Non-binary Symbols Research Articles

Dynamic Injection and Permutation Coding for Enhanced Data Transmission.

In this paper, we propose a novel approach to enhance spectral efficiency in communication systems by dynamically adjusting the mapping between cyclic permutation coding (CPC) and its injected form. By monitoring channel conditions such as interference levels and impulsive noise strength, the system optimises the coding scheme to maximise data transmission reliability and efficiency. The CPC method employed in this work maps information bits onto non-binary symbols in a cyclic manner, aiming to improve the Hamming distance between mapped symbols. To address challenges such as low data rates inherent in permutation coding, injection techniques are introduced by removing δ column(s) from the CPC codebook. Comparative analyses demonstrate that the proposed dynamic adaptation scheme outperforms conventional permutation coding and injection schemes. Additionally, we present a generalised mathematical expression to describe the relationship between the spectral efficiencies of both coding schemes. This dynamic approach ensures efficient and reliable communication in environments with varying levels of interference and impulsive noise, highlighting its potential applicability to systems like power line communications.

Nonbinary LDPC Coded QAM Signals With Optimized Mapping: Bounds and Simulation Results

This paper studies specific properties of nonbinary low-density parity-check (NB LDPC) codes when used in coded modulation systems. The paper is focused on the practically important NB LDPC codes over the Galois extension fields GF(2m) with m ≤ 6 used with QAM signaling. Performance of NB LDPC coded transmission strongly depends on the mapping of nonbinary symbols to signal constellation points. We obtain a random coding bound on the maximum-likelihood decoding error probability for an ensemble of random irregular NB LDPC codes used with QAM signaling for specific symbol-to-signal point mappings. This bound is based on the ensemble average squared Euclidean distance spectra derived for these mappings. The simulation results for the belief-propagation decoding in the coded modulation schemes with the NB quasi-cyclic (QC)-LDPC codes under different mappings are given. Comparisons with the optimized binary QC-LDPC codes in the WiFi and 5G standards, as well as with the new bound, are performed.

Non-Binary PRN-Chirp Modulation: A GNSS Fast Acquisition Signal Waveform

In this letter, we propose a new non-binary modulation which allows both Global Navigation Satellite Systems (GNSS) synchronization and the demodulation of non-binary symbols, without the need of a pilot signal, with the aim to provide a fast first position, velocity and time fix. The waveform is constructed as the product of i) a pseudo-random noise sequence with good auto-correlation and cross-correlation properties, and ii) a chirp spread spectrum family, which allows to demodulate non-binary symbols even if the signal phase is unknown. In order to demodulate the data, a bank of non-coherent matched filters is proposed. Because of the particular modulation structure, the receiver is capable to demodulate the navigation message faster while allowing the basic GNSS signal processing functionalities. Illustrative results are provided to support the discussion.

Quantum digital signatures with smaller public keys

We introduce a variant of quantum signatures in which nonbinary symbols are signed instead of bits. The public keys are fingerprinting states, just as in the scheme of Gottesman and Chuang \cite{GC2001}, but we allow for multiple ways to reveal the private key partially. The effect of this modification is a reduction of the number of qubits expended per message bit. Asymptotically the expenditure becomes as low as one qubit per message bit. We give a security proof, and we present numerical results that show how the improvement in public key size depends on the message length.

“Alle Sternchen mitgemeint1”

In this article, I analyze the grassroots reception of queer theories within queer DIY music festivals in Post-Wende Germany. I argue that queer theories develop not only in academic settings, and look at the ways in which festival organizers, participants and musicians construct a form of queerness shaped by its international circulation and local understandings. In Section 1, I observe how the invention of non-binary symbols and pronouns allows for a queering of German linguistics. Then, I consider the impact of this process on festivals’ atmosphere and rituals. In the final section, I look at the manifestation of queer theories and identities in songwriting through the study of Sookee’s rap music.

Non-Binary Constrained Codes for Two-Dimensional Magnetic Recording

The two-dimensional magnetic recording (TDMR) technology promises storage densities of $10$ terabits per square inch. However, when tracks are squeezed together, a bit stored in the two-dimensional (TD) grid suffers inter-symbol interference (ISI) from adjacent bits in the same track, and inter-track interference (ITI) from nearby bits in the adjacent tracks. A bit is highly likely to be read incorrectly if it is isolated in the middle of a $3 \times3$ square; surrounded by its complements, horizontally and vertically. We improve the reliability of TDMR systems by designing two-dimensional constrained codes that prevent these square isolation patterns. We exploit the way TD read heads operate to design our codes, and we focus on TD read heads that collect signals from three adjacent tracks. We represent the two-dimensional square isolation constraint as a one-dimensional constraint on an alphabet of eight non-binary symbols. We use this new representation to construct a non-binary lexicographically-ordered constrained code where one third of the information bits are unconstrained. Our TD constraint codes are capacity-achieving, and the data protection is achieved with redundancy less than $3\%$ and at modest complexity.

Parallel Generation of Most Reliable LLRs of a Non-Binary Symbol

The first task of non-binary (NB) decoders [low-density parity check (LDPC) or turbo] is to generate the log-likelihood ratio (LLR) of the received NB symbols defined over Galois fields GF( $q>2$ ). In the extended min-sum decoding algorithm, the intrinsic information associated to a given received symbol is a sorted list of the $n_{m}$ most reliable NB GF symbols along with their associated reliability values. In this letter, we present a fully parallel LLR generation algorithm, an enabler for very high throughput decoding that processes one received symbol per clock cycle. We provide complexity figures for LLR architectures designed over GF( $q$ ) of sizes 64, 256, and 1024, as well as different values of $n_{m}$ . Compared with the related state-of-the-art architecture, field-programmable gate array (FPGA) synthesis results show that the proposed parallel architecture improves the hardware efficiency by a factor ranging from 7 up to 15.

A 2.4-mm2 130-mW MMSE-Nonbinary LDPC Iterative Detector Decoder for 4 $\times$ 4 256-QAM MIMO in 65-nm CMOS

Iterative detection and decoding (IDD) employs a soft-in soft-out (SISO) detector and an SISO forward error correction (FEC) decoder in an iterative loop to improve the receiver performance in multiple-input multiple-output (MIMO) wireless communications. This paper describes a 256-QAM 4 × 4 prototype IDD design made up of a minimum mean square error (MMSE) detector and a nonbinary low-density parity-check (NBLDPC) decoder with the symbol size of the NBLDPC code matched to the modulation to enhance performance. By directly translating between nonbinary symbols and constellation points, the detector-decoder interface is simplified. We present a Gb/s MMSE detector using a shortened tandem scheduling, a low-latency dual-lookup reciprocal unit, an optimized interleaved microarchitecture, and a Gb/s NBLDPC decoder with efficient internal skipping paths and memory allocation. The designs were demonstrated in a 0.7-mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 1.38-Gb/s MMSE detector and a 1.7-mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 1.02-Gb/s-NBLDPC decoder that are integrated in a 65-nm CMOS test chip. The chip is measured to achieve 19.2 pJ/b in detection and 20.1 pJ/b/iteration in decoding.

Channel Decoding for Nonbinary Physical-Layer Network Coding in Two-Way Relay Systems

In this paper, we propose a generalized channel decoding scheme for nonbinary physical-layer network coding (CD-NC) in two-way relay channels (TWRCs), where two source nodes A and B exchange their nonbinary symbols via a relay. The two sources use the same nonbinary low-density parity-check (LDPC) channel code over the integer ring Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> and M-pulse-amplitude modulation, respectively. The existing channel decoding schemes for nonbinary network coding suffer severe rate loss compared with the cut-set bound of TWRC, especially in the low-to-medium signal-to-noise ratio regime. The proposed CD-NC can decrease the rate loss. Our contributions are as follows: 1) We develop a generalized nonbinary sum product algorithm (G-SPA) for CD-NC according to the principle of virtual encoding of the superimposed symbols. Simulation results show that our CD-NC can achieve significant performance gains over the conventional nonbinary network coding for both additive white Gaussian noise and fading channels; and 2) We exploit two-dimensional fast-Fourier-transform-based belief propagation (2-D-FFT-BP) and extended min-sum (EMS) decoding algorithms to reduce the decoding complexity of G-SPA. Simulation results show that the 2-D-FFT-BP has the same performance as G-SPA, while EMS can greatly reduce the decoding complexity of G-SPA at the cost of slight performance degradation.

Construction of high rate $$\mathcal {T}$$ T - $${{\varvec{direct}}}$$ direct codes

This paper presents a class of multi-user codes named as \(\mathcal {T}\)-direct tertiary codes. These classes of codes have higher information rate in comparison to the existing \(\mathcal {T}\)-direct codes. A coding scheme (encoding and decoding) is proposed for the class of \(\mathcal {T}\)-direct codes, that increases the information rate of the constituent codes in comparison to the existing coding scheme for the class of \(\mathcal {T}\)-direct codes: the coding scheme first drives the code \(\mathcal {C}_{i}'(n', k', d')\) from the constituent code \(\mathcal {C}_{i}(n, k, d)\) of existing \(\mathcal {T}\)-direct code \((\mathcal {C}_{1}, \mathcal {C}_{2}, \ldots , \mathcal {C}_{\mathcal {T}})\) with \(\frac{k'}{n'} \ge \frac{k}{n}\), and eventually forms a \(\mathcal {T}\)-direct code \((\mathcal {C}_{1}', \mathcal {C}_{2}', \ldots , \mathcal {C}_{\mathcal {T}}')\) with an overall increase in sum-rate. Just like \(\mathcal {T}\)-direct codes, the newly constructed class of codes can be used for simultaneous transmission of non-binary symbols over an \(\mathbb {F}\)-adder channel. Further, an associated decoding technique for the constructed class of codes is also proposed.

Low Delay Single Symbol Error Correction Codes Based on Reed Solomon Codes

To avoid data corruption, error correction codes (ECCs) are widely used to protect memories. ECCs introduce a delay penalty in accessing the data as encoding or decoding has to be performed. This limits the use of ECCs in high-speed memories. This has led to the use of simple codes such as single error correction double error detection (SEC-DED) codes. However, as technology scales multiple cell upsets (MCUs) become more common and limit the use of SEC-DED codes unless they are combined with interleaving. A similar issue occurs in some types of memories like DRAM that are typically grouped in modules composed of several devices. In those modules, the protection against a device failure rather than isolated bit errors is also desirable. In those cases, one option is to use more advanced ECCs that can correct multiple bit errors. The main challenge is that those codes should minimize the delay and area penalty. Among the codes that have been considered for memory protection are Reed-Solomon (RS) codes. These codes are based on non-binary symbols and therefore can correct multiple bit errors. In this paper, single symbol error correction codes based on Reed-Solomon codes that can be implemented with low delay are proposed and evaluated. The results show that they can be implemented with a substantially lower delay than traditional single error correction RS codes.

멀티 레벨 낸드 플래시 메모리용 연판정 복호를 수행하는 이진 ECC 설계를 위한 EM 알고리즘

멀티 레벨 낸드 플래시 메모리는 한 셀에 2 비트 이상의 정보를 저장하는 구조이고, 비트 위치별 채널 LLR의 밀도 함수 l-밀도가 비대칭 특성을 가지고 있다. 이런 특성은 이진 무기억 대칭 채널 조건에서 설계된 오류 정정부호의 성능이 제대로 발휘되지 못하게 할 뿐만 아니라, 멀티 레벨 낸드 플래시 메모리용 연판정 복호를 수행하는 이진 오류 정정 부호의 설계도 어렵게 한다. 본 논문에서 밀도 미러링과 EM 알고리즘을 이용하여 오류 정정 부호 설계를 위한 차선책을 소개한다. 밀도 미러링은 EM 알고리즘을 적용하기 전에 0 부호어를 전송한 경우로 가정할 수 있도록 하기 위해서 채널 LLR을 처리하는 과정이고, 이후 채널 LLR l-밀도를 EM 알고리즘을 적용하여 K개의 성분으로 이루어진 대칭 가우시안 혼합 밀도로 근사화하는 방법을 소개한다. In this paper, we present two signal processing techniques for designing binary error correction codes for Multi-Level Cell(MLC) NAND flash memory. MLC NAND flash memory saves the non-binary symbol at each cell and shows asymmetric channel LLR l-density which makes it difficult to design soft-decision binary error correction codes such as LDPC codes and Polar codes. Therefore, we apply density mirroring and EM algorithm for approximating the MLC NAND flash memory channel to the binary-input memoryless channel. The density mirroring processes channel LLRs to satisfy roughly all-zero codeword assumption, and then EM algorithm is applied to l-density after density mirroring for approximating it to mixture of symmetric Gaussian densities. These two signal processing techniques make it possible to use conventional code design algorithms, such as density evolution and EXIT chart, for MLC NAND flash memory channel.

Reliable Communication over Non-Binary Insertion/Deletion Channels

We consider the problem of reliable communication over non-binary insertion/deletion channels where symbols are randomly deleted from or inserted in the received sequence and all symbols are corrupted by additive white Gaussian noise. To this end, we utilize the inherent redundancy achievable in non-binary symbol sets by first expanding the symbol set and then allocating part of the bits associated with each symbol to watermark symbols. The watermark sequence, known at the receiver, is then used by a forward-backward algorithm to provide soft information for an outer code which decodes the transmitted sequence. Through numerical results and discussions, we evaluate the performance of the proposed solution and show that it leads to significant system ability to detect and correct insertions/deletions. We also provide estimates of the maximum achievable information rates of the system, compare them with the available bounds, and construct practical codes capable of approaching these limits.

Joint nonbinary low-density parity-check codes and modulation diversity over fading channels

A joint exploitation of coding and diversity techniques to achieve efficient, reliable wireless transmission is considered. The system comprises a powerful non-binary low-density parity-check (LDPC) code that will be soft-decoded to supply strong error protection, a quadratic amplitude modulator (QAM) that directly takes in the non-binary LDPC symbols and a modulation diversity operator that will provide power- and bandwidth-efficient diversity gain. By relaxing the rate of the modulation diversity rotation matrices to below 1, we show that a better rate allocation can be arranged between the LDPC codes and the modulation diversity, which brings significant performance gain over previous systems. To facilitate the design and evaluation of the relaxed modulation diversity rotation matrices, based on a set of criteria, three practical design methods are given and their point pairwise error rate are analyzed. With EXIT chart, we investigate the convergence between demodulator and decoder.A rate match method is presented based on EXIT analysis. Through analysis and simulations, we show that our strategies are very effective in combating random fading and strong noise on fading channels.

Joint Estimation and Compression of Correlated Nonbinary Sources Using Punctured Turbo Codes

We propose a system to perform data compression of correlated nonbinary sources when the correlation between sources is not known, either at the encoder or the decoder. The sequences of nonbinary symbols are transformed into sequences of bits, and then source coded using punctured turbo codes, with the puncturing adjusted to achieve the desired compression rate. Each source is compressed without knowledge about the other source, and the correlation model is not assumed to be known at the encoder. The source decoder uses iterative schemes over the compressed binary sequences, and recovers the nonbinary symbol sequences from both sources. The correlation model between sources does not need to be known at the decoder, since it can be estimated jointly with the iterative decoding process. Compared with the case in which the correlation is known at the decoder, no significant performance loss is observed. The performance of the proposed scheme is close to the Slepian-Wolf theoretical limit.

Vector symbol decoding with list inner symbol decisions

Prior work showed the ability of a randomly chosen outer code of a concatenated code to correct large numbers of nonbinary symbol errors in a simple manner, provided that the error symbols as vectors are linearly independent. This paper extends the technique in two ways: 1) it is shown how to correct a large class of dependent errors with little additional complexity; and 2) if the inner code supplies a list of two (or more) candidates in some or all decisions, a slightly modified vector symbol decoder can directly reveal most correct alternatives, allowing more powerful and often simpler correction.

Probability density functions of soft information

It is well known that soft decoding algorithms are based on the use of reliability measures for binary and nonbinary symbols. We consider an analytical estimation of the distributions of reliability measures after soft decoding of block and convolutional codes or intersymbol interference (ISI) channels. This estimation, based on a simple geometric interpretation of reliability measures, allows the observation of different distributions for correct and incorrect symbols, and could be helpful for the analysis of soft decoding algorithms.

Error correction in nonbinary words

A simple error detecting and correcting procedure is described for nonbinary symbol words; here, the error position is located using the Hamming method and the correct symbol is substituted using a modulo-check procedure.

Character-correcting convolutional self-orthogonal codes

A class of convolutional, character-error-correcting codes with limited error propagation is presented. This class of codes is derived from binary convolutional self-orthogonal codes (BCSOC). By character-error-correcting, we mean that the code is character oriented, where each character can be thought of as a string of binary or higher base symbols of fixed length or as a single nonbinary symbol of correspondingly higher base. It is shown that, given a t -error-correcting BCSOC of rate b —1/ b , a character-error correcting convolutional self-orthogonal code (CCSOC) of rate k ( b —1)/( k ( b —1) + 1) can be constructed for any integer k , the rate expansion factor. The CCSOC so constructed corrects t character errors, and also possesses large simultaneous burst-error-correcting capabilities. Lower bounds on the burst-error-correcting capability for both BCSOC and CCSOC are found. Decoding consists of a mixture of majority logic decoding and algebraic computation. The decoding algorithm seems practical if either the rate expansion factor k or the number of errors corrected t are not large. Such codes are most suitable for channels with both random and burst noise, and also effect a compromise between the cost of terminal equipment and the efficient use of channels.

Convolutional Reed-Solomon Codes

We derive a new family of convolutional character-error-correcting codes which are a convolutional form of the Reed-Solomon block codes and as such have nonbinary symbols. We also derive a bound on the error correcting capabilities of these codes in which the error-correcting capability per constraint length grows approximately with the square root of the constraint length. When these codes are used on a binary channel they are effective for both random and burst error correction because a single character spans several channel digits. These codes have greater error-correcting capabilities than the Robinson-Bernstein self-orthogonal codes but are harder to decode. The single-character-error-correcting codes, when interleaved, are shown to be more powerful than the equivalent Hagelbarger code and appear to be simpler to implement. They are also slightly better than the interleaved version of Berlekamp's code. We discuss encoding and decoding algorithms and illustrate a simple decoding algorithm for some of the codes. These codes are closely related to the Bose-Chaudhuri-Hocquenghem block codes and share with them the decoding simplification for character erasures in place of errors. Any Bose-Chaudhuri-Hocquenghem decoding algorithm can be used to decode these codes.