Efficient Implementation of Density Evolution for Punctured Polar Codes

Polar codes asymptotically achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under low-complexity sequential decoding algorithms such as successive cancellation decoding. However, in their original formulation, the block length of polar codes is limited to integer powers of the dimension of the underlying polarization kernel used, thus imposing strict constraints on possible application scenarios. While leeway in the choice of kernel or concatenation with other codes mitigates this drawback to a certain extent, puncturing presents a promising approach to specify the target length of a polar code with much greater flexibility. In this paper, we present an efficient implementation of the construction of punctured polar codes based on density evolution, a crucial tool in the construction of both regular, i.e., unpunctured, as well as punctured polar codes. Our implementation of density evolution covers the construction of both regular and punctured polar codes and allows for treating the construction of both code classes in a unified framework. Using our implementation, we achieve substantial reductions in the number of density convolutions necessary for the construction of punctured polar codes and obtain tight upper bounds on the block error rates.