Code Construction Method Research Articles

Algebraic and Geometric Methods for Construction of Topological Quantum Codes from Lattices

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices Λ′/Λ of index m combined with geometric technique of the projections of vector basis Λ′ over vector basis Λ, we reproduce surface codes found in the literature with parameter [[2m2,2,|a|+|b|]] for the case Λ=Z2 and m=a2+b2, where a and b are integers that are not null, simultaneously. We also obtain a new class of surface code with parameters [[2m,2,|a|+|b|]] from the Λ=A2-lattice when m can be expressed as m=a2+ab+b2, where a and b are integer values. Finally, we will show how this technique can be extended to the construction of color codes with parameters [[18m,4,6(|a|+|b|)]] by considering honeycomb lattices partition A2/Λ′ of index m=9(a2+ab+b2) where a and b are not null integers.

DMMC: A Polar Code Construction Method for Improving Performance in TLC NAND Flash

3D TLC NAND flash memory significantly increases the storage capacity but makes data more prone to errors. To address the reliability problem, Polar Code is one coding method that can reach the channel capacity1. However, the 3D NAND flash memory channel is uncertain due to the variation of retention time and program/erase(P/E) cycle. So it is difficult to construct Polar Code in NAND flash memory. This paper proposes a new Polar Code construction method called Dynamic Merge Monte Carlo(DMMC). DMMC uses the Monte Carlo method to obtain the optimal Polar Code construction and then reduces the space overhead by combining similar construction so that it can be used in flash memory systems. Compared with the conventional method, the uncorrectable bit error rate(UBER) of the proposed method can reduce by up to 90.5%, and the read latency can reduce by more than 50%.

Weakly mutually uncorrelated codes with maximum run length constraint for DNA storage

DNA storage systems have begun to attract considerable attention as next-generation storage technologies due to their high densities and longevity. However, efficient primer design for random-access in synthesized DNA strands is still an issue that needs to be solved. Although previous studies have explored various constraints for primer design in DNA storage systems, there is no attention paid to the combination of weakly mutually uncorrelated codes with the maximum run length constraint. In this paper, we first propose a code design by combining weakly mutually uncorrelated codes with the maximum run length constraint. Moreover, we also explore the weakly mutually uncorrelated codes to satisfy combinations of maximum run length constraint with more constraints such as being almost-balanced and having large Hamming distance, which are also efficient constraints for random-access in DNA storage systems. To guarantee that the proposed codes can be adapted to primer design with variable length, we present modified code construction methods to achieve different lengths of the code. Then, we provide an analysis of the size of the proposed codes, which indicates the capacity to support primer design. Finally, we compare the codes with those of previous works to show that the proposed codes can always guarantee the maximum run length constraint, which is helpful for random-access for DNA storage.

Application research of polar coded OFDM underwater acoustic communications

To further accelerate the practical process of polar codes in underwater acoustic communication (UWC) and improve the reliability and efficiency of underwater data transmission, we first propose a polar code construction method suitable for the underwater acoustic channel, which drastically reduces the complexity and meets the need for practical UWC. Then, we establish a practical polar coded UWC scheme by devising a two-step procedure based on the Orthogonal Frequency Division Multiplexing technique. Moreover, through simulation, the scheme's performances with the unimproved and the proposed polar code construction method are analyzed. Moreover, its performances with four different polar decoders are also given. The performance comparison with the LDPC coded UWC system is studied as well. Furthermore, lake-trial results under two different channel conditions demonstrate that the proposed construction method and the scheme effectively guarantee the reliability of data transmission, achieving error-free transmission at a transmission distance of 1718 m with the signal-to-noise-ratio (SNR) of 21 dB, and a transmission distance of 749 m with the SNR of 16.5 dB, better than the LDPC coded system with same conditions. The semi-experimental results show that the proposed polar coded system under the Cyclic Redundancy Check-Aided Successive Cancellation List decoder outperforms the LDPC coded system by approximately 2.1–3.7 dB.

Metode Reversible Self-Dual untuk Konstruksi Kode DNA atas Lapangan Hingga GF(4)

The DNA molecule chain consists of two complementary strands composed of a sequence of four nucleotide bases, namely adenine (A), cytosine (C), guanine (G) and thymine (T). DNA code is a set of codewords with a fixed length of the alphabet {A, C, T, G}. DNA coding is one application of coding theory over a finite field. The set {A, C, T, G} is identified as finite field GF(4) = {0, 1, w, w2} with w2 + w + 1 = 0. The reversible self-dual (RSD) code over the finite field GF(4) is a code whose dual is itself and the reverse of each codeword contained in the code. This study aims to obtain an algorithm to construct a DNA code derived from the RSD C code on the field to GF(4) which is called the Reversible Self-Dual Method. The aspects studied include the characteristics that form the basis properties of the theory in compiling the DNA code algorithm over the RSD code over GF(4). The compiled algorithm is a DNA code construction method of codeword length even that conforms to the Hamming distance constraint, reverse-complement constraint, and GC-content constraint. The input of the algorithm is a generator matrix of RSD code C with a minimum distance of d and the output is a DNA code that satisfies these three constraints.

Construction and Characterization of Optical Orthogonal Codes (n,w,1) for Fast Frequency Hopping-Optical Code Division Multiple Access

This paper investigates a new combinatorial construction approach based on cyclic difference packings for the construction of an optical orthogonal code that is suitable as a time spreading sequence for fast frequency hopping optical code division multiple access (OCDM) systems. The proposed construction provides a solution to the electrical decoding delay problem when compared to its optical counterpart, as well as other optimization issues. The criteria for optimizing the performance of such an OCDMA system are provided, namely: the flexibility and simplicity in constructing the optimal code for any length and weight, a reduction in the encoding/decoding complexity that complies with changes in the fast frequency hopping system, and the provision of system transfer transparency, a decoding rule that exploits the embedded asymmetric error correcting capability of the code. Our neighbor difference approach refers to the partition of different solutions into sub-classes, given in the correlation matrix form. It takes into account both direct and recursive combinatorial code construction methods, and the resulting characteristics are evaluated accordingly. A performance analysis based on the OCDMA channel is given, and a comparison with other difference family solutions is performed.

A New Hybrid Prime Code for OCDMA Network Multimedia Applications

This paper presents a new family of spreading code sequences called hybrid prime code (HPC), to be used as source code for the optical code division multiple access (OCDMA) network for large network capacity. The network capacity directly depends on the number of available code sequences provided and their correlation properties. Therefore, the proposed HPC is designed based on combining two or more different code words belonging to two or more different prime numbers. This increases the number of code sequences generated. The code construction method utilized allows the generation of different code sets, each with different code length and weight, according to the number of prime numbers used. In addition, the incoherent pulse position modulation (PPM) OCDMA system is proposed based on the HPC code. Furthermore, the bit error rate (BER) performance analysis is introduced versus the received optical power and the number of active users. Moreover, the error vector magnitude (EVM) is calculated versus the optical signal-to-noise ratio. This work proves that using two prime numbers simultaneously generates far more codes than using prime numbers separately. It also achieved an OCDMA system capacity higher than the system that uses the optical orthogonal codes (OOCs), modified prime codes (MPCs) families, and two code families with separate simultaneously prime numbers, at a BER below 10−9 which is the optimum level.

Construction of Multi-Kernel Polar Codes With Kernel Substitution

Traditional polar codes are constructed based on a size-2 kernel and the code length is thus restricted to an integer power of 2. To achieve flexible code lengths, punctured and shortened schemes are used in the existing 5G standard. Recently, an alternative approach for flexible code lengths, multi-kernel (MK) polar code, has been proposed, which is constructed based on both size-2 and other larges kernels. However, the error-correction performance is not as good as that of the punctured and shortened schemes. In this letter, an MK polar code construction method based on a kernel substitution scheme is proposed, which is a practical scheme to construct the codes using two types of large kernels with the same size so that the error-correction performance can be improved. Compared with the existing MK polar codes, simulation results show that 0.15 - 0.25 dB performance gain can be achieved by SC decoding for most code settings at a block error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> without complexity overhead.

On equivalency of zero-divisor codes via classifying their idempotent generator

In 2009, Ted and Paul Hurley proposed a code construction method using group rings. These codes with single generator are termed group ring codes and in particular zero-divisor codes when using zero-divisors as generators. In this paper, we mainly study the equivalency of zero-divisor codes in $$F_2G$$ having generator from I(G), the set of all idempotents in $$F_2G$$ . For abelian G, our previous notion of generated idempotents completely classified I(G) by serving as its basis. Here, we first extend the notion of generated idempotents to study and classify some elements in I(G) for non-abelian G. Later, the study is generally done on equivalency of zero-divisor codes in $$F_2G$$ , then concentrating on those with idempotent generator. In particular, we affirm the conjecture “Every group ring code in $$F_2D_{2n}$$ is equivalent to some in $$F_2C_{2n}$$ ” in the cases where the generators are our classified idempotents. We also show that the equivalency of zero-divisor codes in $$F_2C_n$$ with generated idempotent as generators can be established sufficiently on the generator property.

Design and performance of spectral efficient modified new diagonal code for a spectral amplitude code-optical code division multiple access system

Our study contains performance and design of a spectral amplitude code-optical code division multiple access (SAC-OCDMA) system using proposed modified new diagonal (MND) code. The proposed MND code has advantages as very low cross-correlation value compared to other code sequences. Also the code has a short length for the higher code weight value. Furthermore, due to its enormous code property, it also introduces very low multiple access interference. Thus the MND code-based SAC-OCDMA system permits a higher number of users. The MND code construction method is also explained, which is very easy using diagonal matrices. Based on the theoretical analysis, bit error rate (BER) performance of MND code AND-subtraction scheme compare with conventional codes is better than random diagonal code, Pascal’s triangle matrix code (PTMC), and multidiagonal (MD) code, which makes it spectral efficient. It provides 5 dB lower BER compared to MD code and PTMC for 90 users. Simulation of the MND code-based system for four users with 2 Gbps data rate has been successfully demonstrated using the AND-subtraction scheme and the direct detection scheme.

Performance improvement of multi access OCDMA system based on a new zero cross correlation code

In this paper, a novel zero cross correlation (ZCC) code is proposed for spectral amplitude coding-optical code division multiple access (SAC-OCDMA) systems. The code construction method starts from the identity matrix and has several benefits summarized in both the code simplicity and flexibility. The proposed code presents also an adapted (or accepted) code length for SAC-OCDMA systems and a high SNR value that reaches respectively 3.18 and 1.84 times of the system capacity comparing to other existed codes such as modified quadratic congruence (MQC) and modified double weight (MDW) codes.

A Linear-Complexity Channel-Independent Code Construction Method for List Sphere Polar Decoder

Besides the main decoding methods for polar codes such as successive cancellation (SC) decoding, SC list decoding (SCL), and belief propagation (BP) decoding, list sphere decoding (LSD) is an alternative for short codes regarding its lower complexity. Though the improved LSD (ILSD) can achieve better performance than LSD by a synchronous determination scheme, its performance is still not satisfactory with an inappropriate code construction. Good polar code construction is to pick the K most reliable bits from N bits. However, existing polar construction methods are designed for SC or BP decoders. These construction methods might not be suitable for ILSD, which searches based on the Euclidean distance. In this paper, a linear-complexity channel-independent polar code construction method for ILSD is proposed to improve the performance. Considering the Hamming distance as the reliability for ILSD, the proposed method arranges more frozen bits in the early synchro sets and improves the evaluation of the paths at each decoding level. With linear complexity, this method is channel-independent, which is efficient for offline constructions. Numerical results show the proposed construction with low code rates outperforms the state-of-the-art by 2 dB with ILSD.

Multi-Rack Distributed Data Storage Networks

The majority of works in distributed storage networks assume a simple network model with a collection of identical storage nodes with the same communication cost between the nodes. In this paper, we consider a realistic multi-rack distributed data storage network and present a code design framework for this model. Considering the cheaper data transmission within the racks, our code construction method is able to locally repair the nodes failure within the same rack by using only the survived nodes in the same rack. However, in the case of severe failure patterns when the information content of the survived nodes is not sufficient to repair the failures, other racks will participate in the repair process. By employing the criteria of our multi-rack storage code, we establish a linear programming bound on the size of the code in order to maximize the code rate.

Polar Codes and Their Quantum-Domain Counterparts

Arikan's polar codes are capable of achieving the Shannon's capacity at a low encoding and decoding complexity, while inherently supporting rate adaptation. By virtue of these attractive features, polar codes have provided fierce competition to both the turbo as well as the Low Density Parity Check (LDPC) codes, making its way into the 5G New Radio (NR). Realizing the significance of polar codes, in this paper we provide a comprehensive survey of polar codes, highlighting the major milestones achieved in the last decade. Furthermore, we also provide tutorial insights into the polar encoder, decoders as well as the code construction methods. We also extend our discussions to quantum polar codes with an emphasis on the underlying quantum-to-classical isomorphism and the syndrome-based quantum polar codes.

Construction of Bio-Constrained Code for DNA Data Storage

With extremely high density and durable preservation, DNA data storage has become one of the most cutting-edge techniques for long-term data storage. Similar to traditional storage which impose restrictions on the form of encoded data, data stored in DNA storage systems are also subject to two biochemical constraints, i.e., maximum homopolymer run limit and balanced GC content limit. Previous studies used successive process to satisfy these two constraints. As a result, the process suffers low efficiency and high complexity. In this letter, we propose a novel content-balanced run-length limited code with an efficient code construction method, which generates short DNA sequences that satisfy both constraints at one time. Besides, we develop an encoding method to map binary data into long DNA sequences for DNA data storage, which ensures both local and global stability in terms of satisfying the biochemical constraints. The proposed encoding method has high effective code rate of 1.917 bits per nucleotide and low coding complexity.

Analysis of a Code Construction Method for Non-Binary Quasi-Cyclic Irregular Low Density Parity Check Decoder

Non-binary LDPC codes is a class of linear block code outperform in the short and moderate block length, closer to Shannon's limit. The numerical strength of the decoder is proportional to the sparsity of the parity check matrix. A Hierarchically Diagonal Parity Check Matrix (HDPCM) with better sparseness is constructed to optimize the computation complexity and decoding performance. The decoder based on the proposed matrix using FFT-SP decoding algorithm is analyzed, with different modulation schemes and channel environment. The positive impact of the constructed matrix is elaborated, concludes with the suitability of the HDPCM structure in AWGN channel and BPSK modulation environment.

Nonbinary Quasi-Regular QC-LDPC Codes Derived From Cycle Codes

Nonbinary ultra sparse codes, particularly regular cycle codes, are known to approach Shannon-limit performance as the Galois field $ \mathrm {GF}(q)$ order is sufficiently large. Good cycle codes can result from a class of algebraically defined graphs called cages. Meanwhile, when smaller $q$ is desirable, the cycle codes are outperformed by quasi-regular codes. In this letter, we propose a code construction method that takes a cage as a starting point and then progressively inserts a few additional edges into the graph. The edge insertion is terminated as soon as the code performance stops improving. Our simulation results show that the obtained quasi-regular codes outperform cyclic codes for fields up to GF(64) and its performance is slightly better than the quasi-regular improved-Progressive Edge Growth-based codes. The proposed algorithm preserves the block-circulant structure of the initial cage-based graph; therefore, it can be used for structured or quasi-cyclic codes design.

Partial parallel encoding and algorithmic construction of non-binary structured IRA codes

The non-binary (NB) Irregular Repeat Accumulate (IRA) codes, as a subclass of NB LDPC codes, potentially have an excellent error-correcting performance. They are also known to provide linear complexity of encoding, but the basic encoding method with the serial rate-1 accumulator significantly limits the encoder throughput. Then the objective of the research presented in this paper is to develop an encoding method providing significantly increased throughput of an NB-IRA encoder altogether with a flexible code construction methods for the structured (S-NB-IRA) codes eligible for the proposed encoding method. For this purpose, we reformulate the classic encoding algorithm to fit into the partial parallel encoder architecture. We propose the S-NB-IRA encoder block diagram and show that its estimated throughput is proportional to the submatrix size of the parity check matrix, which guarantees a wide complexity-throughput tradeoff. Then, in order to facilitate the S-NB-IRA coding systems design, we present a computer search algorithm for the construction of good S-NB-IRA codes. The algorithm aims at optimizing the code graph topology along with selecting an appropriate non-binary elements in the parity check matrix. Numerical results show that the constructed S-NB-IRA codes significantly outperform the binary IRA and S-IRA codes, while their performance is similar to the best unstructured NB-LDPC codes.

Linear unequal error protection codes based on terminated convolutional codes

Convolutional codes which are terminated by direct truncation (DT) and zero tail termination provide unequal error protection. When DT terminated convolutional codes are used to encode short messages, they have interesting error protection properties. Such codes match the significance of the output bits of common quantizers and therefore lead to a low mean square error (MSE) when they are used to encode quantizer outputs which are transmitted via a noisy digital communication system. A code construction method that allows adapting the code to the channel is introduced, which is based on time-varying convolutional codes. We can show by simulations that DT terminated convolutional codes lead to a lower MSE than standard block codes for all channel conditions. Furthermore, we develop an MSE approximation which is based on an upper bound on the error probability per information bit. By means of this MSE approximation, we compare the convolutional codes to linear unequal error protection code construction methods from the literature for code dimensions which are relevant in analog to digital conversion systems. In numerous situations, the DT terminated convolutional codes have the lowest MSE among all codes.

Generalised array low‐density parity‐check codes

In this study, using Group Permutation Low-Density Parity-Check (GP-LDPC) codes, the authors generalise the concept of array Low-Density Parity-Check (LDPC) codes from fields of prime order to those of prime power order. In fact, they consider the additive group of the finite field GF( q ), q a prime power, as the underlying group for the GP-LDPC code construction and since when q is a prime, the author's code construction method coincides with that of quasi-cyclic array LDPC codes, they call their codes, generalised array LDPC (GA-LDPC) codes. First, they prove that, like array LDPC codes, GA-LDPC codes are quasi-cyclic codes. Then, they analyse the girth of GA-LDPC codes in a way similar to that for array LDPC codes and introduce some shortened GA-LDPC codes with girths 8, 10 and 12. For many values of g , J and L , the lengths of ( J , L )-regular shortened GA-LDPC codes of girth g and rate at least 1 - J / L , constructed in this study, are smaller than the lengths of ( J , L )-regular LDPC codes of girth g and rate at least 1 - J / L , constructed in the literature. Also, simulation results show that GA-LDPC codes perform well with the iterative message-passing decoding.