Novel Recursive Kernel Construction for Polar Codes with Practical Codeword Lengths

Abstract

The basic approach introduced by Arikan in [1] to polarize equal capacity channels to unequal capacities, can be used to design only codewords of length $N=2^{n}$, which is clearly a major limitation when codewords of length $N\neq 2^{n}$ are required. In order to systematically construct polarization kernels of size 3 and higher, a recursive construction technique is developed in this paper, that describes each kernel as a concatenation of smaller kernels. Optimally designing and selecting kernels over the underlying system parameters is based on the evolution of z-parameter values (i.e. effect of channel polarization) of the bit channels, which are used to define a design metric $\zeta$. The complex process of optimizing kernel designs, which depends on a wide range of parameters particularly the coderate, is simplified by this metric and its effectiveness is validated by comparative error rate performance.

Authors 5

1. Souradip Saha

   Fraunhofer Institute for Communication, Information Processing and Ergonomics

   Affiliation as printed

   Fraunhofer FKIE, Wachtberg, Germany

2. Marc Adrat

   Fraunhofer Institute for Communication, Information Processing and Ergonomics

   Affiliation as printed

   Fraunhofer FKIE, Wachtberg, Germany

3. Luis Masny Aachen Institute of Communication Systems

   RWTH Aachen University

   Affiliation as printed

   Institute of Communication Systems, RWTH Aachen University, Aachen, Germany

4. Matthias Schrammen Aachen Institute of Communication Systems

   RWTH Aachen University

   Affiliation as printed

   Institute of Communication Systems, RWTH Aachen University, Aachen, Germany

5. Peter Jax Aachen Institute of Communication Systems

   RWTH Aachen University

   Affiliation as printed

   Institute of Communication Systems, RWTH Aachen University, Aachen, Germany