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

Reducing the computation time of scalar multiplication for elliptic curve cryptography is a significant challenge. This study proposes an efficient method curves over finite fields GF(2m). The proposed first converts into binary number. Then, using Horner’s rule, number divided fixed-length bit-words. Each bit-word undergoes repeating point doubling, which can be precomputed. However, doubling typically involves numerous inverse operations. To address this, effort has been made to develop...

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) to determine error-locator polynomial. In this paper, based on ideas of mentioned above, decoder (89, 45, 17) binary code, last one not decoded yet length less than 100 , is proposed. It was also verified theoretically all error patterns within error-correcting capacity code. Moreover,...

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

A composite number can be factored into either N=mp or N=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where p is an odd prime and m, n ≥ 2 are integers. This paper proposes a method for constructing degree-3 degree-4 perfect Gaussian integer sequences (PGISs) of arbitrary length utilizing upsampling technique the base sequence concept proposed by Hu, Wang, Li. In PGISs, degree defined as distinct nonzero elements within one period...

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, sequence defined as perfect if only it has an ideal periodic autocorrelation function. This paper proposes method for constructing sparse sequences (SPGISs) in which most of the elements zero. The proposed SPGISs obtained by linearly combining four base or their cyclic-shift equivalents using nonzero coefficients equal magnitudes. Each contains belonging to set {±1, ±j}. constructed depends on...

In this letter, some perfect Gaussian integer sequences of period 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> - 1 are proposed based on the trace representations Legendre sequences, Hall's sextic residue m-sequences, and Gordon-Mills-Welch (GMW) over finite field \BBF <sub xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . Moreover, energy efficiency these is approximately for sufficiently large m.

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 been verified by software simulation that programs in C++ language executed to check possible error patterns of both codes.

In this paper, three algebraic decoding algorithms are proposed for the binary quadratic residue (QR) codes generated by irreducible polynomials. The polynomial relations among syndromes and coefficients of 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, iteration number first two is at most one. processes in algorithm calculation consecutive syndromes,...

A complex number whose real and imaginary parts are both integers is called a Gaussian integer . sequence said to be perfect if it has an ideal periodic autocorrelation function (PACF) where all out‐of‐phase values zero. Further, the degree of defined as distinct non‐zero within one period sequence. Recently, sequences have been found important practical applications signal processing tools for orthogonal frequency‐division multiplexing systems. The present article generalises authors’...

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

Recently, the perfect Gaussian integer sequences have been widely used in modern wireless communication systems, such as code division multiple access and orthogonal frequency-division multiplexing systems. This letter presents two different methods to generate long with ideal periodic auto-correlation functions. The key idea of proposed is use a short sequence together polynomial or trace computation over an extension field construct family sequences. period resulting not that sequence,...

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

In this letter, two weak general error locator polynomials are proposed to improve the one-step decoding of (23, 12, 7) binary Golay code. Experimental results show that presented decoders significantly reduce area compared existing decoders.

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

The algebraic decoding of a p-ary cyclic code consists four steps: 1) computation the known syndromes using received word; 2) unknown from syndromes; 3) error positions by use Berlekamp-Massey (BM) algorithm and Chien's search; 4) values solving linear system. This paper addresses two problems determining computing syndromes. To solve first problem, new matrix, together with Gaussian elimination instead BM search, is proposed. In this simplified decoder, finding an error-locator polynomial...

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

A multivariate interpolation formula (MVIF) over finite fields is presented by using the proposed Kronecker delta function. The MVIF can be applied to yield polynomial relations base field among homogeneous symmetric rational functions. Besides property that all coefficients are coming from field, there also a significant one on degrees of obtained polynomial; namely, degree each term satisfies certain condition. Next, for any cyclic codes unknown syndrome representation provided and has...

This letter presents two modified algorithms to decode up actual minimum distance for binary cyclic codes with irreducible generator polynomials. The key ideas behind these decoding are the utilization of extended Euclid's algorithm univariate polynomials evaluate unknown syndromes and coefficients general error locator polynomial, which has not been developed before. advantage is particularly suitable software hardware implementations.

Cyclic codes have been widely used in many applications of communication systems and data storage systems. This paper proposes a new procedure for decoding cyclic up to actual minimum distance. The consists two steps: 1) computation known syndromes 2) error positions values simultaneously. To do so, matrix whose all entries are is called syndrome matrix. A either or the elements finite field said be partial In this paper, novel methods presented determine simultaneously directly. first...

A novel, fast decoder for the binary quadratic residue code of length 23, or equivalently famous Golay is proposed. The core a new idea to determine syndrome weight. decoding algorithm can be implemented with parallel design, and based on this not only very efficient but also low area cost. It promises which faster than available decoders.

In this letter, Zech logarithmic decoding method is proposed for triple-error-correcting binary cyclic codes, which have not been developed before, whose generator polynomials at most three irreducible factors.

A Bose-Chaudhuri-Hocquenghem (BCH) is called quasi-reversible if there are consecutive elements a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,⋯,a xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> in the defining set, where negative integer and positive integer. matrix band form presented this article involves use of Newton identities symmetric polynomial identities. This coefficient linear system. The solution to such system...

A cyclic code is called single-syndrome decodable if its decoding up to error-correcting capability completely based on the single syndrome. This letter proposes a construction method of unusual general error locator polynomial (GELP) for triple- and quadruple-error-correcting (SSDC) codes gives an upper bound computational complexity GELP. Both theoretical experimental results show that GELP has lower than conventional triple-error-correcting SSDC codes.