Codes on Planar Graphs

TL;DR

This paper investigates codes on planar Tanner graphs, establishing an upper bound on their minimum distance relative to code rate, and concludes that high-rate, high-distance codes cannot be planar, limiting their asymptotic performance.

Contribution

It derives a specific upper bound on the minimum distance of codes on planar graphs as a function of rate, revealing fundamental limitations of planar graph-based codes.

Findings

Planar graph codes cannot be asymptotically good.

High-rate, high-distance codes require non-planar graphs.

The minimum distance bound is explicitly related to the code rate.

Abstract

Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several intriguing problems in the intersection of coding theory and graph theory. In this paper, we study codes defined by planar Tanner graphs. We derive an upper bound on minimum distance $d$ of such codes as a function of the code rate $R$ for $R \ge 5/8$. The bound is given by $$d\le \lceil \frac{7-8R}{2(2R-1)} \rceil + 3\le 7.$$ Among the interesting conclusions of this result are the following: (1) planar graphs do not support asymptotically good codes, and (2) finite-length, high-rate codes on graphs with high minimum distance will necessarily be non-planar.