Random access codes (RACs) encode a $q$-ary input string into fewer $q$-ary symbols while allowing any requested input symbol to be recovered with probability above the random-guessing value $1/q$. However, it remains unclear what asymptotic guarantees on compression and recovery probability should be required of RACs. We address this question by introducing the notion of asymptotically good RACs and give two polynomial-time constructions. First, an $[n,k,d]_q$ linear code with $d\ge2$, together with an encoder based on its orthogonal-array property, yields an $(n,n-k)_q$-RAC with worst-case decoding success probability \[ p_{\rm wc}=\frac{1}{q}+\left(1-\frac{1}{q}\right)\frac{d-2}{n}. \] These OA-RACs are asymptotically good whenever the underlying linear codes are, and Garcia--Stichtenoth algebraic-geometry codes give a concrete polynomial-time realization. Second, a direct-sum strategy using OA-RACs derived from $q$-ary Hamming codes and the trivial $(1,1)_q$-RAC yields asymptotically good RAC families at every prescribed RAC rate $R_{\rm RAC}=(n-k)/n\in(0,1)$. Numerical results demonstrate favorable finite-length performance of both constructions.

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