Strange Attractor for Efficient Polar Code Design

Abstract: This paper presents a definition of a construction for long polar codes. Recently, we know that partial order is a universal property of the construction with a sublinear complexity for polar codes. In order to describe the partial order, addition and left-swap operators are only defined as universal up to now. In this study, we first propose $1+\log_2 \log_2 N$ universal operators to describe multiple partial order for the block length $N=2^n$. By using these operators, some known antichains can be universally ordered. Furthermore, by using a simple geometric property of Gaussian approximation, we define an attractor that is a pre-defined subset of synthetic channels. They are universally less reliable than the natural channel $W$. Then, we show that the cardinality of this attractor is $(n+2)$-th Fibonacci number which is a significantly large number of channels for long codes. The main contribution is that there are significant number of synthetic channels explicitly defined as almost useless by the help of attractor and multiple partial order. As a result, proposed attractor with multiple partial order can be seen as an efficient tool to investigate and design extremely long codes.