Yaotsu Chang

Algebraic decoding of (71, 36, 11), (79, 40, 15), and (97, 49, 15) quadratic residue codes

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.

Algebraic Decoding of the

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.

(89, 45, 17)

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.

Quadratic Residue Code

In this paper, an algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey algorithm. By a modification of the technique developed by He et al., one can express the unknown syndromes as functions of the known syndromes. The unknown syndromes are determined by an efficient algorithm also developed in this paper. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the Berlekamp-Massey algorithm. The decoding scheme, developed here, is easier to implement than the previous decoding algorithm developed for the Golay code and the (47, 24, 11) QR code. Moreover, it can be extended to decode all codes of the family of binary quadratic residue codes with irreducible generating polynomials.

Algebraic decoding of (103, 52, 19) and (113, 57, 15) quadratic residue codes

This paper proposes an improved voice activity detection (VAD) algorithm using wavelet and support vector machine (SVM) for European Telecommunication Standards Institution (ETSI) adaptive multi-rate (AMR) narrow-band (NB) and wide-band (WB) speech codecs. First, based on the wavelet transform, the original IIR filter bank and pitch/tone detector are implemented, respectively, via the wavelet filter bank and the wavelet-based pitch/tone detection algorithm. The wavelet filter bank can divide input speech signal into several frequency bands so that the signal power level at each sub-band can be calculated. In addition, the background noise level can be estimated in each sub-band by using the wavelet de-noising method. The wavelet filter bank is also derived to detect correlated complex signals like music. Then the proposed algorithm can apply SVM to train an optimized non-linear VAD decision rule involving the sub-band power, noise level, pitch period, tone flag, and complex signals warning flag of input speech signals. By the use of the trained SVM, the proposed VAD algorithm can produce more accurate detection results. Various experimental results carried out from the Aurora speech database with different noise conditions show that the proposed algorithm gives considerable VAD performances superior to the AMR-NB VAD Options 1 and 2, and AMR-WB VAD.

Algebraic Decoding of Quadratic Residue Codes Using Berlekamp-Massey Algorithm *

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-Salas 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.

Improved voice activity detection algorithm using wavelet and support vector machine

In this letter, some perfect Gaussian integer sequences of period 2m - 1 are proposed based on the trace representations of Legendre sequences, Halls 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.

Algebraic Decoding of a Class of Binary Cyclic Codes Via Lagrange Interpolation Formula

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.

Perfect Gaussian Integer Sequences of Odd Period

For algebraic decoding of the (23, 12, 7) Golay code, this letter proposes a new error locator polynomial, called the unusual general error locator polynomial, whose coefficients are expressed as a sum of powers of their previous ones. Because of this special property, the determination of such a polynomial can be terminated earlier, and the number of errors occurred can be recognized at the same time.

{2^m} - 1

A modified Euclidean decoding algorithm to solve the Berlekamps key equation of Reed-Solomon code for correcting errors, is presented in this paper. It is derived to solve the error locator and evaluator polynomials simultaneously without performing the operations of polynomial division and field element inversion. In this algorithm, the number of iterations used to solve the equation is fixed, and also the weights used to reduce the degree of the error evaluator polynomial at each iteration can be extracted from the coefficient of fixed degree. Therefore, this proposed algorithm saves many controlling circuits, and provides module architecture with regularity. As a result it is simple and easy to implement, and in addition it can be easily configured for various applications.