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.

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.

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 novel shift-sum decoding method for non-binary cyclic codes, which only requires finite field operations but yields advanced decoding performance. Using the cyclically different minimum-weight dual codewords (MWDCs) and their proper shifts, a frequency matrix can be obtained as a reliability metric for identifying the error positions and magnitudes. By analyzing the statistical distributions of the matrix entries, the rationale for the shift-sum decoding’s advanced error-correction capability is revealed. Based on this decoding method, a hard-decision iterative shift-sum (HISS) decoding algorithm is first proposed. It can correct errors beyond half of the code’s minimum Hamming distance. By further utilizing the reliability information obtained from the channel, a soft-decision iterative shift-sum (SISS) decoding algorithm is then proposed to improve the decoding performance. Both the HISS and the SISS algorithms are realized only with polynomial multiplications and numerical comparisons, which are hardware-friendly. To further improve the error-correction performance, the HISS and SISS algorithms can be integrated in a Chase decoding mechanism for handling the test-vectors. Simulation results on Reed-Solomon (RS) and non-binary BCH (NB-BCH) codes show that the proposed algorithms yield a competent decoding and complexity performances in comparison with the existing decoding algorithms.

Using the minimum weight dual codewords (MWDCs) of a cyclic code, the shift-sum decoding can correct errors beyond half of the code's minimum Hamming distance. It utilizes the frequency of the syndrome polynomials' coefficients to identify the erroneous positions and correct the errors. This paper analyzes the plausibility of the shift-sum decoding for both binary and non-binary cyclic codes. It first determines the probability distributions of the frequency of the syndrome polynomials' coefficients as well as their expected values for the erroneous and non-erroneous positions. Based on these characterizations, this work further provides an analysis for the iterative shift-sum decoding, unveiling the statistical rationale on the shift-sum decoding's capability of correcting errors beyond the half distance bound.