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Color Multiset Codes based on Sunmao Construction

Wing Shing Wong, Chung Shue Chen, Yuan-Hsun Lo

TL;DR

This work develops color multiset codes on $n$-D grids using the sunmao construction, motivated by sensor-based object tracking where order of symbols cannot be preserved. It introduces braid codes, especially unitary braid codes, and extends them from 1D to $n$-D grids via product codes and unitary sunmao decomposition, proving asymptotic color-efficiency and providing fast, codebook-free decoding algorithms. The main contributions include establishing $m$-distinguishability under sunmao, analyzing asymptotic color usage, and detailing decoding procedures with complexity $O(n_0^2)$, as well as outlining error-correction properties for 1D braid codes. The results have practical implications for high-dimensional proximity sensing and related applications, offering scalable constructions with provable optimality in the large-grid limit and actionable decoding strategies. Open problems remain in finding exact minimal-color codes for finite grids and in developing robust error-correcting color codes.

Abstract

We present results on coding using multisets instead of ordered sequences. The study is motivated by a moving object tracking problem in a sensor network and can find applications in settings where the order of the symbols in a codeword cannot be maintained or observed. In this paper a multiset coding scheme is proposed on source data that can be organized as a flat or cyclic multi-dimensional integer lattice (grid). A fundamental idea in the solution approach is to decompose the original source data grid into sub-grids. The original multiset coding problem can then be restricted to each of the sub-grid. Solutions for the sub-grids are subsequently piece together to form the desired solution. We name this circle of idea as sunmao construction in reference to woodwork construction method with ancient origin. Braid codes are specific solutions defined using the sunmao construction. They are easy to define for multi-dimensional grids. Moreover for a code of a given code set size and multiset cardinality, if we measure coding efficiency by the number of distinct symbols required, then braid codes have asymptotic order equal to those that are optimal. We also show that braid codes have interesting inherent error correction properties.

Color Multiset Codes based on Sunmao Construction

TL;DR

This work develops color multiset codes on -D grids using the sunmao construction, motivated by sensor-based object tracking where order of symbols cannot be preserved. It introduces braid codes, especially unitary braid codes, and extends them from 1D to -D grids via product codes and unitary sunmao decomposition, proving asymptotic color-efficiency and providing fast, codebook-free decoding algorithms. The main contributions include establishing -distinguishability under sunmao, analyzing asymptotic color usage, and detailing decoding procedures with complexity , as well as outlining error-correction properties for 1D braid codes. The results have practical implications for high-dimensional proximity sensing and related applications, offering scalable constructions with provable optimality in the large-grid limit and actionable decoding strategies. Open problems remain in finding exact minimal-color codes for finite grids and in developing robust error-correcting color codes.

Abstract

We present results on coding using multisets instead of ordered sequences. The study is motivated by a moving object tracking problem in a sensor network and can find applications in settings where the order of the symbols in a codeword cannot be maintained or observed. In this paper a multiset coding scheme is proposed on source data that can be organized as a flat or cyclic multi-dimensional integer lattice (grid). A fundamental idea in the solution approach is to decompose the original source data grid into sub-grids. The original multiset coding problem can then be restricted to each of the sub-grid. Solutions for the sub-grids are subsequently piece together to form the desired solution. We name this circle of idea as sunmao construction in reference to woodwork construction method with ancient origin. Braid codes are specific solutions defined using the sunmao construction. They are easy to define for multi-dimensional grids. Moreover for a code of a given code set size and multiset cardinality, if we measure coding efficiency by the number of distinct symbols required, then braid codes have asymptotic order equal to those that are optimal. We also show that braid codes have interesting inherent error correction properties.

Paper Structure

This paper contains 26 sections, 24 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. System Model
  3. Mathematical Notation
  4. Grids and Product Grids
  5. Color Multiset Code
  6. The Sunmao Construction
  7. Braid Code for 1D Grids
  8. Construction Based on Repetitive Code
  9. Performance Comparison of Braid Codes
  10. Construction for a General Size Unitary Braid Code
  11. Braid Code for General Dimension
  12. Product Color Code
  13. $n$-dimensional Unitary Sunmao Code
  14. $n$-Dimensional Repetitive Code
  15. $n$-dimensional Unitary Braid Code
  16. ...and 11 more sections

Key Result

Lemma 1

Consider a 1D grid $G^c_M$ with block size $m$ such that $m$ divides $M$. Given a sunmao decomposition of $G^c_M$ into sub-grids $\{G^c_{M_1},\ldots,G^c_{M_I}\}$, the following properties hold:

Figures (1)

  • Figure 1: A 2D unitary braid code example. The color set for $\Phi_{0,0}$ is $\{\color{cyan}{a_0},\ldots,\color{cyan}{a_9},\color{cyan}{a'_0},\color{cyan}{a'_1}\}$, for $\Phi_{0,1}$, $\{\color{blue}{b_0},\ldots,\color{blue}{b_7}\}$, for $\Phi_{1,0}$, $\{ \color{violet}{c_0},\ldots,\color{violet}{c_7} \}$, and for $\Phi_{1,1}$, $\{\color{magenta}{d_0},\ldots,\color{magenta}{d_9},\color{magenta}{d'_{0}},\color{magenta}{d'_{1}}\}$.

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Proposition 1
  • proof
  • Definition 4
  • Proposition 2
  • Definition 5
  • Definition 6
  • ...and 41 more