Gilbert-Varshamov Bound for Codes in L₁ Metric using Multivariate Analytic Combinatorics

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 L₁ metric. Several different code spaces are analyzed, including the simplex and the hypercube in Zⁿ, 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.