Binary polar codes, proposed by Arıkan , were proved to be able to achieve the symmetric-capacity of binary-input discrete memoryless channels (DMCs) with explicit coding structures as well as low encoding and decoding complexity. Polarization effect exists by using nested transformations with the binary kernel (called Arıkan’s kernel) in conjunction with a successive cancellation (SC)-based decoder. As the code length tends to infinity, bit-channels polarize to be either nearly noiseless or useless, and the capacity can be approached by employing only the noiseless bit-channels for data transmission. It was shown that bit-channels can also be polarized with such a generator matrix that none of its column permutations is an upper triangle matrix 
. Thus the original polar codes can be generalized by applying different kernels. The probability of block error for polar codes under SC decoding is found to befor any , where denotes the exponent of the kernel matrix . The exponent of Arıkan’s kernel is , which can be improved by considering larger or nonbinary matrices . Therefore, BCH-generator-matrices as polarization kernels were considered in , which yields an exponent greater than at size . Binary kernels with maximum exponents of dimensions up to sixteen were provided in .
Another way to increase the exponent is to extend binary polar codes to a nonbinary setting. In , Şaşoǧlu et al. proved that nonbinary polar codes with arbitrary finite input-alphabet sizes can achieve symmetric-capacity by decomposing underlying symbol channels into a set of subchannels with prime input alphabet sizes, and they also showed that all DMCs can be polarized by randomized constructions. In 2014, Chiu  proposed a new approach, by using channel symbol permutations, to prove that polar codes polarize arbitrary -ary input randomized channels. Nonbinary polar codes based on an -ary Reed-Solomon (RS) matrix were proposed by Mori and Tanaka , which have exponents of order for all and can be made arbitrarily close to 1 as becomes large. In 2016, Cheng  et al. applied a four-dimensional RS matrix over the finite field GF(4) as the kernel and showed that better error-correcting performance can be achieved compared with binary polar codes.
In this paper, we propose a new class of nonbinary polar codes with two-stage polarization. The outer polarization for symbol-level, referred to as multiplicative repetition-based construction, is based on a -ary matrix , where acts as a multiplier and may vary for nested Kronecker operations. The inner polarization for bit-level, referred to as polarization mapping, is then applied to each coded symbol to further intensify the polarization. With this two-stage polarization, a -ary input channel is treated as several binary input channels. As a result, partially-frozen symbols may arise. That is, there may exist symbols containing both frozen bits and active bits. This is different from existing constructions, in which a symbol is either fully-frozen or fully-active. An initial motivation of such a hybrid construction is to disperse information bits to more symbols, which may benefit from high order modulations. The partially-frozen symbol can be further utilized to improve the error-correcting performance by an active-check construction. An efficient successive cancellation list (SCL) decoding is also introduced for the proposed polar codes, which has a similar structure to that of the Arıkan’s polar codes , . This is an obvious advantage over the existing RS-generator-matrix based polar codes, which require much more computations than ours due to their very different decoding structure from that of the original polar codes. In addition, with the proposed code structure, lower latency can be achieved compared with binary polar codes. Simulation results show that the proposed codes perform better than binary polar codes with BPSK. Simulation results also show that the performance gain can be increased with the increase of the field order. Moreover, the proposed codes also exhibit an outstanding performance with high-order modulations.
Ii System Model
Let GF() be the finite field with , where is a positive integer greater than unity. Let be a primitive element of GF(). Then form all the elements of GF(). Each element GF() can be represented by an -tuple over GF.
The system model under consideration is depicted in Fig. 1. The -ary polar encoder accepts as input a -ary sequence of length or, equivalently, a binary sequence of length . This input sequence is formed as follows. First, a binary sequence of length is encoded by a CRC-encoder; the checked sequence is then formatted as a binary sequence of length by inserting properly frozen bits and active-check bits according to the code construction; finally, the binary sequence is converted into the symbol sequence with GF() () as usual. To intensify the polarization, a polarization mapper is applied to each coded symbol GF() () before the signal mapping. Apparently, the overall code rate .
Suppose that a -dimensional signal constellation of size is used. A signal mapper maps the polar coded symbols to modulated symbols . Without loss of generality, we assume that is a multiple of
. If necessary, zero bits are padded to the coded symbols. With the assumption thatare transmitted over the AWGN channel, the received symbols are given by
where andper dimension. At the receiver, the demapper computes posteriori probabilities (APPs) and delivers them to the SCL decoder. In the SCL decoder, best paths are considered at each decoding stage. Finally, the
candidates from SCL decoder are checked by a CRC decoder successively, and the one with the best metric among the candidates which can pass the CRC detection is determined as the estimate.
Denote the average transmitted energy per symbol by . Then the average received SNR is given by SNR = , where denotes the average energy per information bit.
Iii Nonbinary Polar Codes with Two-Stage Polarization
Iii-a Symbol-Level Polarization with MR Construction
We extend binary polar codes to nonbinary polar codes by considering a -ary kernel given by
where GF(). With as the generator matrix, for each GF(), we have the coded sequence . In the case of , has the same form as Arıkan’s, and is repeated once and then superimposed on . In the nonbinary case with , we see is a multiplicative repetition of . For this reason, we call the construction multiplicative repetition (MR)-based construction for convenience.
Let and represent the multiplier for the -th Kronecker operation. Then the generator matrix with the code length is given by
where is a permutation matrix for bit-reversal operation, and the multiplier () is randomly chosen from the nonzero elements set of GF(). All multipliers are known both to the transmitter and to the receiver.
The MR-based construction is similar to the permutation construction proposed in . Thus, with the Proposition 1 in , we can prove (omitted here due to space limitation) that the MR-based transformation polarize -ary discrete memoryless channels. This is referred to as symbol-level polarization.
Iii-B Bit-Level Polarization with Polarization Mapping
Different from conventional nonbinary polar codes, in which only symbol-level polarization is discussed and the construction is based on the symbol-level, we introduce the bit-level polarization by a polarization mapping to each coded symbol.
The mapping pattern is based on an binary polarization matrix with the maximum polarization rate among the matrices of the same size. Decompose each -ary input channel into bit-channels by treating each coded symbol GF(
) as a binary vector. With the matrix , each coded symbol is mapped to . For example, assuming that and , we have . With this process, bits in the same symbol become dependent. Similar to binary polar codes, it can be proved that polarization exists in the decomposed bit-channels by regarding the mapping process as a binary encoding with the polarization matrix. It is worth noting that, without the polarization mapping, the choice of information index set via bit-channel reliabilities will be equivalent to that via symbol-channel reliabilities.
Iii-C Code Construction
Our code construction is different from the conventional -ary code construction in the sense that we sort bit-channel reliabilities. This effective construction is attributed to the existence of the polarization mapping. Assume that , GF(), represents the channel conditional probabilities given by the demapper. Due to the similar structure to binary polar codes, the recursive formulas for the SC decoding are given as
where , , , and .
A genie-aided SC decoder is used to compute the bit-channel reliabilities for -ary polar codes with , where information bits are uniformly generated. For and , denote . The -th bit-channel within the -th symbol-channel is then referred to as the -th bit-channel, whose reliability is computed under the assumption that symbols () are available at the SC decoder. Denote the index set of bit-channels with the largest capacities by , whose size is determined by the code rate. The bit-channels indexed by will be used to transmit active bits. Given this, frozen bits will be typically distributed as shown in Fig. 2(b).
We distinguish the symbols into three types, i.e., information symbols, partially-frozen symbols, and frozen symbols in terms of the number of active bits in a symbol. Both information symbols and partially-frozen symbols are known as unfrozen symbols, collectively denoted by . Then denote the set of frozen symbol indices, where for any , all corresponding .
The frozen bits in partially-frozen symbols can be used as active-check bits, facilitating the SCL decoder to detect and prune error paths in time. For this end, consider the symbols with only one active bit, referred to as one-bit symbols, collectively denoted by . Denote by and () the bit-channel indices with lowest reliability and highest reliability for the -th symbol, respectively. Let , followed by , the next adjacent one-bit symbol (if any). We retransmit the -th bit (active bit in the -th symbol) once as an active-check bit at the -th bit-channel. Then, at the decoder, the -th symbol is split by considering the constraint that the -th bit is equal to the -th decoded (estimated) bit. We call such a construction active-check construction.
Iii-D Decoding Algorithm
We consider CRC-aided SCL decoding for the proposed -ary polar codes. Assume that the maximum decoding paths are kept at each decoding stage. Since a partially-frozen symbol can only take values from a subset of GF(), the decoding path for an unfrozen symbol is split into at most branches. Then we choose the most likely paths from all the possible extended paths () according to their path metrics. After SCL decoding, the decoder outputs the estimated information sequence given by the decoding path with the maximum path metric among the paths which can pass the CRC. The main CRC-aided SCL decoding algorithm for two-stage polarization-based polar codes is shown in Algorithm 1.
Iii-E Decoding Latency
Let us first consider the latency of SC decoding. Denote by functions and the node update equations (2) and (3), respectively. According to , for binary polar codes with , clock cycles (CLKs) are required for the SC decoder, where both functions and can be carried out in a single CLK. For the proposed -ary polar codes, CLKs are needed for the function due to additions for each unit , and the number of CLKs required for the function is the same as that of binary cases. Thus, a total of CLKs are required for the proposed -ary SC decoder.
An -size SCL decoder can be viewed as the combination of copies of SC component decoders. In addition, the SCL decoder needs to sort (for worst case) path metrics and selects the largest metrics for each decoded bit, thus, extra CLKs are required to carry out sorting and selecting functions. Using Bitonic sorter , the total number of stages is , where each stage contains compare-and-select (CAS) units consisting of one comparator and a 2-to-2 MUX. Generally, an intermediate variable (or register) is required to swap two numbers resulting in 3 CLKs . Since only unfrozen symbols require sorting and selecting functions, at most CLKs are needed for -ary polar decoder to sort and select paths. For binary decoder, the number of unfrozen bits is . The CLKs required by SC-decoder and SCL-decoder for binary and -ary polar codes are tabulated in Table I for comparison.
Iv Numerical Results and Analysis
In this section, error-correcting performances are provided for the proposed nonbinary (NB) polar codes, where both BPSK and high-order modulations over AWGN channels are considered.
The polarization matrices
are applied for the polarization mapping of GF, GF, GF and GF, respectively.
Example 1: Consider the proposed -ary polar codes with BPSK over the AWGN channel. Two rate- codes with equivalent code length of =512 and =768 are simulated. In Fig. 3(a), both -ary polar codes and -ary polar codes are constructed by Monte-Carlo method at dB. In Fig. 3(b), both -ary polar codes and -ary polar codes are constructed by Monte-Carlo method at dB. The FER performance are given under both SC decoding and 8-bit CRC-aided SCL decoding method.
For reference, the performance of comparable binary polar codes, constructed by GA method at dB, are also given in Fig. 3. The quasi-uniform puncturing (QUP) method in  is used to adapt the code length.
From Fig. 3, we can see that our proposed nonbinary polar codes outperform binary polar codes with BPSK over the AWGN channel. The coding gain is increased with the increase of the field order. Moreover, from Table I, it can be checked that the latency of our proposed nonbinary polar codes is lower than that of binary cases, and that can be decreased with the increase of the field order.
Example 2: Consider the proposed -ary polar codes with -ary modulations over the AWGN channel. Rate-1/2 codes with two modulation schemes -QAM and -QAM are simulated. In Fig. 4(a), the proposed -ary polar codes with are constructed by Monte-Carlo method at dB. In Fig. 4(b), the proposed -ary polar codes with are constructed by Monte-Carlo method at dB. The 16-bit CRC-aided SCL decoding with various list sizes are applied.
For comparison, the performance of comparable modified RS-based -ary polar codes in  are also given in Fig. 4, where the codes are constructed by Monte-Carlo method at the aforementioned design-SNRs. Also considered for comparison are bit-interleaved coded-modulation (BICM) scheme  and multilevel coding (MLC) scheme  using comparable binary polar codes with -ary QAM, where the codes are constructed by Monte-Carlo method at dB for -QAM and dB for -QAM. For the BICM-based scheme, the bit-interleaver designed in  is applied, and the QUP method is used to adapt the code length. For the MLC-based scheme, individual coding levels are employed. The rate allocations are obtained via the capacity rule in . The SP labeling is applied to the MLC-based scheme, and Gray labeling is employed to other schemes.
From Fig. 4, we see that our proposed nonbinary polar codes combined with high-order modulations exhibit better performance than BICM-based scheme and can achieve similar performance but with a smaller list size compared with modified RS-based polar codes. Under -QAM, compared with MLC-based scheme, about dB gain can be obtained by our proposed polar codes with at FER.
We have presented a new class of nonbinary polar codes constructed with two-stage polarization, where the symbol-level polarization is based on a -ary kernel, which is a variation of Arıkan’s kernel, and the bit-level polarization is considered by using a polarization mapping with a designed binary matrix. Simulation results show that the proposed nonbinary polar codes perform better than binary polar codes with BPSK and high-order modulations over the AWGN channel. Furthermore, compared with modified RS-based polar codes, the proposed polar codes can achieve similar performance but with a smaller list size for high-order modulations.
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