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Codes defined on graphs and their properties have been subjects of intense recent research. In this work, we are concerned with codes that have planar Tanner graphs. When the Tanner graph is planar, message-passing decoders can be efficiently implemented on chips without any issues of wiring. Also, recent work has shown the existence of optimal decoders for certain planar graphical models. The main contribution of this paper is an explicit upper bound on minimum distance $d$ of codes that have planar Tanner graphs as a function of the code rate $R$ for $R \geq 5/8$. The bound is given by
\begin{equation*}
d\le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil + 3\le 7.
\end{equation*}
As a result, high-rate codes with planar Tanner graphs will result in poor block-error rate performance, because of the constant upper bound on minimum distance.

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