Abstract
In this article we propose a technique for polar codes (PC) construction for any code length. By default, PC construction is limited to code length proportional to the power of two. To construction the code length arbitrary, puncturing, shortening and extension techniques must be applied. However, performance is degraded with the use of these techniques. Other ways to design polar codes with arbitrary code length but which have encoding and decoding with higher complexity such as multi-kernel, concatenated codes and specific constructions for belief propagation (BP) or successive cancellation list (SCL) decoding. The polarization theory is generalized for non-uniform channels (NUC) and with this approach we can construction rate-compatible PC and variable code length. We developed an implementation algorithm based on the of PC construction by Gaussian approximation (NUPGA). In a scenario where the transmission is over an additive white Gaussian noise (AWGN) channel and under successive cancellation (SC) decoding, the PC construction of arbitrary code length can be implemented with NUPGA. With NUPGA we re-polarize the projected synthetic channels by choosing more efficiently the positions of the information bits. In addition, we present a generalization of the Gaussian approximation (GA) for the polarization and re-polarization processes and an extension technique for PC. The PC construction based on NUPGA present better performance than the existing techniques as shown in the simulations of this work.
Highlights
We present a generalization of the Gaussian approximation (GA) algorithm, which is used for both polarization of the initial channel and re-polarization of the shortened PC, and a simplified construction technique for extended polar codes
The results show that the NUPAG extension technique outperforms the Non-Uniform Polarization based on Gaussian Approximation (NUPGA) shortening and the PD algorithms using list decoding with Cyclic Redundancy Check (CRC)
In this work, we propose an efficient technique for PC construction of arbitrary length through the theory of non-uniform channel polarization
Summary
Polar codes and channel polarization theory were introduced by Arikan [1] in 2009. Such codes constitute a powerful channel coding scheme, with a low complexity encoder and decoder. In [38] a shortening method is presented that produces good results and its technique is based on using column weight (CW) to reduce the size of the generating matrix. The main technique for reducing the generating matrix used in the shortened PC was proposed by [39] and is known as reversal quasi-uniform puncturing (RQUP). The main objective of this work is to demonstrate the use of non-uniform channel (NUC) polarization theory in the PC construction of arbitrary length [52]. A new technique for constructing polar codes of arbitrary length based on NUPGA along with a proof that it achieves capacity;.
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