Color Multiset Codes based on Sunmao Construction

TL;DR

This paper introduces a multiset coding scheme called sunmao construction for source data on multi-dimensional grids, with applications in sensor networks and error correction, offering efficient and robust code solutions.

Contribution

It proposes a novel sunmao construction method for multiset coding on multi-dimensional grids, including braid codes with optimal asymptotic efficiency and inherent error correction capabilities.

Findings

Braid codes are easy to define for multi-dimensional grids.

Braid codes achieve asymptotic optimality in symbol efficiency.

Braid codes possess inherent error correction properties.

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.