A cyclic redundancy check aided encoding construction method for list sphere polar decoder

Abstract

Polar codes are the only error-correcting codes that have been mathematically proven to achieve the Shannon limit to date, playing a crucial role in the control channels of 5G mobile communication systems. For control channels, although the sphere decoding (SD) algorithm boasts excellent performance, its high computational complexity and significant latency present clear limitations in practical applications. In contrast, the list sphere decoding (LSD) algorithm strikes a balance between performance and complexity. This paper proposes a construction method that delays the decoding of specific information bits with the minimum row weight to mitigate the impact of error propagation. For scenarios where the total number of delay-decodable bits is limited, we introduce a segmented construction strategy. Through mathematical analysis, this strategy effectively increases the number of delay-decodable bits, thereby significantly reducing the impact of error propagation without changing the number of transmitted bits. Simulation results show that the decoding performance of the proposed algorithm is comparable to that of the SD algorithm at medium and low code rates (with a difference of less than 0.2 dB), but it abandons the concept of search radius in the SD algorithm and does not require a backtracking process. Furthermore, under high code rate conditions, such as P(32, 20), the proposed algorithm also maintains excellent performance.

Introduction

In the field of communications, with the continuous increase in data transmission rate and system complexity, the demand for efficient and reliable channel coding schemes is also increasing. Against this backdrop, polar codes emerged. This coding theory was proposed by Turkish scientist Erdal Arikan in 2008^1,2, and it has attracted widespread attention because it can theoretically achieve the Shannon capacity of symmetric binary-input discrete memoryless channel (B-DMC). For this reason, polar codes were ultimately selected as the control channel coding scheme in the 5G communication standard². Compared to the heavy data transmission relying on long codes over data channels, the payload of control channels is relatively small, which requires new constructions for short codes and improvements to their corresponding decoding algorithms.

The successive cancellation (SC) decoding algorithm is the most basic decoding algorithm for polar codes¹. Subsequently, the successive cancellation list (SCL) decoding algorithm and the cyclic redundancy check (CRC) aided SC list (CA-SCL) decoding algorithm have been introduced^3–5. CA-SCL performs well on long codes but has limited performance in short-code application scenarios. It is based on the concept of channel polarization, starting from the least reliable bit and decoding bit by bit, using the results as auxiliary information for the decoding of the next bit. The advantage of the SC decoding algorithm is its simple structure, ease of implementation, and performance that approaches channel capacity as the code length increases. However, the disadvantages of the SC decoding algorithm are also apparent; its error propagation problem leads to a sharp decline in performance when errors occur, and for short codes, its performance is not optimal.

Sphere decoding (SD) is a nonlinear decoding method that achieves decoding by searching for the codeword closest to the received signal in the signal space. SD can be considered as an implementation of maximum likelihood (ML) decoding⁶, with the objective of finding the codeword with the minimum Euclidean distance from the received vector among all possible codewords in a higher-dimensional space. In ref.⁷, a stack SD (SSD) algorithm was proposed for polar codes. This algorithm organizes candidate paths in a stack sorted by descending path metrics, achieving ML performance while significantly reducing computational complexity. Reference⁸ introduced a multiple sphere decoder search (MSDS) method, which executes sequential/hybrid tree traversals with early termination criteria to significantly reduce decoding latency and complexity. Additionally, an efficient software stack sphere decoder (ESSD) based on synchronized determination was developed in ref.⁹, reducing polar code decoding complexity while preserving error-correction performance comparable to the conventional SD algorithm for both low-rate and high-rate polar codes. In ref.¹⁰, an SD algorithm based on symmetric encoding was proposed to reduce complexity at an acceptable performance loss. The advantage of the SD algorithm is its ability to achieve near-optimal decoding performance, especially in the case of short code lengths and medium code rates. However, the computational complexity of the SD algorithm is high, as it requires traversal of the entire signal space for decoding, which restricts its applicability for practical communication applications.

List SD (LSD) is an improvement upon the traditional SD algorithm¹¹. The LSD algorithm discards the search radius and backtracking process inherent in SD, opting instead to maintain a list of candidate codewords to reduce computational complexity, thereby striking a balance between performance and complexity. In ref.¹², it was proposed that combining LSD with the SCL can effectively reduce algorithmic complexity. Reference¹³. introduced a matrix reordering technique that significantly enhances the performance of the LSD algorithm. Furthermore, ref.¹⁴ presented a synchronized LSD algorithm, which defines a synchronization set based on the generator matrix of polar codes. By calculating the sum of all Euclidean distances within the synchronization set, the performance of LSD is improved.

Aiming for further performance enhancement, this paper initially proposes a construction method suitable for the LSD algorithm. Specifically, it aims to delay the decoding of as many information bits corresponding to rows with smaller weight as possible, thereby reducing the impact of decoding failures on error propagation throughout the entire codeword. Secondly, for cases where direct delay decoding construction cannot be applied to the codeword and where the number of delay-decodable bits is limited, this paper proposes a segmented construction strategy, which allows for better utilization of delayed decoding constructions. Furthermore, through mathematical analysis and experimental validation, we provide recommendations on whether to use segmented construction for polar codes under different circumstances. Finally, for decoding, this paper proposes the integration of segmented CRC bits during the segmented construction process, thereby incorporating an early-stopping mechanism without affecting decoding performance. Simulation results show that the algorithm reduces the required number of decoding operations, thereby lowering the decoding complexity of the algorithm while keeping performance loss moderate.

The remainder of this paper is organized as follows. “Preliminaries” section reviews the fundamental knowledge of SD and LSD. “Improved list sphere decoding algorithm” section illustrates the improved algorithms for the proposed delayed decoding construction and segmentation strategy. “Performance and complexity evaluation” section analyzes the performance and complexity. Finally, “Conclusion” section concludes the paper.

Preliminaries

Polar codes

As a linear block code, polar codes utilize the phenomenon of channel polarization. For a binary polar code P(N, K) with code rate $R=K/N$ and input sequence $U_1^n=(u_1^{},u_2^{},...,u_N^{})$, which includes information bits and known frozen bits–the values of which are fixed and known by both the transmitter and the receiver–a polar code of length N can be generated as follows: $c_1^N=u_1^{N}G_N^{}$, where $G_N^{}$ is the generator matrix, which can be constructed through the nth Kronecker product of the kernel matrix $F=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$, namely $G_N^{}=F_{}^{\otimes n}$.

The polarization effect will produce N polarized sub-channels with different reliabilities, within which K of the most reliable sub-channels are used to transmit information bits, and the indices of these sub-channels are placed in the information set A. The remaining $N-K$ sub-channels transmit frozen bits, and the indices of these sub-channels are placed in the frozen set $A_c^{}$. For binary erasure channels (BECs), the Bhattacharyya parameter is typically calculated to assess channel reliability. For other types of channels, such as additive white Gaussian noise (AWGN) channels, methods including density evolution (DE)^15,16 or its simplified version termed Gaussian approximation (GA)¹⁷, and partial ordering of independent channels¹⁸ can be used to evaluate channel reliability.

This paper considers binary input additive white Gaussian noise (BI-AWGN) channels and employs binary phase-shift keying (BPSK) modulation. After modulation, encoded vector $c_1^N=(c_1^{},c_2^{},...,c_N^{})$ is modulated into the transmission signal $x_i^{}=1-2c_i^{}$, in which $1\le i\le N$. Then, received vector $y_1^N=(y_1^{},y_2^{},...,y_N^{})$ is obtained through the signal transmission over the BI-AWGN channel, and each entry $y_i^{}$ can be written as follows:

$$\begin{array}{c}\hfill y_i^{}=x_i^{}+n_i^{},\end{array}$$ 1

where $n_i^{}$ is an AWGN term with a mean of zero and variance ${\sigma }^2$.

Sphere decoding algorithm

The SD algorithm traverses all paths and selects the path with the minimum Euclidean distance (ED) as the decoding result. For polar codes applied over BI-AWGN channels, the SD algorithm can be interpreted as the following optimization problem:

$$\begin{array}{c}\hfill {\widehat{u}}_1^N=arg\underset{u_1^N\in {\left\{0,,,1\right\}}^N}{min}||y_1^N-(1-2u_1^{N}G_N^{}){||}^2,\end{array}$$ 2

where ${\widehat{u}}_1^N$ represents the decoding estimate, and the decoding algorithm essentially enumerates all possible input information vectors to find a vector $y_1^N$ that minimizes the ED value. Let $D_i$ be the ED between $y_i$ and $x_i$, which can be calculated as

$$\begin{array}{c}\hfill D_i=||y_i-x_i{||}^2={\left[y_i,-,\left(1,-,2,\sum _{j=1}^N,u_j,g_{j,i}\right)\right]}^2,\end{array}$$ 3

where $1\le i\le N$, and $g_{j,i}$ denotes the element at the ith row and jth column of $G_N$. Considering the lower triangular matrix structure of $G_N$, the value of $D_i$ can be simplified to

$$\begin{array}{c}\hfill D_i={\left[y_i,-,\left(1,-,2,\sum _{j=i}^N,u_j,g_{j,i}\right)\right]}^2.\end{array}$$ 4

Thus, the partial ED value between vectors $y_i^N$ and $x_i^N$ can be calculated as:

$$\begin{array}{c}\hfill d_i^N=\sum _{k=i}^N{D}_k=\sum _{k=i}^N{\left[y_k,-,\left(1,-,2,\sum _{j=k}^N,u_j,g_{j,k}\right)\right]}^2.\end{array}$$ 5

The above formula satisfies the following recursive computation constraint:

$$\begin{array}{c}\hfill d_i^N=d_{i+1}^N+D_i\le r_0^2,\end{array}$$ 6

where $1\le i\le N-1$, and $r_0$ is the radius of SD, which is initialized to its maximum value.

List sphere decoding algorithm

Although SD can achieve the maximum likelihood bound, the high complexity of the algorithm limits its application in short codes. Moreover, the decoding performance of the SD algorithm is highly dependent on the choice of the initial radius. Therefore, to address the aforementioned issues, researchers have proposed the LSD algorithm reference required here. The LSD algorithm abandons the concept of search radius and does not require a backtracking process, making its time complexity independent of the channel conditions. Additionally, the LSD algorithm continues to use ED as the path metric from the SD algorithm. Its decoding process is still a reverse decoding, starting from the Nth bit of the codeword and gradually decoding to the 1 st bit. Let $Q_{}^j(1\le j\le L)$ represent the decoding estimate of the jth candidate path, and let $Q_i^j(1\le k\le N)$ represent the ith bit of $Q_{}^j$; these values are initialized to 0. Let $M_i^j$ represent the cumulative ED of candidate path j at decoding depth i. If $i\in A_{}^{}$, then it is also necessary to initialize $Q_i^{j+L}(1\le k\le N,1\le j\le L)$ to 1. $M_i^j$ and $M_i^{j+L}(1\le j\le L)$ can be updated by the following formula:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{M}_i^j=M_{i+1}^j+(y_i^{}-(1-2{\sum }_{k=i}^N(g(k,i)Q_k^j)))_{}^2\hfill \\ M_i^{j+L}=M_{i+1}^j+(y_i^{}-(1-2{\sum }_{k=i}^N(g(k,i)Q_k^{j+L})))_{}^2.\hfill \end{array}\right)\end{array}$$ 7

If $i\in A_{}^c$, $M_i^j(1\le j\le L)$ can be updated by the following formula:

$$\begin{array}{c}\hfill M_i^j=M_{i+1}^j+\{y_i^{}-[1-2\sum _{k=i}^N(g(k,i)Q_k^j)]\}_{}^2.\end{array}$$ 8

After calculating the cumulative ED of all candidate paths for the current layer, we select the L paths with the smallest EDs. Finally, when the decoding process reaches the first layer, we choose the decoding estimate sequence with the smallest distance metric from the L candidate paths as the final output result.

Improved list sphere decoding algorithm

Delayed decoding construction algorithm

The LSD algorithm proposed in ref.¹¹ abandons the search radius and backtracking process in SD, striking a balance between performance and complexity. However, the performance of this algorithm is greatly influenced by the length of the list. Simulations show that when the list length is limited, the performance gap between this algorithm and SD is significant^10,14. To enhance the performance of the algorithm, we hereby correct and improve upon it, with specific descriptions as follows.

Firstly, taking the polar code P(8, 8) as an example, its generator matrix is

$$\begin{array}{c}\hfill G_8=\left[\begin{array}{c}10000000\\ 11000000\\ 10100000\\ 11110000\\ 10001000\\ 11001100\\ 10101010\\ 11111111\end{array}\right].\end{array}$$ 9

Let $g\left(r,,,c\right)$ represent the element at the rth row and cth column of the generator matrix, where $g\left(r,,,c\right)=1$ represents the partial ED $D_c$ whose rth bit affects during the decoding process¹⁴. Due to the lower triangular structure of the generator matrix, it can be seen that the bit value at the rth position not only affects the calculation of partial ED $D_r$ at the rth position but may also influence the partial ED calculations of other bits. Clearly, the larger the row weight of the rth row in the generator matrix is, the more significant impact the value of $u_r$ has on the total ED of the codeword. For instance, the row weight of the 8th row in the generator matrix is equal to 8, meaning that the value of $u_8$ affects the calculation of partial ED $D_t(1\le t\le 8)$. Considering the special structure of the generator matrix, we propose and prove the following theorem:

Theorem 1

For any polar code P(N, K) generator matrix, let $\overline{k}$ represent the average of the column indices of non-zero elements in the ith row, and let $\overline{m}$ represent the average of the column indices of non-zero elements in the jth row. For any two indices i and j that satisfy $1\le i<j\le N$, if the row weight of the ith row is equal to that of the jth row, then the following relation holds: $\overline{k}<\overline{m}$.

Proof

Let $w_r$ represent the row weight of the rth row of the generator matrix, and let $k_d^{}(1\le d\le w_i)$ represent the column index of the dth non-zero element in the ith row, and $m_t^{}(1\le t\le w_j)$ represent the column index of the tth non-zero element in the jth row. Firstly, we assume that $\overline{k}\ge \overline{m}$ holds. Since $w_i=w_j$, we have:

$$\begin{array}{c}\hfill \sum _{d=1}^{w_i}{k}_d\ge \sum _{t=1}^{w_j}{m}_t.\end{array}$$ 10

Note that the generator matrix is of lower triangular structure. Therefore, for any $i<j$, to satisfy Eq.(10), it is necessary to satisfy:

$$\begin{array}{c}\hfill \sum _{d=1}^{w_i-1}{k}_d+i\ge \sum _{t=1}^{w_j-1}{m}_t+j.\end{array}$$ 11

This is equivalent to:

$$\begin{array}{c}\hfill \sum _{d=1}^{w_i-1}{k}_d\ge \sum _{t=1}^{w_j-1}{m}_t+j-i\ge \sum _{t=1}^{w_j-1}{m}_t+1.\end{array}$$ 12

Since generator matrix $G_N$ is constructed through the nth Kronecker product of $F=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$. Thus, in the generator matrix, for any two rows with equal row weights, after removing the last non-zero element of these two rows, in the row with the larger row number, its remaining last non-zero element’s column index will not be less than the last non-zero element’s column index of the row with the smaller row number. Accordingly, by continuously eliminating the last non-zero elements of two rows with equal row weights, we obtain:

$$\begin{array}{c}\hfill \sum _{d=1}^{w_i-1}{k}_d\le \sum _{t=1}^{w_j-1}{m}_t.\end{array}$$ 13

This contradicts Eq.(12). Therefore, the original assumption does not hold, and the opposite is thereby proven.

Common methods for constructing polar codes include construction based on equivalent Bhattacharyya parameters, density evolution algorithms, and Gaussian approximation constructions^15–17. These construction methods all select K reliable bits out of N bit positions to transmit information bits, with the remaining $N-K$bits used to transmit frozen bits.The LSD algorithm, in contrast, selects the path with the minimum ED as the decoding result, but it does not fully utilize the polarization characteristics of the channel. Therefore, during the decoding process, bits corresponding to rows with equal weights in the generator matrix exhibit small differences in reliability. As indicated in ref¹⁴., $g\left(r,,,c\right)$ represents the impact of the value of the rth bit on partial ED $D_c$ during the decoding process. According to Theorem 1, for any two indices i and j that satisfy $1\le i<j\le N$ in the generator matrix, if the row weight of the ith row is equal to that of the jth row, then the average column index of non-zero elements is positively correlated with the row number. Based on this, we can conclude that for two bits with the same row weight, if the decoding of the bit with the smaller row number $u_i$ fails, it will only affect the bits with smaller indices, and vice versa. In the LSD algorithm, the decoding judgment for each bit is made based on the cumulative ED. Obviously, when the bit with the larger index received is incorrectly decoded, it will inevitably propagate the error to the subsequent bit, leading to a degradation in decoding performance.

Based on the above observation, this paper proposes to place information bits with smaller row weights in the generator matrix at the front of bits with the same row weight, viz., to swap the information bits with larger indices to positions with the same row weight but smaller indices, thereby delaying the decoding of these bits. If the decoding of these bits fails, the failure will only affect the bits with smaller indices, thereby reducing the scope of error propagation and reducing its impact on decoding performance.

It is found in ref.¹⁹ that Reed-Muller (RM) codes and polar codes have the same generator matrix. When the polar code length is $N=2^n$, the number of rows with equal row weight $2^i(0\le i\le n)$ can be determined by the combination number $\left(\begin{array}{c}n\\ n-i\end{array}\right)$. Based on the value of the information bit length K, we can discuss two situations in particular:

(1) When $K={\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)+m\left(0,\le ,j,\le ,n,-,1,,,0,<,m,<,\left(\begin{array}{c}n\\ j+1\end{array}\right)\right)$, we first select ${\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)\left(0,\le ,j,\le ,n,-,1\right)$ bits with larger row weights as information bits as per the RM constraint rule. Let these ${\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)$ information bits correspond to the smallest row weight $w_{}^{_{}^{\prime }}$ in the generator matrix, then the next step is to select m bits from the remaining $\left(\begin{array}{c}n\\ j+1\end{array}\right)$ bits with row weight $\frac{w_{}^{_{}^{\prime }}}{2}$ and set them as information bits. Taking the polar code P(8, 3) as an example, we first select $\left(\begin{array}{c}3\\ 0\end{array}\right)=1$ bits as information bits ($w_8=8$), at this time $w_{}^{_{}^{\prime }}=8$. Next, we need to select $m=2$ bits from the remaining $\left(\begin{array}{c}3\\ 1\end{array}\right)=3$ bits with row weight $w=\frac{w_{}^{_{}^{\prime }}}{2}=4$, viz., to choose two bits from the three bits with indices 4, 6, and 7 as information bits. The Gaussian approximation construction chooses the 6th and 7th bits as information bits, while the scheme proposed in this paper chooses the 4th and 6th bits as information bits. The difference between the two methods lies in the choice between the 4th and 7th bits. From the generator matrix $G_8$, it can be seen that if the 4th information bit is decoded incorrectly, it will have an impact on the calculation of the Euclidean distance for the 4th, 3rd, 2nd, and 1 st bits; if the 7th information bit is decoded incorrectly, it will affect the decoding of the 7th, 5th, 3rd, and 1 st bits. Obviously, the value of the 4th bit will only have an impact on bits with smaller indices. Therefore, compared with the 7th bit, choosing the 4th bit as an information bit can reduce the negative impact of error propagation on the performance of the LSD algorithm. Figure 1 shows the construction differences and bit error rate (BER) performance of P(8,3) and P(16,8). It is evident that the proposed delayed decoding construction exhibits superior BER performance compared to the Gaussian approximation construction algorithm. When the BER is at level ${10}^{-2}$, the delayed decoding construction algorithm achieves approximately a performance gain of 0.5 dB.

Fig. 1.

Construction differences and BER performance comparison of P(8,3) and P(16,8).

Remark

The proposed delayed decoding method, by postponing the decoding of rows with lighter weights, effectively mitigates the impact of error propagation, thereby enhancing the decoding performance of the LSD algorithm. However, there are scenarios where the direct application of the delayed decoding strategy is not feasible. To address this limitation, we introduce a novel approach termed the segmentation strategy. This strategy offers a flexible alternative for managing decoding processes in situations where the standard delayed decoding cannot be applied, ensuring robust performance across a broader range of conditions.

(2) When $K={\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)\left(0,\le ,j,\le ,n\right)$, it is possible to exactly select K bits with larger row weights as information bits. Let $w_{}^{_{}^{\prime }}$ represent the minimum row weight of these K information bits. Clearly, the row weights of the remaining $N-K$ frozen bits are all less than $w_{}^{_{}^{\prime }}$, hence we cannot directly apply the delayed decoding strategy to the positions $w_{}^{_{}^{\prime }}$ in the information bits. In this regard, this paper proposes a segmented construction strategy, by which the specific operation is to transform P(N, K) into two sub-codes $P_1(N/2,K_1)$ and $P_2(N/2,K_2)$ for transmission, where $K_1+K_2=K$, and $|K_1-K_2|$ should be kept within its minimum. Through this operation, we can convert situations where the delayed decoding strategy cannot be applied into situations where the delayed decoding strategy can be used, with specific analysis as follows:

In the first scenario, when $m\approx \frac{1}{2}\left(\begin{array}{c}n\\ j+1\end{array}\right)$, the number of delay-decodable bits $min_{}^{}\left\{m,,,\left(\begin{array}{c}n\\ j+1\end{array}\right),-,m\right\}$ reaches the maximum, and the difference between the delayed decoding construction proposed in this paper and the Gaussian approximation construction is the most significant. Therefore, our construction goal for situation $K={\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)\left(0,\le ,j,\le ,n\right)$ is to increase the number of delay-decodable bits as many as possible, thereby reducing the impact of error propagation and optimizing the overall decoding performance without changing the number of transmitted bits. At the same time, we notice

$$\begin{array}{c}\hfill K=\sum _{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)=\left(\begin{array}{c}n\\ 0\end{array}\right)+\left(\begin{array}{c}n\\ 1\end{array}\right)+...+\left(\begin{array}{c}n\\ j\end{array}\right)=\left(\begin{array}{c}n-1\\ 0\end{array}\right)+\left(\begin{array}{c}n-1\\ 0\end{array}\right)+\left(\begin{array}{c}n-1\\ 1\end{array}\right)+...+\left(\begin{array}{c}n-1\\ j-1\end{array}\right)+\left(\begin{array}{c}n-1\\ j\end{array}\right).\end{array}$$ 14

Dividing the above by 2, we obtain

$$\begin{array}{c}\hfill \frac{K}{2}=\left(\begin{array}{c}n-1\\ 0\end{array}\right)+\left(\begin{array}{c}n-1\\ 1\end{array}\right)+...+\left(\begin{array}{c}n-1\\ j-1\end{array}\right)+\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)=\sum _{i=0}^{j-1}\left(\begin{array}{c}n-1\\ i\end{array}\right)+\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right).\end{array}$$ 15

Letting $N^{\prime }=2^{n-1}=\frac{N}{2}$, $K^{\prime }=\frac{K}{2}$, we can see that in polar code $P(N^{\prime },K^{\prime })$, the number of delay-decodable bits reaches $\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)$. Therefore, in this situation, by dividing the codeword into two equal parts as described above, we can effectively increase the number of delay-decodable bits, thereby effectively reducing the impact of error propagation without changing the number of transmitted bits.

Taking polar code P(64, 22) as an example, where $K=22=\left(\begin{array}{c}6\\ 0\end{array}\right)+\left(\begin{array}{c}6\\ 1\end{array}\right)+\left(\begin{array}{c}6\\ 2\end{array}\right)$, the number of rows with row weights of 64, 32, and 16 in the generator matrix are 1, 6, and 15, respectively. After selecting the bits corresponding to these positions as information bits, we find that there are no other rows with a row weight of 16 in the generator matrix. Therefore, we divide the codeword into two segments; that is, we divide P(64, 22) into two sub-codes $P_1(32,11)$ and $P_2(32,11)$ for transmission. In which, the two sub-codes use the same encoder. Obviously, the code length and the number of information bits of each sub-code become half of the original, i.e., $\frac{N}{2}=32$, $\frac{K}{2}=11=\left(\begin{array}{c}5\\ 0\end{array}\right)+\left(\begin{array}{c}5\\ 1\end{array}\right)+\frac{1}{2}\left(\begin{array}{c}5\\ 2\end{array}\right)$. It can be known that the number of delay-decodable bits for each sub-code is 5; that is, the total number of delay-decodable bits for the entire codeword has increased from 0 to 10, thereby effectively reducing the impact of error propagation. Let $P(N^{\prime },K^{\prime })\times 2$ represent the use of our segmented strategy for $P\left(N,,,K\right)$, and the performance comparison of the two schemes for P(64, 22) is shown in Figure 2. As can be seen from this figure, when the code length $N=64$ and the length of information bits $K=22$, the operation of increasing the number of delay-decodable bits through segmentation can bring a significant performance improvement.

Fig. 2.

FER performance comparison of two constructions for P(64,22) and P(64,28).

Therefore, we analyze the codewords under different transmission scenarios, with the goal of maximizing the number of delay-decodable bits under different transmission conditions. Firstly, for the two scenarios mentioned above, we combine them into the following situations for ease of subsequent discussion, transmitting code length $N=2^n$ and information bit length $K={\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)+m\left(0,\le ,j,\le ,n,-,1,,,0,\le ,m,\le ,\left(\begin{array}{c}n\\ j+1\end{array}\right)\right)$. Depending on whether the segmented strategy is used, the following cases can be discussed:

(1) If we do not use the segmented strategy, when $0\le m<\frac{1}{2}\left(\begin{array}{c}n\\ j+1\end{array}\right)$, the total number of delay-decodable bits is m; when $\frac{1}{2}\left(\begin{array}{c}n\\ j+1\end{array}\right)\le m\le \left(\begin{array}{c}n\\ j+1\end{array}\right)$, the total number of delay-decodable bits is $\left(\begin{array}{c}n\\ j+1\end{array}\right)-m$.

(2) If we use the segmented strategy, when $0\le m<\left(\begin{array}{c}n-1\\ j\end{array}\right)$, the total number of delay-decodable bits is $\left(\begin{array}{c}n-1\\ j\end{array}\right)-m$; when $\left(\begin{array}{c}n-1\\ j\end{array}\right)\le m\le \left(\begin{array}{c}n\\ j+1\end{array}\right)$, the total number of delay-decodable bits is $m-\left(\begin{array}{c}n-1\\ j\end{array}\right)$.

The calculation of the total number of delay-decodable bits under segmented and non-segmented strategies is as follows:

Consider a polar code with code length $N=2^n$ and information bits $K={\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)+m\left(0,\le ,j,\le ,n,-,1,,,0,\le ,m,\le ,\left(\begin{array}{c}n\\ j+1\end{array}\right)\right)$. Depending on whether the segmented strategy is used, there are two possible scenarios given as follows::

If the non-segmented strategy is not used, it is clear that when $0\le m<\frac{1}{2}\left(\begin{array}{c}n\\ j+1\end{array}\right)$, the total number of delay-decodable bits is:

$$\begin{array}{c}\hfill min\left\{m,,,\left(\begin{array}{c}n\\ j+1\end{array}\right),-,m\right\}=m.\end{array}$$ 16

When $\frac{1}{2}\left(\begin{array}{c}n\\ j+1\end{array}\right)\le m\le \left(\begin{array}{c}n\\ j+1\end{array}\right)$, the total number of delay-decodable bits is:

$$\begin{array}{c}\hfill min\left\{m,,,\left(\begin{array}{c}n\\ j+1\end{array}\right),-,m\right\}=\left(\begin{array}{c}n\\ j+1\end{array}\right)-m.\end{array}$$ 17

If the segmented strategy is used, then:

$$\begin{array}{c}\hfill \frac{K}{2}=\frac{{\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)+m}{2}=\frac{\left(\begin{array}{c}n-1\\ 0\end{array}\right)+\left(\begin{array}{c}n-1\\ 0\end{array}\right)+\left(\begin{array}{c}n-1\\ 1\end{array}\right)+...+\left(\begin{array}{c}n-1\\ j-1\end{array}\right)+\left(\begin{array}{c}n-1\\ j\end{array}\right)+m}{2}=\sum _{i=0}^{j-1}\left(\begin{array}{c}n-1\\ i\end{array}\right)+\frac{m+\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}.\end{array}$$ 18

Note there is $m\ge 0$, leading to $\frac{m+\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}\ge \frac{\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}$. When $0\le m<\left(\begin{array}{c}n-1\\ j\end{array}\right)$, then $\frac{\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}\le \frac{m+\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}<\left(\begin{array}{c}n-1\\ j\end{array}\right)$, and the total number of delay-decodable bits is:

$$\begin{array}{c}\hfill 2\times \left\{\left(\begin{array}{c}n-1\\ j\end{array}\right),-,\frac{m+\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}\right\}=\left(\begin{array}{c}n-1\\ j\end{array}\right)-m.\end{array}$$ 19

when $\left(\begin{array}{c}n-1\\ j\end{array}\right)\le m\le \left(\begin{array}{c}n\\ j+1\end{array}\right)$, then $\frac{m+\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}\ge \left(\begin{array}{c}n-1\\ j\end{array}\right)$, resulting in:

$$\begin{array}{c}\hfill \frac{K}{2}=\frac{{\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)+m}{2}=\frac{\left(\begin{array}{c}n-1\\ 0\end{array}\right)+\left(\begin{array}{c}n-1\\ 0\end{array}\right)+\left(\begin{array}{c}n-1\\ 1\end{array}\right)+...+\left(\begin{array}{c}n-1\\ j-1\end{array}\right)+\left(\begin{array}{c}n-1\\ j\end{array}\right)+m}{2}=\sum _{i=0}^j\left(\begin{array}{c}n-1\\ i\end{array}\right)+\frac{m-\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}.\end{array}$$ 20

Note there is $\left(\begin{array}{c}n-1\\ j\end{array}\right)\le m\le \left(\begin{array}{c}n\\ j+1\end{array}\right)$, and we can thus have $0\le \frac{m-\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}\le \frac{\left(\begin{array}{c}n-1\\ j\end{array}\right)+\left(\begin{array}{c}n-1\\ j+1\end{array}\right)-\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}=\frac{\left(\begin{array}{c}n-1\\ j+1\end{array}\right)}{2}$. Therefore, in this case, the total number of delay-decodable bits is:

$$\begin{array}{c}\hfill 2\times \frac{m-\left(\begin{array}{c}n-1\\ j\end{array}\right)}{2}=m-\left(\begin{array}{c}n-1\\ j\end{array}\right).\end{array}$$ 21

Table 1 presents a comparison of the total number of delay-decodable bits for the two strategies under different circumstances, from which we draw the following conclusions:

Table 1.

Strategy recommendation based on the range of m.

the range of m	Comparison of Delayed Decoding Numbers	Recommended Strategy
$0\le m<\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)$	TDDN-SS > TDDN-NS	Segmentation Strategy
$\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)<m<\frac{1}{2}(\left(\begin{array}{c}n\\ j+1\end{array}\right)+\left(\begin{array}{c}n-1\\ j\end{array}\right))$	TDDN-SS < TDDN-NS	Non-Segmentation Strategy
$\frac{1}{2}(\left(\begin{array}{c}n\\ j+1\end{array}\right)+\left(\begin{array}{c}n-1\\ j\end{array}\right))<m<\left(\begin{array}{c}n\\ j+1\end{array}\right)$	TDDN-SS > TDDN-NS	Segmentation Strategy

TDDN-SS:Total delayed decoding numbers with segmentation strategy.

TDDN-NS:Total delayed decoding numbers with non-segmentation strategy.

When $\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)\le m\le \frac{1}{2}\left(\left(\begin{array}{c}n\\ j+1\end{array}\right),+,\left(\begin{array}{c}n-1\\ j\end{array}\right)\right)$, the total number of delay-decodable bits without using the segmented strategy is greater than or equal to the number of delay-decodable bits when using the segmented strategy; hence, it is recommended to use the non-segmented strategy for P(N, K) in this case.

When $0\le m<\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)$ or $\frac{1}{2}\left(\left(\begin{array}{c}n\\ j+1\end{array}\right),+,\left(\begin{array}{c}n-1\\ j\end{array}\right)\right)<m\le \left(\begin{array}{c}n\\ j+1\end{array}\right)$, the total number of delay-decodable bits with the segmented strategy is greater than the number of delay-decodable bits without using the segmented strategy; hence, it is recommended to use the segmented strategy for P(N, K) in these cases.

For example, when $n=6$ and $j=2$, at this time, the information bit length K varies in the following range: $K={\sum }_{i=0}^j\left(\begin{array}{c}n\\ i\end{array}\right)+m=\left(\begin{array}{c}6\\ 0\end{array}\right)+\left(\begin{array}{c}6\\ 1\end{array}\right)+\left(\begin{array}{c}6\\ 2\end{array}\right)+m=22+m\left(0,\le ,m,\le ,\left(\begin{array}{c}6\\ 3\end{array}\right),=,20\right)$. We take m as the independent variable and the total number of delay-decodable bits as the dependent variable and plot the total number of delay-decodable bits under the use of segmented strategy and non-segmented strategy as shown in Figure 3.

Fig. 3.

Total number of delay-decodable bits for two strategies in P(64,22+m) for $0\le m<20$.

It can be seen from the figure that when $0\le m<5$ or $15<m\le 20$, the total number of delay-decodable bits with the segmented strategy exceeds that without the segmented strategy.Therefore, in this case, it is recommended to use the segmented strategy proposed in this paper for the construction of polar codes. When $5\le m\le 15$, the total number of delay-decodable bits with the segmented strategy is less than or equal to that without the segmented strategy; in this case, we recommend using the non-segmented strategy for the construction of polar codes.

The following two experiments verify the above conclusions.

Experiment 1 considers the transmission code length $N=64$ and information bit length $K=22$, corresponding to $m=0$. Obviously, we have $0\le m<\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)=\frac{1}{2}\left(\begin{array}{c}5\\ 2\end{array}\right)$. Therefore, our recommendation is to use the segmented strategy. The decoding performance of polar code P(64, 22) with and without the segmented strategy is shown in Figure 2. It can be seen from this figure that under the segmented strategy, the decoding performance is effectively improved by using the delay decoding construction. When the frame error rate (FER) is at level ${10}^{-1}$, this strategy exhibits an approximate 1.7 dB performance gain over the original algorithm.

Experiment 2 considers the transmission code length $N=64$ and information bit length $K=28$, corresponding to $m=6$. Obviously, at this time, we have $\frac{1}{2}\left(\begin{array}{c}n-1\\ j\end{array}\right)=5\le m\le \frac{1}{2}\left(\left(\begin{array}{c}n\\ j+1\end{array}\right),+,\left(\begin{array}{c}n-1\\ j\end{array}\right)\right)=15$ is satisfied, and our recommendation is to use the non-segmented strategy. Figure 2 provides a performance comparison of polar code P(64, 28) between using segmentation strategy and not using segmentation strategy. It can be seen from this figure that in the medium and low signal-to-noise ratio region, the performance without using the segmented construction is better; while in the high signal-to-noise ratio region, the performance with the segmented strategy and using delay decoding construction is better. Overall, it is better to use the non-segmented strategy in this case.

The above LSD construction selection strategy is described in Algorithm 1.

Algorithm 1.

Proposed construction method for LSD

Algorithm 2.

Improved CA-LSD with proposed strategy

Improved Cyclic Redundancy Check-Aided List Sphere Decoding (CA-LSD)

In this section, we propose the integration of segmented CRC bits within the segmented construction to introduce an early stopping mechanism. Our aim is to reduce the decoding complexity to a certain extent while still benefiting from the performance gains associated with the segmented construction strategy. The specific description of the algorithm is as follows: Place the CRC bits at the end of the segmented information sequence in the CA-LSD decoding process. If the first segment of the codeword fails the CRC check, we terminate the decoding process early and abandon the transmission of subsequent codewords. The system block diagram of the improved CA-LSD algorithm is shown in Figure 4, and the specific decoding process is described in Algorithm 2.

Fig. 4.

Flowchart of the segmented CA-LSD decoding early stopping strategy.

Let $P(N,k+r)$ represent the polar code concatenated with CRC bits, where k is the length of the information bits, and r is the length of the CRC check bits. For example, for $P(64,16+8)$, if constructed without the segmented strategy under Gaussian approximation, the number of delay-decodable bits is 3, located in the 51 st, 53rd, and 57th bit positions, which can be delayed to the 8th, 12th, and 14th bit positions for decoding, respectively; if the segmented strategy is used, we divide the 8-bit CRC check bits into two segments and place them at the end of the two information sequences for transmission. If the front segment of the codeword fails the CRC check, we will terminate the decoding process early. Note that, in this way, the total number of delay-decodable bits increases from 3 to $\left(\begin{array}{c}n-1\\ j\end{array}\right)-m=\left(\begin{array}{c}5\\ 2\end{array}\right)-2=8$.

Performance and complexity evaluation

This section examines the FER performance characteristics of polar codes constructed using the algorithm presented in this paper over a binary input additive white Gaussian noise channel (BI-AWGNC) under different code lengths and rates through simulations. The total number of frames for our experimental simulation is $T\_total=10_{}^6$, with the termination condition being when the number of error frames exceeds 2, 000 or the total number of frames exceeds $T\_total$.

Performance analysis

Consider the polar codes P(64, 22), P(64, 24), and P(32, 20) with code rates of 11/32, 12/32, and 20/32, respectively. Figures 5, 6, and 7 illustrate the FER performance of these three codes, where ‘Proposed’ denotes the implementation of a delayed decoding strategy, and ‘early stop’ signifies the incorporation of an early termination mechanism.We have the following observations.

• When the code length N is 64 and the list size Lis 32, the CA-LSD algorithm, which is based on the delayed decoding segmented construction proposed in this paper, performs comparably to the SD algorithm at low code rates (within a 0.2 dB difference). It also matches the performance of the improved SD algorithm in ref.¹⁰ and outperforms the improved LSD algorithm in ref.¹⁴. Moreover, by eliminating the backtracking process and decoding radius of the SD algorithm, the time complexity is adjusted to $O(L{N}^2)$, which is in consistent with the derivation in ref.²⁰.

• With the inclusion of the early stopping strategy, if the first segment of the codeword fails the CRC check, the transmission of the second segment is aborted. Simulations demonstrate that its performance is on par with that of the original algorithm (within a 0.1 dB difference).

• Considering a code length N of 32 and non-frozen bits K of 20, at a frame error rate (FER) equal to ${10}^{-2}$, the CA-LSD algorithm, which is based on the delayed decoding construction, achieves approximately a 1.4 dB performance gain over the original CA-LSD algorithm, a 1 dB gain over the SD algorithm, and a 0.2 dB gain over the CA-SCL algorithm.This indicates that through the delayed decoding construction strategy, the improved CA-LSD algorithm can also perform well at high code rates. These findings highlight the potential of short polar codes in scenarios where low latency and high reliability are critical, such as in URLLC applications and resource-constrained devices.

Fig. 5.

FER performance of P(64,22).

Fig. 6.

FER performance of P(64,24).

Fig. 7.

FER performance of P(32,20).

Complexity analysis

For the delayed decoding construction scheme, once the transmission code length and the number of non-frozen bits are determined, its encoding complexity is fixed. The number of non-frozen bits with the smallest row weight is determined based on the structure of the generator matrix. The number of frozen bits with the same row weight is identified. By comparing the two values mentioned above, the algorithm in this paper selects the smaller one as the number of delay-decodable bits. Therefore, the construction method proposed in this paper has lower complexity O(N). Additionally, to determine whether a segmented strategy is needed, preliminary recommendations need to be made for scenarios with different code lengths and numbers of non-frozen bits.

In the proposed early stopping strategy, we employ a specific metric $AverageListSize=\frac{ActualTotalNumberofDecodings}{CodeLength}$ to measure the complexity optimization brought about by the early stopping mechanism²¹. Figure 8 presents the average decoding complexity of P(64, 24) with list lengths $L=32$ and $L=16$. We have the following observations from the data presented in this figure.

• Given decoding list length $L=16$ and the signal-to-noise ratio to be 0.5 dB, the complexity of the decoder with the early stopping mechanism is reduced by 7.61% compared to the original scheme.

• Given decoding list length $L=32$ and the signal-to-noise ratio to be 0.5 dB, the complexity of the decoder with the early stopping mechanism is reduced by 0.99% compared to the original scheme. At the same time, the performance loss is within an acceptable range.

Fig. 8.

Average list lengths of CALSD P(64,24) decoders with different list sizes.

Conclusion

This paper presents a construction method referred to as delayed decoding, which enhances the performance of LSD by delaying the decoding of bits with the minimum row weight to mitigate the impact of error propagation. Based on the presence or absence of bits with the same minimum row weight as the information bits among the frozen bits, we have conducted a detailed discussion and proposed an improved strategy for segmented construction. Simulation results indicate that, by employing the segmented construction method combined with the delayed decoding strategy proposed in this paper, the CA-LSD algorithm exhibits decoding performance that is comparable to the SD algorithm at medium and low code rates, but it eliminates the concept of the search radius in the SD algorithm and does not require a backtracking process. Furthermore, under high code rate conditions, this scheme is also able to maintain excellent performance. Besides, it should be noted that the segmented strategy proposed in this paper only transmits the next segment of data after the first segment of the codeword passes the CRC check, and therefore, it is only suitable for communication scenarios that do not require real-time feedback.

Author contributions

Conceptualization: W.H., and H.C.. Methodology: W.H., and R.W.. Software and Validation: W.H., H.C., R.W., Q.G., Y.S., and X.L.. Writing (original draft): W.H., and R.W.. Writing (review and editing): H.C., and S.D..

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-15936-3.