Cyclic Codes Research Articles

An efficient high throughput BCH module for multi-bits error correction mechanism on hardware platform

The bose-chaudhuri-hocquenghem (BCH) codes are a cyclic error correction codes (ECC) class. The BCH is constructed by using a polynomial over the Galois field. The BCH codes can detect and correct the multi-bits with an easy decoding mechanism. The BCH codes are used in most of the storage device's cryptography, disk drives, and satellite applications. This manuscript presents an efficient high-throughput BCH module with an encoding and decoding mechanism for multi-bit corrections. The BCH code of (15, k) is used to construct the encoder and decoder architectures. The BCH encoder decoder (ED) module with single error correction (SEC), double error correction (DEC), and triple-error correction (TEC) are discussed in detail. The BCH encoder module uses a linear feedback shift register (LFSR). The BCH decoder with SEC and DEC is constructed using the syndrome generator module (SGM) and chien search module (CSM). The BCH decoder with TEC is designed using SGM, inversion-based berlekamp-massey-algorithm (BMA), and CSMs. The BCH-ED module with SEC, DEC, and TEC utilizes <1 % chip area on Artix-7 FPGA. The BCH-ED with SEC, DEC, and TEC achieves a throughput of 7.13 Gbps, 1.2 Gbps, and 0.803 Gbps, respectively. Lastly, the BCH module is compared with existing BCH approaches with better improvement in chip area, frequency, and throughput parameters.

Improved Upper Bounds on the Number of Non-Zero Weights of Cyclic Codes

Improved Upper Bounds on the Number of Non-Zero Weights of Cyclic Codes

Another Infinite Family of Binary Cyclic Codes With Best Parameters Known

Another Infinite Family of Binary Cyclic Codes With Best Parameters Known

Some results on the Hamming distances of cyclic codes

Some results on the Hamming distances of cyclic codes

On reversibility problem in DNA bases over a class of rings

Let R k = Z 4 [ u 1 , u 2 , … , u k ] / ⟨ u i 2 − u i , u i u j − u j u i ⟩ be a non-chain ring of characteristic 4, where 1 ≤ i , j ≤ k and k ≥ 1 . In this article, we discuss reversible cyclic codes of odd lengths over the ring R k . We construct bijections between the elements of the ring R k and DNA- 2 k bases for k = 1, 2 in such a manner that the reversibility problem is solved. Employing these bijections, reversible complement cyclic codes of odd lengths are generated. Furthermore, we construct a Gray map Φ : R k n → Z 4 2 k n and as an application of the Gray map Φ, we obtain the GC-content of cyclic codes of arbitrary odd length over the ring R k . Meanwhile, we provide some examples of reversible cyclic codes of odd lengths over the ring R k for different values of k, and also obtain the Lee distances of these codes.

Comparative Analysis of Channel Coding in Communication Technology

The performance analysis of Binary Phase Shift Keying (BPSK) and Quadrature Phase Shift Keying (QPSK) employing error-correcting code is highlighted in this work. Several error-correcting code types were employed via an Additive White Gaussian Noise (AWGN) channel to determine the bit error rate. Two techniques were utilized as encoders and decoders: hamming code and cyclic code. Source coding used Huffman coding. Essentially, signal energy to noise power density ratio (Eb/No) and the bit rate error (BER) were used to measure performance. Comparisons were also made between BPSK and QPSK in terms of symbol error capability. MATLAB R2022b was the program used for all the simulations. The conclusion is that different coding methods have a certain error correction range, and the signal-to-noise ratio (SNR) can only reduce the BER in the error correction range, and if it is not within the error correction range, it will increase the bit error rate.

Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet

In this note, we study skew cyclic and skew constacyclic codes over the mixed alphabet R=FqR1R2, where q=pm, p is an odd prime with m odd and R1=Fq+uFq with u2=u, and R2=Fq+uFq+vFq with u2=u,v2=v,uv=vu=0. Such codes consist of the juxtaposition of three codes of the same size over Fq,R1, and R2, respectively. We investigate the generator polynomial for skew cyclic codes over R. Furthermore, we discuss the structural properties of the skew cyclic and skew constacyclic codes over R. We also study their q-ary images under suitable Gray maps.

Several Families of Ternary Negacyclic Codes and Their Duals

Several Families of Ternary Negacyclic Codes and Their Duals

Study of cyclic codes from low differentially uniform functions and its consequences

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7].

Constacyclic locally recoverable codes from their duals

A code has locality r if a symbol in any coordinate of a codeword in the code can be recovered by accessing the value of at most r other coordinates. Such codes are called locally recoverable codes (LRCs for short). Since LRCs can recover a failed node by accessing the minimum number of the surviving nodes, these codes are used in distributed storage systems such as Microsoft Azure. In this paper, constacyclic LRCs are obtained from their parity-check polynomials. Constacyclic codes with locality r≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\le 2$$\\end{document} and dimension 2 are obtained and a sufficient and necessary condition for these codes to have locality 1 is given. Then, the construction is generalized. Distance optimal constacyclic LRCs with distance 2 are obtained. Also, constacyclic codes with locality 1 are constructed. They may be so useful in practice thanks to their minimum locality. Constacyclic codes whose locality is equal to their dimension are given. Furthermore, constacyclic LRCs are obtained from cyclotomic cosets.

Obtaining new classes of optimal linear codes by puncturing and shortening optimal cyclic codes

AbstractIn this paper we use the puncturing and shortening techniques on two already-known classes of optimal cyclic codes in order to obtain three new classes of optimal linear codes achieving the Griesmer bound. The weight distributions for these codes are settled. We also investigate their dual codes and show that they are either optimal or almost optimal with respect to the sphere-packing bound. Moreover, these duals contain classes of almost maximum distance separable codes which are shown to be proper for error detection. Further, some of the obtained optimal linear codes are suitable for constructing secret sharing schemes with nice access structures.

Derivative Descendants of Cyclic Codes and Derivative Decoding

Derivative Descendants of Cyclic Codes and Derivative Decoding

An Infinite Family of Binary Cyclic Codes With Best Parameters

An Infinite Family of Binary Cyclic Codes With Best Parameters

Large optimal cyclic subspace codes

Constant dimension codes, a specialized subclass of subspace codes, have garnered significant attention in recent years due to their applications in random network coding. Among these codes, large cyclic constant dimension codes, endowed with optimal minimum distance, offer efficient solutions for encoding and decoding algorithms. This paper is dedicated to the design of cyclic constant dimension codes that achieve the dual goals: maximizing code size while maintaining optimal minimum distance. We first explore a new form of Sidon spaces and employ this new form to construct several families of cyclic subspace codes. Our codes have optimal minimum distances and have more codewords than previous constructions in the literature.

A tactical data link model based on CCSK-FMSK

In collaborative combat operations, the challenges brought by the communication adversities of tactical data links are becoming increasingly prominent. Ensuring the reliable connection of tactical data links has always been a key issue, especially considering the increasing impact of external interference on the signal at the receiving end of the system, which in turn affects the bit error rate (BER) of the tactical data link system. To address this challenge, this paper proposes a tactical data link anti-interference scheme based on cyclic code shift keying (CCSK) spread spectrum and minimum shift keying (MSK) modulation fusion frequency hopping technology, named CCSK-FMSK, and introduces turbo code and cyclic redundancy check (CRC) coding to construct a tactical data link model based on CCSK-FMSK. Specifically, CCSK-FMSK disperses interference power by expanding the signal spectrum, enhancing the anti-interference ability of tactical data links. CRC coding and Turbo coding achieve error detection and correction by introducing supervised symbols. In addition, this paper proposes a soft despreading and a soft demodulation algorithm corresponding to CCSK spread spectrum and MSK modulation to achieve cascade with Turbo decoder and improve the error rate performance and robustness of the model. Compared with existing models, the tactical data link based on CCSK-FMSK has better BER performance.

Cyclic Codes over a Non-Local Non-Unital Ring

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Decoding of cyclic codes over quaternion integers by modified Berlekamp–Massey algorithm

Decoding of cyclic codes over quaternion integers by modified Berlekamp–Massey algorithm

Gray Images of Constacyclic Codes over a Non-chain Extension of Z 4

Let [Formula: see text] be the ring of integers modulo 4. We study the [Formula: see text]-constacyclic and [Formula: see text]-cyclic codes over the non-chain ring [Formula: see text] for a unit [Formula: see text] in [Formula: see text]. We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over [Formula: see text] are cyclic, quasi-cyclic or permutation equivalent to this code. For an odd positive integer [Formula: see text], we determine the generator polynomials of cyclic and [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text]. Further, we prove that a [Formula: see text]-cyclic code of length [Formula: see text] is a [Formula: see text]-constacyclic code if [Formula: see text] is odd, and a [Formula: see text]-quasi-twisted code if [Formula: see text] is even. A few examples are also incorporated, in which two parameters are new and one is best known to date.

Optimal quinary cyclic codes with three zeros

Optimal quinary cyclic codes with three zeros

Shift-Sum Decoding of Non-Binary Cyclic Codes

Shift-Sum Decoding of Non-Binary Cyclic Codes