Gotowa bibliografia na temat „Bose-Chaudhuri-Hocquenghem Codes”

Several families of nonbinary asymmetric quantum Bose-Chaudhuri-Hocquenghem (BCH) codes are presented in this paper. These quantum codes have parameters better than the ones available in the literature. Additionally, such codes can be applied in quantum systems where the asymmetry between qudit-flip and phase-shift errors is large.

The Bose – Chaudhuri – Hocquenghem type of linear cyclic codes (BCH codes) is one of the most popular and widespread error-correcting codes. Their close connection with the theory of Galois fields gave an opportunity to create a theory of the norms of syndromes for BCH codes, namely, syndrome invariants of the G-orbits of errors, and to develop a theory of polynomial invariants of the G-orbits of errors. This theory as a whole served as the basis for the development of effective permutation polynomial-norm methods and error correction algorithms that significantly reduce the influence of the selector problem. To date, these methods represent the only approach to error correction with non-primitive BCH codes, the multiplicity of which goes beyond design boundaries. This work is dedicated to a special error-correcting code class – generic Bose – Chaudhuri – Hocquenghem codes or simply GBCH-codes. Sufficiently accurate evaluation of the quantity of such codes in each length was produced during our work. We have investigated some properties and connections between different GBCH-codes. Special attention was devoted to codes with constructive distances of 3 and 5, as those codes are usual for practical use. Their almost complete description is given in the range of lengths from 7 to 107. The paper contains a fairly clear theoretical classification of GBCH-codes. Special attention is paid to the corrective capabilities of the codes of this class, namely, to the calculation of the minimal distances of these codes with various parameters. The codes are found whose corrective capabilities significantly exceed those of the well-known GBCH-codes with the same design parameters.

By studying the properties of [Formula: see text]-cyclotomic cosets, the maximum designed distances of Hermitian dual-containing constacyclic Bose–Chaudhuri–Hocquenghem (BCH) codes with length [Formula: see text] are determined, where [Formula: see text] is an odd prime power and [Formula: see text] is an integer. Further, their dimensions are calculated precisely for the given designed distance. Consequently, via Hermitian Construction, many new quantum codes could be obtained from these codes, which are not covered in the literature.

The classic Bose – Chaudhuri – Hocquenghem (BCH) codes is famous and well-studied part in the theory of error-correcting codes. Generalization of BCH codes allows us to expand the range of activities in the practical correction of errors. Some generic BCH codes are able to correct more errors than classic BCH code in one message block. So it is important to provide appropriate method of error correction. After our investigation it was found that polynomial-norm method is most convenient and effective for that task. The result of the study was a model of a polynomial-norm decoder for a generic BCH code at length 65.

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

Error correction coding (ECC) has become one of the most important tasks of flash memory controllers. The gate count of the ECC unit is taking up a significant share of the overall logic. Scaling the ECC strength to the growing error correction requirements has become increasingly difficult when considering cost and area limitations. This work presents a configurable encoding and decoding architecture for binary Bose–Chaudhuri–Hocquenghem (BCH) codes. The proposed concept supports a wide range of code rates and facilitates a trade-off between throughput and space complexity. Commonly, hardware implementations for BCH decoding perform many Galois field multiplications in parallel. We propose a new decoding technique that uses different parallelization degrees depending on the actual number of errors. This approach significantly reduces the number of required multipliers, where the average number of decoding cycles is even smaller than with a fully parallel implementation.

A novel algorithm of blind recognition of Bose-Chaudhuri-Hocquenghem (BCH) codes is proposed to solve the problem of Adaptive Coding and Modulation (ACM) in cognitive radio systems. The recognition algorithm is based on soft decision situations. The code length is firstly estimated by comparing the Log-Likelihood Ratios (LLRs) of the syndromes, which are obtained according to the minimum binary parity check matrixes of different primitive polynomials. After that, by comparing the LLRs of different minimum polynomials, the code roots and generator polynomial are reconstructed. When comparing with some previous approaches, our algorithm yields better performance even on very low Signal-Noise-Ratios (SNRs) with lower calculation complexity. Simulation results show the efficiency of the proposed algorithm.

In this paper, the theoretical lower-bound on the success probability of blind reconstruction of Bose–Chaudhuri–Hocquenghem (BCH) codes is derived. In particular, the blind reconstruction method of BCH codes based on the consecutive roots of generator polynomials is mainly analyzed because this method shows the best blind reconstruction performance. In order to derive a performance lower-bound, the theoretical analysis of BCH codes on the aspects of blind reconstruction is performed. Furthermore, the analysis results can be applied not only to the binary BCH codes but also to the non-binary BCH codes including Reed–Solomon (RS) codes. By comparing the derived lower-bound with the simulation results, it is confirmed that the success probability of the blind reconstruction of BCH codes based on the consecutive roots of generator polynomials is well bounded by the proposed lower-bound.

This work is the further development of the theory of norms of syndromes: the theory of polynomial invariants of G-orbits of errors expands with the group G of automorphisms of binary cyclic BCH codes obtained by joining the degrees of cyclotomic permutation to the group Γ and practically exhausting the group of automorphisms of BCH codes. It is determined that polynomial invariants, like the norms of syndromes, have a scalar character and are one-to-one characteristics of their orbits for BCH codes with a constructive distance of five. The paper introduces the corresponding vector polynomial invariants for primitive cyclic BCH codes with a constructive distance of seven, next to the norms of the syndromes that are already vector quantities; the basic properties of the vector polynomial invariants are investigated. It is established that the property of mutual unambiguity is violated: there are G-orbit-isomers, which are different, but have the same vector polynomial invariants. It is substantiated and demonstrated by examples that this circumstance greatly complicates error decoding algorithms based on polynomial invariants

The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of [Formula: see text]-ary BCH codes with length [Formula: see text] (also called antiprimitive BCH codes in the literature), where [Formula: see text] is a power of 2 and [Formula: see text]. By a detailed analysis of some useful properties about [Formula: see text]-ary cyclotomic cosets modulo [Formula: see text], Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of [Formula: see text]-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.

Error correcting codes (ECCs) are essential to transmission and data storage sys-tems to protect the information from errors introduced by noisy communication channels. There are two main classes of ECCs, namely algebraic and iterative ECCs. While iterative ECCs like low-density parity-check (LDPC) codes provide improved performance in the waterfall region albeit exhibiting flooring effect for not so well-designed codes, algebraic ECCs like Bose–Chaudhuri–Hocquenghem (BCH) and Reed Solomon (RS) codes provide guaranteed error correction capability irrespective of the waterfall or error floor regions. Due to recent advancements in higher-dimensional data storage technologies like shingled and 2-D magnetic recording (TDMR), 3-DNAND flash memories, and holographic memories, native 2-Dsignal processing and coding techniques are re-quired to overcome inter-symbol interference (ISI) and noise leading to 2-Dburst and random errors. With high data densities beyond 2 Tb/in2 in practical TDMR channels, reliable information storage and retrieval require highly efficient ECCs. The primary motivation of this dissertation is to design efficient hardware architectures for error correcting codes pertaining to 1-Dand 2-Dstorage channels. The focus topics are as follows: (i) First, we designed a high-throughput 1-DLDPC decoder using layered and non-layered min-sum algorithm based on non-uniform quantization on a field programmable gate array (FPGA) kit. Unlike the standard state-of-the-art uniform quantization used in virtually all decoder circuits, our non-uniform quantization technique achieves a slight performance improvement in the signal-to-noise ratio (SNR) using the same bit budget as the uniform case. Using 1 bit lesser than uniform quantization, it yields area savings for the block RAMs used for storing intermediate check node and variable node messages. (ii) We proposed efficient encoding and decoding hardware architectures for (n,k), t-error correcting BCH product codes in the frequency domain. Using the properties of conjugate classes over a finite field, we reduced the algorithmic complexity of the encoder, leading to a significant reduction in the hardware complexity. v vi A low-latency (2t + 2) decoder for the above encoder is also designed. For a particular case of n = 15 and t = 2, the architectures were implemented on a FPGA kit, giving high throughputs of 22.5 Gbps and 5.6 Gbps at 100 MHz for the encoder and decoder respectively. (iii) We proposed fast and efficient hardware architectures for a 2-D BCH code of size n × n, with a quasi-cyclic burst error correction capability of t × t, in the frequency domain for data storage applications. A fully parallel encoder with the ability to produce an output every clock cycle was designed. Using conjugate class properties of finite fields, the algorithmic complexity of the encoder was significantly reduced, leading to a reduction in the number of gates by about 94% compared to the brute force implementation per 2-Dinverse discrete finite field Fourier transform (IDFFFT) point for a 15 × 15, t = 2, 2-DBCH code. We also designed a pipelined, low-latency decoder for the above encoder. The algorithmic complexities of various pipeline stages of the decoder were reduced significantly using finite field properties, reducing the space complexity of the entire decoder. For a particular case of n = 15 and t = 2, the architectures were implemented targeting a Kintex 7 KC-705 FPGA kit, giving high throughputs of 22.5 Gbps and 5.6 Gbps at 100 MHz for the encoder and decoder, respectively. (iv) We developed an efficient design architecture for finding the roots of a bi-variate polynomial over GF(q) by extending the Chien search procedure to two-dimensions. The complexity of the Chien search was reduced to an order of the number of conjugacy classes over GF(qλ), leading to a significant reduction in the computational complexity. We provided an efficient design architecture for our algorithm towards a circuit realization, useful for decoding of 2-Dalgebraic ECCs. v) Native 2-DLDPC codes provide 2-Dburst erasure correction capability and have promising applications in TDMR technology. Though carefully constructed rastered 1-DLDPC codes can provide 2-Dburst erasure correction, they are not as efficient as 2-Dnative codes constructed for handling 2-Dspan of burst erasures. Our contributions are two-fold: (a) We propose a new 2-DLDPC code with girth greater than 4 by generating a parity check tensor through stacking permutation tensors of size p×p×p along the i,j,k axes. The permutations are achieved through circular shifts on an identity tensor along different co-ordinate axes in such a way that it provides a burst erasure correction capability of at least p×p. (b) We propose a fast, efficient, and scalable hardware architecture for a parallel 2-DLDPC decoder based on the proposed code construction for data storage applications. Through efficient indexing of the received messages in a RAM, we propose novel routing mechanisms for messages between the check nodes and variable nodes through a set of two barrel shifters, producing shifts along two axes. Through simulations, we show that the performance of the proposed 2-D LDPC codes match a 1-DQC-LDPC code, with a sharp waterfall drop of 3-4 orders of magnitude over ∼0.3 dB, for random errors over code sizes of ∼32 Kbits or equivalently ∼180×180 2-Darrays. Further, we prove that the proposed native 2-DLDPC codes outperform their 1-Dcounterparts in terms of 2-Dcluster erasure correction ability. For p = 16 and code arrays of size 48 × 48, we implemented the proposed design architecture on a Kintex-7 KC-705 FPGA kit, achieving a significantly high worst case throughput of 12.52 Gbps at a clock frequency of 163 MHz.