A Bose-Chaudhuri-Hocquenghem (BCH) is called quasi-reversible if there are consecutive elements a1,⋯,a2in the defining set, where a1is a negative integer and a2is a positive integer. A matrix in band form presented in this article involves the use of the Newton identities and symmetric polynomial identities. This matrix is the coefficient matrix of a linear system. The solution to such a linear system is the coefficients of an error-locator polynomial for quasi-reversible BCH codes. Its computational complexity is significantly lower than BCH decoding using the linear system with coefficient matrix in column echelon form. This article also proposes a new partial syndrome matrix with radical locators and a lot of zeros. Such a matrix is still in band form. The expansion of its matrix determinant is exactly a radical-locator polynomial. Compared to the former radical-locator polynomials in the literature, the newly discovered radical-locator polynomial not only requires less storage memory but also results in computational benefits. Surprisingly, the resulting polynomials are much sparser than those for the narrow-sense BCH codes. Finally, a complete algebraic decoding algorithm for quasi-reversible BCH codes is provided. The error-locator and radical-locator polynomials of degree less than or equal to 5 are listed.

It is well-known that the decoding of algebraic codes with error locators have been extensively conceived in the literature over a half century. The radical locator, which is an ω th root of error locator, has recently been discovered. This paper focuses on the two classes of radical locators. The first is complete radical locators, where all the locators are assigned to radical locators. The second is partial radical locators in that there are only a few locators being radical locators. In the former case based on complete radical locators, a new square matrix whose determinant is a univariate radical-locator polynomial is proposed. In particular, this matrix is modified to allow a large square matrix. It can be transformed by Gaussian elimination to the matrix in a row-echelon form in which zeros appearing in the diagonal entries of the current matrix are able to determine errors implicitly. Furthermore, this work further extends the previous results from the univariate case to multivariate cases. The matrix methods found herein enable one to decode a large class of cyclic codes efficiently. In the latter case, the cyclotomic cosets are a practical approach to select a small subset of positive integers. They are employed to develop partial radical locators. Finally, in contrast with the complete radical locators, the algebraic decoding methods make a natural use of partial radical locators more widely and flexibly to arbitrary cyclic codes.

An efficient decoding of quadratic residue (QR) codes utilising hashing search to find error patterns is presented in this study. The key idea behind the proposed decoding method is theoretically based on the existence of a one‐to‐one mapping between the single primary known syndrome and correctable error patterns. Compared with the binary search time approach, one of the advantages of utilising this method presented in this study is that the hashing search time can be reduced by a factor of two. This method would help reduce the binary search time for finding error patterns when decoding the (23, 12, 7), (41, 21, 9), and (47, 24, 11) QR codes. Furthermore, it would reduce the decoding time by 45% using the fast lookup table decoding method to decode the (71, 36, 11) QR code. Ultimately, the proposed decoding algorithm for QR codes can be made regular, simple, and suitable for software implementations.

This study extends the authors’ earlier work to show that the Gaussian integer sequences of period p m − 1 with p − 2 non‐zero out‐of‐phase autocorrelation values can be constructed from the known families of two‐tuple‐balanced p ‐ary sequences over the finite field , where p is an odd prime and m ≥ 2. The proposed Gaussian integer sequences have high energy efficiency and are superior to the perfect Gaussian integer sequences (introduced by Hu et al. in 2012) for the peak‐to‐average power ratio reduction in orthogonal frequency‐division multiplexing systems.

The algebraic decoding of a p-ary cyclic code consists of four steps: 1) computation of the known syndromes using the received word; 2) computation of the unknown syndromes from the known syndromes; 3) computation of the error positions by a use of the Berlekamp-Massey (BM) algorithm and Chien's search; and 4) computation of the error values by solving a linear system. This paper addresses two problems of determining the error positions and computing the unknown syndromes. To solve the first problem, a new matrix, together with Gaussian elimination instead of the BM algorithm and Chien's search, is proposed. In this new simplified decoder, finding an error-locator polynomial is completely avoided. A main advantage of the presented decoding method is when the Bose-Chaudhuri-Hocquenghem bound is unequal to the minimum distance of the code. Some cyclic codes, which do not have 2t consecutive known syndromes, can be decoded up to their actual minimum distance by using the presented matrix once. To solve the second problem, two algorithms for different square matrices reported recently by Lee et al. are also provided to calculate the value of an unknown syndrome. Finally, an algebraic decoding of the (47,23,15) ternary cyclic code is given.

In this paper, an explicit expression of the weak-locator polynomial for p-ary quadratic residue codes is presented by a modification of the Feng-Tzeng matrix method. The differences between the modified version and the original Feng-Tzeng matrix are that in the new matrix, not every entry is a syndrome, and every syndrome entry is a known syndrome. By utilizing this technique, an algebraic decoding of the ternary (61, 30, 12) quadratic residue code is proposed. This new result has never been seen in the literature to our knowledge. An advantage of the proposed decoding algorithm is that in general the obtained weak-locator polynomials can decode efficiently not only all the error patterns of weights four and five, but also some error patterns of weight six.

A general algorithm for decoding the binary systematic quadratic residue (QR) codes with lookup tables is presented in this study. The algorithm can be applied in decoding the QR codes with either reducible or irreducible generator polynomials. If the generator polynomial of the QR codes is reducible, the number of elements in the Galois field is less than the sum of all correctable error patterns. In other words, the mapping between elements of syndrome set and all correctable error patterns is not one to one. The key idea of decoding based on the mapping between the ordered q-tuples of the primary known syndrome and error patterns is one to one. In addition, the algorithm directly determines the error locations by lookup tables without the operations of multiplication over a finite field. According to the simulation result, the new lookup table decoding algorithm for the (31, 16, 7) QR code and the (73, 37, 13) QR code dramatically reduces the memory required by approximately 90 and 92%, respectively. Moreover, the high speed of decoding procedure could be utilised in modern communication system.

Recently, the general error locator polynomials have been widely used in the algebraic decoding of binary cyclic codes. This paper utilizes the proposed general error locator polynomial to develop an algebraic decoding algorithm for a class of the binary cyclic codes. This general error locator polynomial differs greatly from the previous general error locator polynomial. Each coefficient of the proposed general error locator polynomial is expressed as a binary polynomial in the single syndrome and the degrees of nonzero terms in the binary polynomial satisfy at least one congruence relation.

A new effective lookup table for decoding the binary systematic (41, 21, 9) Quadratic Residue (QR) code up to 4 errors is presented in this paper. The key ideas behind this decoding technique are based on one to one mapping between the syndromes “S1” and the error correctable patterns. Such an algorithm determines the error locations directly by lookup tables without the operations of multiplication over a finite field. Moreover, the methods to dramatically reduce the memory requirement are given. The new algorithm has been verified through a software simulation by C language. The new approach is modular, regular and naturally suitable for SOC software implementation.

In this paper, two algebraic decoders for the (103, 52, 19) and (113, 57, 15) quadratic residue codes, which have lengths greater than 100, are presented. The results have been verified by software simulation that programs in C++ language have been executed to check possible error patterns of both quadratic residue codes.

In the applications of AES, the long-term robustness/reliability during the period of operation should be taken into serious considerations. From such considerations, one may initiate the requirements of the design for diversity against break through from outside. In system design, the use of reconfigurable FPGA can provide higher level of flexibility. In this paper, the proposed system uses different generators, various transforms, modules and algorithms to enhance the randomization of the ciphertext. It is also a challenge to improve the system flexibility and to get a more secure design in the AES system. Several reconfigurable modules are developed on our integrated test-bench.