Solving quadratic equations over finite fields is a fundamental task in algebraic coding theory and serves as a key subroutine for computing the roots of cubic and quartic polynomials. Notably, any quadratic polynomial over binary extension fields can be transformed into the reduced form $x^2+x+c\in \mathbb{F}_{2^m}[x]$, for which existing formula-based methods rely on heavy exponentiation or case distinctions on $m$ (odd/even or powers of two), limiting uniformity and efficiency. This paper presents a unified, formula-based solution for all positive integers $m$ that uses only exclusive-OR operations (XORs). The approach leverages a Reed-Muller matrix characterization of evaluations and transforms the problem into computing a binary matrix-vector multiplication. The total cost is at most $m^2-2m+1$ XORs, and under parallelism, the latency is $\lceil \log_2 m\rceil$ XORs, making the method attractive for low-power, low-latency applications.

Based on a recently proposed fast Fourier transform by Lin, Chung, and Han, this paper presents two fast erasure decoding algorithms for Reed–Solomon (RS) codes over binary extension fields of lengthNand dimensionK. The first algorithm applies to low-rate RS codes (i.e.,K/N≤ 0:5) and achieves a complexity ofO(N log K). The second algorithm applies to high-rate RS codes (i.e.,K/N≥ 0:5) and achieves a complexity ofO(N log(N–K)). Compared to recent state-of-the-art algorithms, both proposed algorithms achieve the best complexity, resulting in significant throughput improvements in Single Instruction Multiple Data (SIMD) based simulations. Besides yielding new fast algorithms for RS codes, this paper also presents a new interpolation formula, as well as related results, which may be of independent interest.

In this paper, a new soft-decision decoding for extended Bose-Chaudhuri-Hocquenghem (eBCH) codes, referred to as CPC-SCL, is proposed, and it can achieve error-correction performance close to that of polarization-adjusted convolutional (PAC) codes in [1] when the code length n and dimension k are 128 and 64, respectively. Specifically, this paper first decomposes the eBCH code into a concatenated structure comprising an outer code and a Reed-Muller inner code. The outer code has a parity-check matrix characterized by a special block structure, which reveals that the positions of all frozen bits (including frozen zero bits and dynamic frozen bits) in the eBCH code are closely related to the index weights of elements from the perspective of the polar code. Subsequently, the CPC-SCL decoding of the eBCH codes is proposed by utilizing the cyclic property of codewords and parity-check-aided successive cancellation list (PC-SCL) decoding. Simulations also demonstrate that, over an additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) modulation, the proposed decoding can achieve near maximum-likelihood (ML) performance at n = 64, k = 24 or 45.

In 2021, Tang et al. proposed an improved construction of Reed-Solomon (RS) erasure codes with four parity symbols to accelerate the computation of Reed-Muller (RM) transform-based RS algorithm. The idea is to change the original Vandermonde parity-check matrix into a systematic Vandermonde parity-check matrix. However, the construction relies on a computer search and requires that the size of the information vector of RS codes does not exceed$52$. This paper improves its idea and proposes a purely algebraic construction. The proposed method has a more explicit construction, a wider range of codeword lengths, and competitive encoding/erasure decoding computational complexity.

Binary array codes are widely used in storage systems to prevent data loss, such as the Redundant Array of Independent Disks (RAID). Most designs for such codes, such as Blaum-Roth (BR) codes and Independent-Parity (IP) codes, are carried out on the polynomial ring F2[x]/⟨ Σp-1i=0xi⟩, where F2is a binary field, andpis a prime number. In this paper, we consider the polynomial ring F2[x]/⟨ Σp-1i=0xiτ⟩, where p > 1 is an odd number and τ ≥ 1 is any power of two, and explore variant codes from codes over this polynomial ring. Particularly, the variant codes are derived by mapping parity-check matrices over the polynomial ring to binary parity-check matrices. Specifically, we first propose two classes of variant codes, termed V-ETBR and V-ESIP codes. To make these variant codes binary maximum distance separable (MDS) array codes that achieve optimal storage efficiency, this paper then derives the connections between them and their counterparts over polynomial rings. These connections are general, making it easy to construct variant MDS array codes from various forms of matrices over polynomial rings. Subsequently, some instances are explicitly constructed based on Cauchy and Vandermonde matrices. In the proposed constructions, both V-ETBR and V-ESIP MDS array codes can have any number of parity columns and have the total number of data columns of exponential order with respect top. In contrast, previous binary MDS array codes only have a total number of data columns of linear order with respect top. This makes the codes proposed in this paper more suitable for application to large-scale storage systems. In terms of computation, two fast syndrome computations are proposed for the Vandermonde-based V-ETBR and V-ESIP MDS array codes, both meeting the lowest known asymptotic complexity among MDS codes. Due to the fact that all variant codes are constructed from parity-check matrices over simple binary fields instead of polynomial rings, they are attractive in practice.

This paper proposes a class of rateless Reed-Solomon (RLRS) codes with near-linear encoding/decoding complexities. Like fountain codes, the RLRS codes can generate a reasonably large number of encoded packets in packet-level transmissions. Furthermore, the RLRS codes are maximum distance separable (MDS) codes that always maintain zero reception overhead. In the proposed RLRS codes, the preservative field extensions are realized through Cantor’s field tower, which avoids searching some quadratic irreducible polynomials as in the prior RLRS codes based on Cauchy generator matrices. Additionally, the proposed RLRS codes are based on Vandermonde generator matrices, whereby the LCH transforms, a variant of fast Fourier transforms (FFTs) over binary extension fields, can be employed to reduce the encoding/decoding complexity. To further improve computational efficiency, this paper also proposes a scheduling scheme for the LCH transforms to generate encoded packets on demand, instead of generating packets whose number must be a power of two. Analysis shows that compared to the prior approach, the used field tower leads to a lower speed of computational complexity growth caused by field extensions. In addition, with the total number of source packets to be transmitted beingk, analysis shows that the proposed RLRS codes have the encoding/decoding complexityO(log2k) per source packet, superior toO(k) in the prior approach.

Reed-Solomon (RS) codes are widely used in storage systems to provide high data reliability. The existing RS codes are constructed over finite field with field size larger than the code length that have high encoding/decoding complexity. In this paper, we propose a new construction for RS codes over a cyclic ring instead of finite field. Our new RS codes only incurs XOR and cyclic-shift operation in the encoding/decoding processes, and therefore have lower encoding/decoding complexity than the existing RS codes. In addition, we show that we can employ the efficient Reed-Muller (RM) transform in our new RS codes that reduces the encoding/decoding complexity. Moreover, we implement our new RS codes by C++ and show that our new RS codes have better encoding/decoding performance than the existing RS codes for the evaluated parameters.

In clinical diagnosis, multimodal medical image fusion is meaningful and necessary, for the reason that some diseases need to be diagnosed in combination with the situation of different tissues of patients. Spiral computed tomography (CT) realizes the high precision and smooth reconstruction of bone tissue, while it can not represent the color and texture information in soft tissue reconstruction with high accuracy. The face scan precisely records the color and shape of the maxillofacial region. The diagnosis of some diseases (like cavernous hemangioma and jaw deformity caused by idiopathic condylar resorption) needs to combine the information of maxillofacial soft tissue and bone, so it is of great significance to fuse spiral CT and face scan images. In this article, a novel intelligent Internet of Things scene is proposed: a multimodal medical images acquisition and fusion system, and by combining CT machine and face scan equipment, the CT and face scan of patients can be synchronously collected. Deep point neural networks are used to extract feature points and a threshold iterative closest point algorithm performing registration with deep feature points and contributed region segmentation is applied. Finally, high-precision fused modal data is output at the mobile terminal to facilitate diagnosis and analysis and improve the efficiency of doctor–patient communication. Quantitative experiments show promising results, and clinical experiments prove that our method enables patients and doctors to better understand the state of an illness and improves the efficiency of doctor–patient communication.

Based on the Reed-Muller (RM) transform, this paper proposes a Reed-Solomon (RS) encoding/erasure decoding algorithm for any number of parities. Specifically, we first generalize the previous RM-based syndrome calculation, which allows only up to seven parities, to support any number of parities. Then we propose a general encoding/erasure decoding algorithm. The proposed encoding algorithm eliminates the operations in solving linear equations, and this improves the computational efficiency of existing RM-based RS algorithms. In terms of erasure decoding, this paper employs the generalized RM-based syndrome calculation and lower–upper (LU) decomposition to accelerate the computational efficiency. Analysis shows that the proposed encoding/erasure decoding algorithm approaches the complexity of$\lfloor \lg T \rfloor + 1$XORs per data bit with$N$increasing, where$T$and$N$denote the number of parities and codeword length respectively. To highlight the advantage of the proposed RM-based algorithms, the implementations with Single Instruction Multiple Data (SIMD) technology are provided. Simulation results show that the proposed algorithms are competitive, as compared with other cutting-edge implementations.

This article describes a fast Reed-Solomon encoding algorithm with four and seven parity symbols in between. First, we show that the syndrome of Reed-Solomon codes can be computed via the Reed-Muller transform. Based on this result, the fast encoding algorithm is then derived. Analysis shows that the proposed approach asymptotically requires 3 XORs per data bit, representing an improvement over previous algorithms. The simulation demonstrates that the performance of the proposed approach improves with the increase of code length and is superior to other methods. In particular, when the parity number is 5, the proposed approach is about two times faster than other cutting-edge methods.