Non-systematic Polar Codes Research Articles

Rate‐compatible systematic polar codes

Puncturing and shortening are two common ways to achieve rate-compatible non-systematic polar codes (NSPCs). Systematic polar codes (SPCs) have been shown to outperform NSPCs with the same encoding and decoding complexity. However, rate-compatible SPCs have never been comprehensively studied in previous work. In this paper, two rate-compatible algorithms for SPCs are first proposed: uniform puncturing (UP) algorithm and uniform shortening (US) algorithm, which are referred to as SPC-UP and SPC-US, respectively. In order to effectively estimate the maximum likelihood decoding performance of punctured and shortened polar codes, subsequently, a distance spectrum calculation algorithm based on successive cancellation list (SCL) decoder for rate-compatible polar codes is proposed. Simulation results show that rate-compatible SPCs yield better bit error rate performance than rate-compatible NSPCs while they have the same frame error rate performance under different code rates and decoding algorithms. Eventually, union bounds that are obtained by the distance spectrum to provide the theoretical explanation for the superiority of rate-compatible SPCs are utilised.

Serially concatenated scheme of polar codes and the improved belief propagation decoding algorithm

In this study, a serially concatenated scheme of polar codes with convolutional codes is proposed to improve the error correction performance. The novel belief propagation (BP) decoding algorithm addresses two issues that are present in the currently available BP decoding algorithms. The first issue is the poor performance of the BP decoding algorithms, in particular the introduction of an error floor. The second is the component codes can only use systematic codes in the traditional concatenated scheme of polar codes with convolutional codes, which inhibits the effective update of the prior information of the redundant check bits. The proposed BP decoding algorithm is based on right-directed message processing, which effectively improves the decoding performance. In addition, the proposed concatenated scheme extends the selection of component codes from the systematic polar codes to the non-systematic polar codes. Hence, the areas of applications and the prior information of information bits for polar codes are expanded and more effectively updated, respectively. The simulation results show that the proposed scheme is much better than the traditional concatenated scheme, and the error floor is no longer introduced in terms of the block error rate, while the storage and computational complexities have not increased obviously.

Channel Estimation With Systematic Polar Codes

Study of polar codes in fading channels is of great importance when applying polar codes in wireless communications. Channel estimation is a fundamental step for communication to be possible in fading channels. For both systematic and non-systematic polar codes, construction of them is based on an information set and the known frozen bits. Efficient implementation of systematic and non-systematic polar codes exists. When it comes to channel estimation or channel tracking, additional pilot symbols are inserted in the codeword traditionally. In this paper, to improve the performance of polar codes in the finite domain, pilot symbols are selected from the coded symbols themselves. In order to keep the existing efficient structure of polar code encoding, pilot selection is critical since not all selections can reuse the existing structure. In this paper, two pilot selections denoted as Uneven Pilot Selection (UEPS) and Even Pilot Selection (EPS) are proposed, which do not change the efficient polar encoding structure. The proposed UEPS and EPS is proven to satisfy the efficient construction condition. The performance of EPS is shown in this paper to outperform both the UEPS and the traditional pilot insertion scheme. Simulation results are provided which verify the performance of the proposed pilot selection schemes.

A Low Complexity Encoding Algorithm for Systematic Polar Codes

Arikan has shown that systematic polar codes (SPC) outperform nonsystematic polar codes (NSPC). However, the performance gain comes at the price of elevated encoding complexity, i.e., compared to NSPC, the available encoding methods for SPC require higher memory and computation. In this letter, we propose an efficient encoding algorithm requiring only $N$ bits of memory and having $\frac{N}{2}\log_2N$ XOR operations. Moreover, the auxiliary variables in the algorithm can share the memory to reduce extra memory requirement. Furthermore, a parallel 2-bit encoding algorithm is also presented to improve the encoding throughput. Remarkably, we show that parallel encoding can be implemented with the same number of XOR operations and memory bits. Finally, the proposed encoding algorithm can be directly used for NSPC with the same complexity.

Secret Key Cryptosystem Based on Non-systematic Polar Codes

Polar codes are a new class of error correcting linear block codes, whose generator matrix is specified by the knowledge of transmission channel parameters, code length and code dimension. Moreover, regarding computational security, it is assumed that an attacker with a restricted processing power has unlimited access to the transmission media. Therefore, the attacker can construct the generator matrix of polar codes, especially in the case of binary erasure channels, on which this matrix can be easily constructed. In this paper, we introduce a novel method to keep the generator matrix of polar codes in secret in a way that the attacker cannot access the required information to decode the intended polar code. With the help of this method, a secret key cryptosystem is proposed based on non-systematic polar codes. In fact, the main objective of this study is to achieve an acceptable level of security and reliability through taking advantage of the special properties of polar codes. The analyses revealed that our scheme resists the typical attacks on the secret key cryptosystems based on linear block codes. In addition, by employing some efficient methods, the key length of the proposed scheme is decreased compared to that of the previous cryptosystems. Moreover, this scheme enjoys other advantages including high code rate, and proper error performance as well.