This paper presents a new structure of product codes formed by combining two polar codes. The encoding performance of these codes is verified by implementing an exhaustive search algorithm, which determines their minimum weight specifications. Conducted analysis and simulations confirm that with the equal code length and rate, the newly proposed codes outperform the conventional polar codes in high energy per bit-to-noise ratios (

Polar codes are promising forward error correction (FEC) codes for the next generation of broadband networks including the fifth generation of wireless networks (5G). These codes are recognized as capacity-achieving codes under a successive cancellation (SC) decoding technique, which were initially constructed for the symmetric binary-input discrete memoryless channel (BI-DMC) [

A serially concatenated code was proposed based on an inner polar code and an outer Reed-Solomon (RS-polar) code [

On the other hand, construction of a parallel concatenated code formed by two polar codes was proposed in [

The main similarity between the above-mentioned codes is the serial combination of a polar code with a nonpolar code. On the other hand, a turbo product code is based on multiple polar codes (polar-TPC) introduced in [

This paper presents a new scheme of serially concatenated codes constituted by two systematic polar codes linked by an interleaver. Codes are constructed on the basis of the message, whose length is the product of the length of constituent codes. This code can be also introduced as a polar code, as every bit of the codeword is properly polarized. Moreover, the puncturing structure of these codes will be verified. Unlike a punctured single polar code, which applies a relatively high number of punctured bits, punctured product polar codes can be constructed by removing a low number of bits applied in one of the constituent codes. This feature will allow decoders to interactively exchange their decoded information with each other and improve the error correcting capability of the code. Conducted simulations confirm that with the same rate and code length, punctured product polar codes have better performance than single punctured polar codes. In addition, it is analysed that the proposed code decoded with BP decoding has similar performance to codes decoded with the SCL-Chase-based algorithm.

The rest of the paper is organized as follows. A brief introduction of polar codes is given in Section

Let

Let

In systematic polar encoding, the codeword is split into two parts [

The minimum weight (

Minimum weight specifications of polar codes with rates 1/2 and 1/4.

Polar code | Polar code | ||
---|---|---|---|

(8, 4) | 4 (14) | (8, 2) | 4 (2) |

(16, 8) | 4 (28) | (16, 4) | 8 (14) |

(32, 16) | 4 (8) | (32, 8) | 8 (12) |

(64, 32) | 8 (662) | (64, 16) | 16 (364) |

(128, 64) | 8 (1724) | (128, 32) | 16 |

(256, 128) | 8 (2330) | (256, 64) | 16 |

(512, 256) | 8 | (512, 128) | 32 |

(1024, 512) | 8 | (1024, 256) | 32 |

(2048, 1024) | 16 | (2048, 512) | 64 |

(4096, 2048) | 16 | (4096, 1024) | 64 |

(8192, 4096) | 16 | (8192, 2048) | 64 |

As a class of linear block codes, performance of polar codes can be analysed by its Input-output weight enumerating function (IOWEF), which is given by [

This leads to expressing the upper bound of the probability of error for the code decoded by a maximum likelihood decoding technique as follows:

Here,

Let

The IOWEF of (8, 4) systematic polar code is given by

The CWEF of this code is

From (

Considering (

Figure

Relative contribution of different weights to the BER performance of (16, 8) and (32, 16) polar codes.

Relative contribution of different weights to the BER performance of (32, 8) and (64, 16) polar codes.

Product codes are represented as one of the most well-known codes, whose high error-correcting capability is guaranteed due to their high minimum weight. Figure

(a) Two-dimensional codeword structure of product polar codes. (b) The encoding structure of product polar codes.

The product code can be viewed as a serially concatenated code, which is formed by

The behaviour of the product code allows us to analyse its performance by utilizing the uniform interleaver with length

Let

The third term is expressed as

In product polar codes with the

For

Similarly,

As shown in Section 1.1, codewords with the minimum weight have the most effect on the performance of the code at the medium to high signal-to-noise ratios. Therefore, equation (

As the lowest order of

On the other hand, the CWEF of a single polar code with rate

Based on

Minimum weight specifications of polar codes with rates 1/2 and 1/4.

Product code | Exponential form of |
||||
---|---|---|---|---|---|

32 | 4 | 35960 | |||

64 | 4 | 635376 | |||

128 | 4 | ||||

512 | 4 | ||||

2048 | 8 | ^{1} |

In particular cases, multiplicity of minimum weight of the product code can be similar or even smaller than the multiplicity of the minimum weight of the single code. This will provide a better condition for the product code to outperform the single code at the error floor region.

Figure

Probability of error for (64, 16) polar code.

The codeword obtained from the product polar encoding can be expressed as a

Table

Reliability of bits for (16, 4) polar codes.

Product polar code | (16, 4) single polar code | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

First encoding | Second encoding | ||||||||||

0.84 | 0.25 | 0.36 | 0.018 | 0.9745 | 0.4410 | 0.5911 | 0.0363 | 0.9994 | 0.6875 | 0.8328 | 0.0713 |

0.84 | 0.25 | 0.36 | 0.018 | 0.7062 | 0.0637 | 0.1300 | 0.0003 | 0.9137 | 0.1233 | 0.2430 | 0.0007 |

0.9745 | 0.4410 | 0.5911 | 0.0363 | 0.9497 | 0.1945 | 0.3493 | 0.0013 | ||||

0.7062 | 0.0637 | 0.1300 | 0.0003 | 0.4987 | 0.0041 | 0.0169 |

This table also gives the reliabilities of bit channels for the single (16, 4) polar code. Similar to the single code, the product code effectively polarizes the channel as either good or bad channels. Such recognition allows us to construct polar codes with arbitrary lengths and

A

In order to have

This means a lower number of bits punctured from one of the constituent codes are required in comparison with the number of punctured bits applied for the single code.

Based on the two-dimensional structure of the codeword,

On the other hand, based on transposing the received information, at the second decoder, some adjacent information is recognized as punctured information. However, as shown in Figure

Structure of interactive iterative decoding of product codes.

By contrast, in a single polar code formed by a high number of punctured bits, it is possible to puncture adjacent bits or those having high reliability. This is mainly evident for codes with

The reliability of the bit channel can effect error performance of the punctured code. Puncturing a bit ultimately deletes its bit-channel reliability. This consequently reduces the overall bit-channel reliability (

As mentioned above, the

Therefore, if

As mentioned before, in product polar codes, a lower of number of bits are punctured from their constituent codes compared to a single punctured code. Hence, less effect of punctured bits on the performance of product code compared to the single code is expected. Table

Comparison of

Product code | Single code | |||
---|---|---|---|---|

48 | 23.5443 | 10.0989 | 23.5443 | 8.0056 |

192 | 94.1771 | 40.3955 | 94.1771 | 30.5400 |

768 | 376.7085 | 161.5818 | 376.7085 | 120.8185 |

3072 | 646.3274 | 482.8405 | ||

12288 |

The bit error rate (BER) and frame error rate (FER) performances of different product and single polar codes transmitted over the AWGN channel are verified. Codes are modulated by BPSK and decoded by the soft-in soft-out belief propagation method mentioned in [

Figure

Performance of (64, 16) codes decoded by different algorithms.

Figure

Performance of product polar codes and their comparison with BCH-polar, RS-polar, and LDPC-polar codes.

Figure

BER and FER performance of (800, 256) product polar code and its comparison with other codes.

Figure

BER and FER performances of (3196, 1024) product and (3072, 1024) single polar codes.

Furthermore, BER performance of the (256^{2}, 239^{2}) product polar code and polar-TPC proposed in [

BER performance of (256^{2}, 239^{2}) codes.

The paper presented a product coding scheme of polar codes formed by two different rates of polar codes. The conducted analysis and simulation results confirmed that the constructed product codes outperform the conventional polar codes. This is mainly evident in the medium to high signal-to-noise ratios, which is applicable for both nonpunctured and punctured codes. In the future work, a modification of the structure of punctured product polar codes will be followed aimed at improving their performance.

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work is supported by the Research Training Program (RTP) scholarship under Charles Darwin University.