Analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert--Varshamov lower bound on the rate of optimal codes in $L_1$ metric. Several different code spaces are analyzed, including the simplex and the hypercube in $\mathbb{Z^n}$, all of which are inspired by concrete data storage and transmission models such as the sticky insertion channel, the permutation channel, the adjacent transposition (bit-shift) channel, the multilevel flash memory channel, etc.