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Gilbert-Varshamov Bound for Codes in $L_1$ Metric using Multivariate Analytic Combinatorics

Keshav Goyal, Duc Tu Dao, Mladen Kovačević, Han Mao Kiah

TL;DR

These tools are applied to determine the Gilbert-Varshamov lower bound on the rate of optimal codes in <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula> metric.

Abstract

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.

Gilbert-Varshamov Bound for Codes in $L_1$ Metric using Multivariate Analytic Combinatorics

TL;DR

These tools are applied to determine the Gilbert-Varshamov lower bound on the rate of optimal codes in <inline-formula> <tex-math notation="LaTeX"> </tex-math></inline-formula> metric.

Abstract

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 metric. Several different code spaces are analyzed, including the simplex and the hypercube in , 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.
Paper Structure (22 sections, 43 theorems, 157 equations, 3 figures, 1 table)

This paper contains 22 sections, 43 theorems, 157 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analytic Combinatorics in Several Variables (ACSV)
  4. GV Bounds for Simplex Spaces
  5. The Standard Simplex
  6. Evaluation of the GV Bound
  7. Evaluation of Marcus and Roth's Improvement of the GV Bound
  8. Numerical Results
  9. The Positive Simplex
  10. Evaluation of the GV Bound
  11. Evaluation of Marcus and Roth's Improvement of the GV Bound
  12. The Inverted Simplex
  13. GV Bound for the Hypercube
  14. Evaluation of the GV Bound
  15. Evaluation of Marcus and Roth's Improvement of the GV Bound
  16. ...and 7 more sections

Key Result

Theorem 1

Let $F(\mathbfsl{z}) = \sum_{\mathbfsl{k}}a_{\mathbfsl{k}} \mathbfsl{z}^\mathbfsl{k}= \frac{G(\mathbfsl{z})}{H(\mathbfsl{z})}$ where $G$ and $H$ are both analytic, $H(\mathbf{0}) \neq 0$, and $a_{\mathbfsl{k}} > 0$. For each $\mathbfsl{k}= (k_1,k_2,\ldots k_\ell)>\mathbf{0}$ there is a unique soluti Furthermore, if $G(\mathbfsl{z}^*) \neq 0$, where $\mathtt{H}$ is the determinant of the Hessian o

Figures (3)

  • Figure 1: Bounds on the highest attainable code rate $\alpha(\Delta_{2},\delta)$ for the standard simplex ($\rho=2$).
  • Figure 2: Lower bounds on the highest attainable code rate for the positive simplex and the inverted simplex.
  • Figure 3: Lower bounds on the highest attainable code rate $\alpha({\mathbb Z}_{4},\delta)$ for the family of hypercubes $\{{\mathbb Z}^n_{4}\}_n$.

Theorems & Definitions (78)

  • Theorem 1: pemantle2008
  • Example 1: Binomial coefficients pemantle2008
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Example 2: Example \ref{['exa:binomial']} continued
  • Corollary 4
  • proof
  • Proposition 5
  • ...and 68 more