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 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 permutation channel, the repetition channel, the adjacent transposition (bit-shift) channel, the multilevel flash memory channel, etc.