For the recently developed quasi-reversible BCH codes with long lengths and high error-correcting capability, this paper is aimed at proposing a new and faster decoding procedure. It consists of four steps: 1) compute the consecutive syndromes; 2) calculate the syndrome functions by the forward and backward recursions; 3) solve a linear subsystem together with one matrix multiplication in order to find an error-locator polynomial; 4) determine the errors from the obtained polynomial by using the root-finding algorithm. This procedure, especially in Steps 2 and 3, differs greatly from the conventional procedures, which determine an error-locator polynomial directly from solving a linear system with the aid of the consecutive syndromes. The key idea behind this decoding technique is that the computational complexity of such a small subsystem instead of an originally large linear system can be significantly reduced, although there are additional forward and backward syndrome calculations with low complexity increasing. Finally, the illustrative examples and numerical simulations can be helpful to demonstrate the accuracy and efficacy of the presented decoding technique at different error-correcting capabilities.

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.

A class of cyclic codes has recently been decoded by using the weak-locator polynomials instead of the conventional error-locator polynomials. In this paper, a generalization of weak-locator polynomials to the multivariate cases, called the multivariate weak-locator polynomials, is defined, and a new matrix whose determinant can be expressed as a multivariate weak-locator polynomial is developed for cyclic codes. Moreover, a modified matrix along with Gaussian elimination found herein enables one to determine error positions precisely. The presented matrices result in a reduction of the number of syndromes when compared with the previously known matrices. The simulation shows that, for example, with the capability of correcting more errors, a considerably sophistical scheme of decoding the (97, 49, 15) binary cyclic code based on the proposed matrices is around 115 times faster than other existing decoders. Therefore, in general, it is developed to facilitate faster decoding of a large class of cyclic codes and is naturally suitable for the software implementation.

In this letter, some perfect Gaussian integer sequences of period 2m- 1 are proposed based on the trace representations of Legendre sequences, Hall's sextic residue sequences, m-sequences, and Gordon-Mills-Welch (GMW) sequences over the finite field \BBF2m. Moreover, the energy efficiency of these sequences is approximately 1 for sufficiently large m.

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.

Recently, the unified unknown syndrome representations to decode a class of binary cyclic codes have been developed by using Lagrange interpolation formula (discussed by Chang and Lee in 2010). In this study, a new method by combining the syndrome matrix search and modified Chinese remainder theorem is proposed to express the unified unknown syndrome representation as a rational function in terms of the known syndromes. A computer simulation has been executed to determine the syndrome matrices for binary cyclic codes of lengths less than or equal to 51. Compared to the Lagrange interpolation method, the method presented here substantially reduces the computational time for binary cyclic codes generated by irreducible polynomials. Finally, a complete decoding of the (31, 16, 7) quadratic residue code with inverse-free Berlekamp–Massey algorithm is given as an illustration.

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.

In this paper, three algebraic decoding algorithms are proposed for the binary quadratic residue (QR) codes generated by irreducible polynomials. The polynomial relations among the syndromes and the coefficients of the error-locator polynomials have been computed with Lagrange interpolation formula (LIF). Unlike some previous QR decoders, which may take several iterations to decode a corrupted word, the iteration number of the first two algorithms is at most one. The processes in the first algorithm are the calculation of consecutive syndromes, inverse-free Berlekamp-Massey algorithm (IFBMA), and the Chien search. One of Orsini-Sala's results on the structure of general error-locator polynomials is generalized and applied to derive the second (respectively, third) algorithm that consists of the determination of general error-locator polynomial (respectively, classical error-locator polynomials) and the Chien search. Finally, the(17, 9, 5), (23, 12, 7), and (41, 21, 9) QR decoders are illustrated and their complexity analyses are given.

In this paper, a new algorithm based on the Tschirnhaus transforms is developed to reduce the computation complexity of the 10-order line spectrum pairs (LSP) frequencies. The first step of the proposed algorithm is to derive a quartic equation from the 1st derivative of the given 5-degree LSP polynomial. Then the extremes of the 5-degree LSP polynomial can be found by applying the Tschirnhaus transform to the above quartic equation. By the use of these extremes as the initial approximations, one can easily solve the roots of the 5-degree LSP polynomial via the Newton's method and get the accurate LSP frequencies. One of the main advantages of the proposed algorithm is the rapid root determination of a quartic equation without complex number operations and resulting in considerable computational saving. Compared to other methods, the proposed algorithm can determine the precise LSP frequencies with the lowest computational complexity.

This letter proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleason's theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distributions of binary (138, 69, 22) and (168, 84, 24) extended quadratic residue codes are given.

This paper proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleason’s theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distribution of binary (168, 84, 24) extended quadratic residue code is given.

Recently, an algebraic decoding algorithm suggested by Truong (2005) for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial. In this paper, based on the ideas of the algorithm mentioned above, an algebraic decoder for the (89, 45, 17) binary quadratic residue code, the last one not decoded yet of length less than 100 , is proposed. It was also verified theoretically for all error patterns within the error-correcting capacity of the code. Moreover, the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations.

Recently, normal bases have been an appealing technique for hardware implementation in many applications, especially when finite fields are very large, such as the public key cryptosystems. In this article, a new viewpoint is introduced for performing the multiplication in the normal basis representation over binary field where the field defining irreducible polynomial is an all-one polynomial. The proposed multipliers carry out the normal basis multiplication by using extended polynomial basis multiplier to construct two architectures. These two classes of architectures are in bit-serial and bit-parallel which reduce the cost of space and time respectively.

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.

The weight distributions of binary quadratic residue codes C can be computed from the weight distribution of a subset of C containing one-fourth (resp., one-eighth) of the codewords in C when the length of the code is congruent to 1 (resp., -1) modulo 8. An algorithm to determine the weight distributions of binary cyclic codes is given. As a consequence, the weight distributions of (73,37,13), (89,45,17), and (97,49,15) quadratic residue codes are determined precisely.

An algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey (BM) algorithm. By applying a technique developed by R. He et al. (see IEEE Trans. Inf. Theory, vol.47, p.1181-6, 2001), one can express unknown syndromes as functions of known syndromes. An efficient algorithm is also developed to determine the unknown syndromes. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the inverse-free BM algorithm. The new decoding scheme can be used to implement the (79,40,15) quadratic residue (QR) code which has not been treated so far. It is verified by a computer program that uses the C++ language.

Recently, a new algebraic decoding algorithm for quadratic residue (QR) codes was proposed by Truong et al. Using that decoding scheme, we now develop three decoders for the QR codes with parameters (71, 36, 11), (79, 40, 15), and (97, 49, 15), which have not been decoded before. To confirm our results, an exhaustive computer simulation has been executed successfully.

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.