Plotkin Construction Research Articles

Polar coded cooperative with Plotkin construction and quasi‐uniform puncturing based on MIMO antennas in half duplex wireless relay network

AbstractRecently, polar code has attracted the attention of many scholars and has been developed as a code technology in coded‐cooperative communication. We propose a polar code scheme based on Plotkin structure and quasi‐uniform punching (PC‐QUP). Then we apply the PC‐QUP to coded‐cooperative scenario and built to a new coded‐cooperative scheme, which is called PCC‐QUP scheme. The coded‐cooperative scheme based on polar code is studied on the aspects of codeword construction and performance optimization. Further, we apply the proposed schemes to space–time block coding (STBC) to explore the performance of the scheme. Monte Carlo simulation results show that the proposed cooperative PCC‐QUP‐STBC scheme can obtain a lower bit error ratio (BER) than its corresponding noncooperative scheme.

Recursive Decoding of Reed-Muller Codes Starting With the Higher-Rate Constituent Code

Recursive list decoding of Reed-Muller (RM) codes, with moderate list size, is known to approach maximum-likelihood (ML) performance of short length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\leq 256)$ </tex-math></inline-formula> RM codes. Recursive decoding employs the Plotkin construction to split the original code into two shorter RM codes with different rates. In contrast to the standard approach which decodes the lower-rate code first, the method in this paper decodes the higher-rate code first. This modification enables an efficient permutation-based decoding technique, with permutations being selected on the fly from the automorphism group of the code using soft information from a channel. Simulation results show that the error-rate performance of the proposed algorithms, enhanced by a permutation selection technique, is close to that of the automorphism-based recursive decoding algorithm with similar complexity for short RM codes, while our decoders perform better for longer RM codes. In particular, it is demonstrated that the proposed algorithms achieve near-ML performance for short RM codes and for RM codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula> and order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m-3$ </tex-math></inline-formula> with reasonable complexity.

Polar Coded Cooperative Differential Spatial Modulation Based on Plotkin Construction and Quasi-Uniform Puncturing

Coded-cooperation combines virtual MIMO transmission and channel coding to achieve forward error correction while achieving diversity gain. It is a promising technology in 5G communications. The polar code is the only channel coding technology that can theoretically achieve Shannon limit transmission, which make it the preferred coding technology in cooperative communication. In this paper, we proposed a new wireless communication scheme, which is a polar coded cooperative spatial modulation schemes based on Plotkin construction and Quasi-uniform puncturing (QUP). First, we adopt Plotkin construction based on the inverted codeword of the polar code to get a cooperation method, which offer a significant improvement in performance on the polar code cooperation under fading channels. We analyze the decoding characteristics of the inverted codewords and then introduces the QUP method to improve the cooperation benefits. At the same time, we extend the cooperation scheme with differential spatial modulation for the proposed coded cooperative schemes. The new coded-cooperation scheme outperforms its coded non-cooperative counterpart schemes under identical conditions, as expected. Monte Carlo simulated results show that the proposed cooperative schemes outperform their corresponding non-cooperative counterpart schemes, by the gain of 2.5 dB with 1024 code length and 0.25 code rate under identical conditions.

Concatenated Codes Based on the Plotkin Construction and Their Soft-Input Decoding

Reed-Muller (RM) codes have recently regained some interest in the context of low latency communications and due to their relation to polar codes. RM codes can be constructed based on the Plotkin construction. In this work, we consider concatenated codes based on the Plotkin construction, where extended Bose-Chaudhuri-Hocquenghem (BCH) codes are used as component codes. This leads to improved code parameters compared to RM codes. Moreover, this construction is more flexible concerning the attainable code rates. Additionally, new soft-input decoding algorithms are proposed that exploit the recursive structure of the concatenation and the cyclic structure of the component codes. First, we consider the decoding of the cyclic component codes and propose a low complexity hybrid ordered statistics decoding algorithm. Next, this algorithm is applied to list decoding of the Plotkin construction. The proposed list decoding approach achieves near-maximum-likelihood performance for codes with medium lengths. The performance is comparable to state-of-the-art decoders, whereas the complexity is reduced.

Distributed Reed-Muller Coded-Cooperative Spatial Modulation

This paper proposes distributed Reed-Muller coded spatial modulation (DRMC-SM) scheme based on Kronecker product (KP) construction. This special construction enabled an effective distribution of classical Reed-Muller (RM) code along source and relay nodes. The proposed DRMC-SM scheme not only offers resilience in bit error rate (BER) performance but also enhances the spectral efficiency due to additional antenna index transmission inculcated by spatial modulation (SM). The usefulness of KP construction over classical Plotkin (CP) construction in coded-cooperation is analysed with and without incorporating SM. An efficient criteria for selecting the optimum bits is adopted at relay node which eventually results in better weight distribution of the mutually constructed (source and relay) RM code. The numerical results show that proposed KP construction outperforms CP construction by 1 dB in signal to noise ratio (SNR) at bit error rate (BER) of $$\,7\times 10^{-7}$$ . Moreover, the proposed DRMC-SM scheme outperforms its non-cooperative Reed-Muller coded spatial modulation scheme as well as distributed turbo coded spatial modulation scheme. This prominent gain in SNR is evident due to the path diversity, efficient selection of bits at the relay node and joint soft-in-soft-out (SISO) RM decoding employed at the destination node.

On the Efficiency of Polar-Like Decoding for Symmetric Codes

The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures based on the Plotkin construction. The decoding algorithm can be applied with slight modifications to Reed-Muller or eBCH codes, that both achieve the capacity of erasure channels, although the list size needed for good performance grows too fast to make the decoding practical even for moderate block lengths. The key ingredient for proving the capacity-achieving property of Reed-Muller and eBCH codes is their group of symmetries. It can be plugged into the concept of Plotkin decomposition to design various permutation decoding algorithms. Although such techniques allow to outperform the straightforward polar-like decoding, the complexity stays impractical. In this paper, we show that although invariance under a large automorphism group is valuable in a theoretical sense, it also ensures that the list size needed for good performance grows exponentially. We further establish the bounds that arise if we sacrifice some of the symmetries. Although the theoretical analysis of the list decoding algorithm remains an open problem, our result provides an insight into the factors that impact the decoding complexity.

Sphere Constraint Based Enumeration Methods to Analyze the Minimum Weight Distribution of Polar Codes

As an important performance metric, the minimum weight distribution (MWD) determines the maximum-likelihood decoding performance. In this paper, we focus on the MWDs of polar codes and concatenated polar codes to analyze the performance and optimize the encoding structure. Existing enumeration methods cannot guarantee to get the MWD exactly and have high search complexity. In order to enumerate all the codewords belonging to the MWD, the sphere constraint property is applied, which means that the codewords with the identical Hamming weight are distribute on a spherical shell. Based on this, three enumeration methods are proposed in this paper. We first propose a sphere constraint based enumeration method (SCEM) to analyze the MWD of polar codes with moderate complexity. Then, according to the SCEM and the Plotkin's construction of polar codes, a sphere constraint based recursive enumeration method (SCREM) is proposed to recursively calculate the MWD with a lower complexity. Finally, we propose a parity-check SCEM (PC-SCEM) to analyze the MWD of concatenated polar codes by introducing the parity-check equations of outer codes. Moreover, due to the sphere constraint property, the proposed three methods can exactly enumerate all the codewords with the minimum Hamming weight. The enumeration results show that the SCREM has about $10^6$ times complexity reduction compared with the existing methods at code length 128 and the PC-SCEM can be used to design high-performance CRC-polar concatenated codes.

Linear codes and the mitochondrial genetic code

The origin of the genetic code can certainly be regarded as one of the most challenging problems in the theory of molecular evolution. Thus the known variants of the genetic code and a possible common ancestry of them haven been studied extensively in the literature. Gonzalez et al. (2012) developed the theory of a primeval mitochondrial genetic code composed of four base codons. These were called tesserae and it was shown that the tesserae code has some remarkable error detection capabilities. In our paper we will show that using classical coding theory we can construct the tessera code as a linear coding of the standard genetic code and at the same time it can be deduced from the code of all dinucleotides by Plotkin's construction. It shows that the tessera model of the mitochondrial code does not just have a biological explanation but also has a clear mathematical structure. This underlines the role that the tessera model might have played in evolution.

Polar coded space–time block coded spatial modulation based on Plotkin's construction for coded cooperative networks

A Plotkin-based polar coded space–time spatial modulation scheme is proposed and its bit error rate (BER) performance over quasi-static Rayleigh fading channel is evaluated. The exacerbated complexity in receiver design is ameliorated by employing a low complexity soft demapper for their proposed Plotkin-based polar coded scheme. As there is a natural split present in the design of the considered Plotkin's construction, therefore the authors effectively extended their proposed scheme to coded-cooperative scenarios. In any coded-cooperative communication, the relay node always plays a pivotal role in determining the BER performance of communication system. Therefore, a rational selection criterion for selecting the information bits at the relay node is presented. The authors have also devised nested polar coded-cooperative space–time spatial modulation (NPCCST-SM) scheme as a useful benchmark. Monte-Carlo simulated results revealed that their proposed coded-cooperative scheme has triumphed conventional NPCCST-SM scheme by a gain of 0.6–0.8 dB under identical conditions. Moreover, the proposed cooperative scheme outperforms its corresponding non-cooperative counterpart scheme by a gain of 1–1.2 dB under identical conditions.

Spherically Punctured Reed-Muller Codes

Consider a binary Reed–Muller code RM $(r,m)$ defined on the $m$ -dimensional hypercube $\mathbb {F}_{2}^{m}$ . In this paper, we study punctured Reed–Muller codes $P_{r}(m,b)$ , whose positions are restricted to the $m$ -tuples of a given Hamming weight $b$ . In combinatorial terms, this paper concerns $m$ -variate Boolean polynomials of any degree $r$ , which are evaluated on a Hamming sphere of some radius $b$ in $\mathbb {F}_{2}^{m}$ . Codes $P_{r}(m,b)$ inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical $b$ -layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes $P_{r}(m,b)$ and find their distances for the arbitrary values of parameters $r,m$ , and $b$ . Finally, we describe recursive (successive cancellation) decoding of these codes.

Jointly Optimized Reed-Muller Codes for Multi-Level Multi-Relay Coded-Cooperative VANETS

In this paper, we present the well-known Reed–Muller (RM) codes for multilevel multirelay vehicular adhoc networks, which can enjoy coded-cooperation among network nodes. At first, we present the code construction principles for a single node/relay scenario and then extend the design principles to the two-level two-relay and, finally, to multilevel multirelay scenarios. The term “level” refers to the fact that different order RM code is used at each relaying node. Plotkin's construction is exploited to utilize RM codes in such coded-cooperative schemes. To achieve an optimum code at the destination node, proper encoding strategy needs to be employed at the relay node. Therefore, we propose a design criteria and an efficient algorithm for proper bit selection at the relay nodes to achieve the best possible code at the destination. It is observed that the increase in the number of levels as well as relays result in better channel code at the destination, as compared to the lesser number of relays, however, at the cost of increased decoding complexity. The channels considered to analyze the bit error rate (BER) performances of proposed coded-cooperative schemes are fast and slow Rayleigh fading channels. At the destination, soft decision maximum likelihood decoding is employed. Numerical simulations show that the single-relay RM coded-cooperative scheme provides significant BER performance gains over the noncooperative and state-of-the-art distributed turbo coded-cooperative schemes under identical conditions.

Network polar coded cooperation with joint SC decoding

A network polar coded-cooperative (NPCC) scheme with joint successive cancellation (SC) decoding at the destination is proposed. Plotkin's construction is employed to build polar codes; however, the proposed construction is opposite to the conventional method where two short codes jointly construct a longer length code. In the proposed construction the two short codes (subcodes) are derived from the longer length code. The proposed construction is then employed to achieve coded cooperation between two users, and also offers a network coding, which not only improves the data rate of the coded cooperative system but also improves the overall bit error rate performance which is shown with the help of numerical simulations.

Spherically Punctured Biorthogonal Codes

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube \BB F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and let all code positions be restricted to the m-tuples of a given Hamming weight b. In this paper, we specify this single-layer construction obtained from the biorthogonal codes RM(1,m) and the Hadamard codes H(m). Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight w of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (w). We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code RM(1,m,b) is achieved at the minimum input weight w = 1 for any . We further refine our codes by limiting the possible weights w of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.

New Coding Techniques for Codes over Gaussian Integers

This work presents block codes over Gaussian integers. We introduce Gaussian integer rings which extend the number of possible signal constellations over Gaussian integer fields. Many well-known code constructions can be used for codes over Gaussian integer rings, e.g., the Plotkin construction or product codes. These codes enable low complexity decoding in the complex domain. Furthermore, we demonstrate that the concept of set partitioning can be applied to Gaussian integers. This enables multilevel code constructions. In addition to the code constructions, we present a low complexity soft-input decoding algorithm for one Mannheim error correcting codes. The presented decoding method is based on list decoding, where the list of candidate codewords is obtained by decomposing the syndrome into two sub-syndromes. Considering all decompositions of the syndrome we construct lists of all possible errors of Mannheim weight two. In the last decoding step the squared Euclidean distance is used to select the best codeword from the list. Simulation results for the additive white Gaussian noise channel demonstrate that the proposed decoding method achieves a significant coding gain compared with hard-input decoding.

List Decoding of Reed-Muller Codes Based on a Generalized Plotkin Construction

List Decoding of Reed-Muller Codes Based on a Generalized Plotkin Construction

Unequal error protection and progressive decoding for JPEG2000

This paper presents an unequal error protection scheme for still and motion image transmissions. The underlying channel codes are created using a Plotkin construction and offer the novel ability of using one long channel codeword to protect an entire image, yet still allowing progressive decoding. Progressive quality improvements occur in two ways: the first is the usual progressive refinement, where image quality is improved as more data are received; the second is that residual error rates of earlier received data are reduced as more data are received.