This paper studies lossy coding of discrete memoryless sources and derives new asymptotic lower and upper
bounds on the rate of optimal fixed-length codes under $epsilon$-fidelity constraints. 
We show that under a regularity condition, the rate of optimal codes is lower-bounded by $R(D) + R_2/sqrt{n} + (log n)/(2n) + underline{R}_4/n + o(1)$ and upper-bounded by $R(D) + R_2/sqrt{n} + (log n)/(2n) + log log n + overline{R}_4/n + o(1)$ where $n$ is the block length, $R(D)$ is Shannon's rate-distortion function, $R_2$ is the second-order coding rate, and $underline{R}_4$ and $overline{R}_4$ are constants that are explicitly identified.