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Register a new userRate-Compatible Punctured Polar Codes: Optimal Construction Based on Polar Spectra
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Added by
Jincheng Dai
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Polar codes are the first class of constructive channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels. But the corresponding code length is limited to the power of two. In this paper, we establish a systematic framework to design the rate-compatible punctured polar (RCPP) codes with arbitrary code length. A new theoretic tool, called polar spectra, is proposed to count the number of paths on the code tree with the same number of zeros or ones respectively. Furthermore, a spectrum distance SD0 (SD1) and a joint spectrum distance (JSD) are presented as performance criteria to optimize the puncturing tables. For the capacity-zero puncturing mode (punctured bits are unknown to the decoder), we propose a quasi-uniform puncturing algorithm, analyze the number of equivalent puncturings and prove that this scheme can maximize SD1 and JSD. Similarly, for the capacity-one mode (punctured bits are known to the decoder), we also devise a reversal quasi-uniform puncturing scheme and prove that it has the maximum SD0 and JSD. Both schemes have a universal puncturing table without any exhausted search. These optimal RCPP codes outperform the performance of turbo codes in LTE wireless communication systems.
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We present a rate-compatible polar coding scheme that achieves the capacity of any family of channels. Our solution generalizes the previous results [1], [2] that provide capacity-achieving rate-compatible polar codes for a degraded family of channels. The motivation for our extension comes from the fact that in many practical scenarios, e.g., MIMO systems and non-Gaussian interference, the channels cannot be ordered by degradation. The main technical contribution of this paper consists in removing the degradation condition. To do so, we exploit the ideas coming from the construction of universal polar codes. Our scheme possesses the usual attractive features of polar codes: low complexity code construction, encoding, and decoding; super-polynomial scaling of the error probability with the block length; and absence of error floors. On the negative side, the scaling of the gap to capacity with the block length is slower than in standard polar codes, and we prove an upper bound on the scaling exponent.
In this paper, we show some applications of algebraic curves to the construction of kernels of polar codes over a discrete memoryless channel which is symmetric w.r.t the field operations. We will also study the minimum distance of the polar codes proposed, their duals and the exponents of the matrices used for defining them. All the restrictions that we make to our curves will be accomplished by the so-called Castle Curves.
We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a~function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within $varepsilon > 0$ of capacity, the code length $n$ often scales as $O(1/varepsilon^{mu})$, where the constant $mu$ is called the scaling exponent. It is known that the optimal scaling exponent is $mu=2$, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the $2times 2$ kernel) on the BEC is $mu=3.63$. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist $elltimesell$ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent $mu(ell)$ that tends to the optimal value of $2$ as $ell$ grows. We furthermore characterize precisely how large $ell$ needs to be as a function of the gap between $mu(ell)$ and $2$. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity $O(n)$ and encoding/decoding complexity $O(nlog n)$.
Polar codes with memory (PCM) are proposed in this paper: a pair of consecutive code blocks containing a controlled number of mutual information bits. The shared mutual information bits of the succeeded block can help the failed block to recover. The underlying polar codes can employ any decoding scheme such as the successive cancellation (SC) decoding (PCM-SC), the belief propagation (BP) decoding (PCM-BP), and the successive cancellation list (SCL) decoding (PCM-SCL). The analysis shows that the packet error rate (PER) of PCM decreases to the order of PER squared while maintaining the same complexity as the underlying polar codes. Simulation results indicate that for PCM-SC, the PER is comparable to (less than 0.3 dB) the stand-alone SCL decoding with two lists for the block length $N=256$. The PER of PCM-SCL with $L$ lists can match that of the stand-alone SCL decoding with $2L$ lists. Two hardware decoders for PCM are also implemented: the in-serial (IS) decoder and the low-latency interleaved (LLI) decoder. For $N=256$, synthesis results show that in the worst case, the latency of the PCM LLI decoder is only $16.1%$ of the adaptive SCL decoder with $L=2$, while the throughput is improved by 13 times compared to it.
In this paper, we propose textit{selectively precoded polar (SPP) code}, built on top of Arikans capacity achieving polar codes. We provide the encoding and decoding scheme for SPP code. Simulation results show that for a target frame erasure rate (FER) of $mathbf{10^{-5}}$, a (128, 64) SPP code is just 0.23 dB away from the information theoretic limit at this blocklength. Further, it is also shown that such codes possess better distance properties compared to other contemporary polar code variants.
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