Hybrid Polar Encoding with Applications in Non-Coherent Channels

In coding theory, an error-correcting code can be encoded either systematically or non-systematically. In a systematic encode, the input data is embedded in the encoded output. Conversely, in a non-systematic code, the output does not contain the input symbols. In this paper, we propose a hybrid encoding scheme for polar codes, in which some data bits are systematically encoded while the rest are non-systematically encoded. Based on the proposed scheme, we design a joint channel estimation and data decoding scheme. We use the systematic bits in the hybrid encoding scheme as pilots for channel estimation. To mitigate the code rate loss caused by the pilots and to provide additional error detecting capability, we propose a dynamic pilot design by building connections between the systematic bits and non-systematic bits. Simulation results show that the performance of the proposed scheme approaches that of the traditional non-systematic polar coding scheme with perfect channel state information (CSI) with the increase of SNR.Comment: 12 pages, 5 figure

An optimized encoding algorithm for systematic polar codes

Many different encoding algorithms for systematic polar codes (SPC) have been introduced since SPC was proposed in 2011. However, the number of the computing units of exclusive OR (XOR) has not been optimized yet. According to an iterative property of the generator matrix and particular lower triangular structure of the matrix, we propose an optimized encoding algorithm (OEA) of SPC that can reduce the number of XOR computing units compared with existing non-recursive algorithms. We also prove that this property of the generator matrix could extend to different code lengths and rates of the polar codes. Through the matrix segmentation and transformation, we obtain a submatrix with all zero elements to save computation resources. The proportion of zero elements in the matrix can reach up to 58.5{\%} from the OEA for SPC when the code length and code rate are 2048 and 0.5, respectively. Furthermore, the proposed OEA is beneficial to hardware implementation compared with the existing recursive algorithms in which signals are transmitted bidirectionally