log[(1-p_{c})d]$. The bound is saturated at $p_c(drightarrowinfty)=1/2$ and provides a reasonable estimate for the qubit transition: $p_c(d=2) le 0.1893$.
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