Algebraic Decoding Algorithm Research Articles

A new algebraic algorithm to decode the ternary Golay code

A new algebraic decoding algorithm for the Ternary (11, 6, 5) Golay code is presented. The algorithm we present is based solely on the computation of the syndrome polynomials S( x) (without the evaluator and locator polynomial) and on the direct use of a Chien search directly over S( x).

Reliability-based syndrome decoding of linear block codes

In this correspondence, various aspects of reliability-based syndrome decoding of binary codes are investigated. First, it is shown that the least reliable basis (LRB) and the most reliable basis (MRB) are dual of each other. By exploiting this duality, an algorithm performing maximum-likelihood (ML) soft-decision syndrome decoding based on the LRB is presented. Contrarily to previous LRB-based ML syndrome decoding algorithms, this algorithm is more conveniently implementable for codes whose codimension is not small. New sufficient conditions for optimality are derived. These conditions exploit both the ordering associated with the LRB and the structure of the code considered. With respect to MRR-based sufficient conditions, they present the advantage of requiring no soft information and thus can be preprocessed and stored. Based on these conditions, low-complexity soft-decision syndrome decoding algorithms for particular classes of codes are proposed. Finally, a simple algorithm is analyzed. After the construction of the LRB, this algorithm computes the syndrome of smallest Hamming weight among o(K/sup i/) candidates, where K is the dimension of the code, for an order i of reprocessing. At practical bit-error rates, for codes of length N/spl les/128, this algorithm always outperforms any algebraic decoding algorithm capable of correcting up to t+1 errors with an order of reprocessing of at most 2, where t is the error-correcting capability of the code considered.

A decoding algorithm for the (23, 12, 7) golay code with error and erasure correction

The problem of complete decoding the binary Golay code over error-and-erasure memoryless channels is addressed. An algebraic description of the erasure channel is given and a nearest neighbor decoding scheme is defined. This decoding scheme is viewed as a standard array decoding by suitably partitioning the set of received words. The analysis of this partition leads to the study of punctured Golay codes and the properties of their cosets. Based on these properties, a complete algebraic decoding algorithm is provided which simultaneously corrects errors and erasures.

Decoding the ternary (23, 12, 8) quadratic residue code

An algebraic decoding algorithm is presented for the ternary (23, 12, 8) quadratic residue code. The algorithm introduces error-counting polynomials which determine the number of errors in a received word. Also a new search technique is developed, which allows the algorithm to be used on the extended code.

On a Class of Reversible Binary Cyclic Codes and their Algebraic Decoding

Abstract A class of reversible binary cyclic codes is presented together with an algebraic decoding algorithm that exploits their error correcting capabilities.

Decoding the (73, 37, 13) quadratic residue code

Algebraic approaches to the decoding of the quadratic residue (QR) codes were studied recently. In Reed et. al. (1992), a decoding algorithm was given for the (41, 21, 9) binary QR code. Here, some new more general properties are found for the syndromes of the subclass of binary QR codes of length n = 8m +1. Using these properties, the new theorems needed to decode this subclass of the QR codes are obtained and proved. As an example of the application of these theorems, a new algebraic decoding algorithm for the (73, 37, 13) binary QR code is presented.

High-speed Reed-Solomon decoder for correcting errors and erasures

A Reed-Solomon decoder for errors-and-erasures correction, based on a new algebraic decoding algorithm, is presented. This high-speed decoder requires only n clock cycles for decoding each received n-symbol block. A serial structure that requires very few multipliers and provides a general expression to calculate the coefficients of the erasure-locator polynomial is also presented. A (15, 11) RS decoder and its shortened version (7, 3) RS decoder are used as design examples to illustrate the operating procedure of the new decoding algorithm.

Successive-erasure decoding of RS-coded MPSK schemes in a Rayleigh fading channel

Earlier analysis has shown that softdecision decoding using the Viterbi algorithm significantly improves the performance of the bandwidth-efficient RS-coded MPSK schemes in a Rayleigh fading channel. The Viterbi decoding, however, is applicable to the decoding of short RS codes with a small amount of redundancy because of its higher complexity. Another soft-decision decoding algorithm, termed successive-erasure decoding (SED), is presented for decoding of RScoded MPSK schemes. This decoding procedure, originally introduced by G.D. Forney in connection with generalised minimum-distance decoding, uses channel measurement information in algebraic decoding algorithm. The performance evaluation of some examples of RS-coded MPSK schemes shows that the use of such information bridges the gap in performance between the harddecision decoding and maximum likelihood decoding (MLD). For example, for RS(7, 5)-coded 8PSK, SED provides 3.6-6 dB coding gain at Pb = 105, compared with errors-and-erasures and errors-only decodings, respectively. This coding gain is only 1.4 dB less than the achievable gain by MLD. The performance of the SED algorithm is improved by an additional 0.7 dB when the channel state information is used.

Algebraic decoding of the ternary (13,7,5) quadratic residue code

An algebraic decoding algorithm for the ternary (13,7,5) quadratic residue code is presented. This seems to be the first attempt to provide an algebraic decoding algorithm for a quadratic residue code over a nonbinary field.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Algebraic decoding of the Zetterberg codes

The Zetterberg codes are one of the best known families of double-error correcting binary linear codes. Unfortunately, no satisfactory decoding algorithm has been known for them until recently when an algebraic decoding algorithm was described by P. Kallquist (1989). It requires, however, to solve a quadratic equation in order to decide whether 2 or 3 errors have occurred. A simple criterion is derived to determine whether 1, 2, or 3 errors have occurred when a Zetterberg code is used for data transmission. Based on criterion a new decoding algorithm is proposed which is faster than the known one.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Algebraic decoding of the (32, 16, 8) quadratic residue code

An algebraic decoding algorithm for the 1/2-rate (32, 16, 8) quadratic residue (QR) code is found. The key idea of this algorithm is to find the error locator polynomial by a systematic use of the Newton identities associated with the code syndromes. The techniques developed extend the algebraic decoding algorithm found recently for the (32, 16, 8) QR code. It is expected that the algebraic approach developed here and by M. Elia (1987) applies also to longer QR codes and other BCH-type codes that are not fully decoded by the standard BCH decoding algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Decoding the (24,12,8) Golay code

A simplified procedure, called the shift-search method, is developed to decode the three possible errors in a (23,12,7) Golay codeword. The algebraic decoding algorithm developed recently by Elia is compared with this algorithm. A computer simulation shows that both algorithms are modular, regular and naturally suitable for either VLSI or software implementation. Both of these algorithms decode efficiently the 1/2-rate (24,12) Golay code for correcting three errors and detecting four errors. The algebraic technique is a slightly faster algorithm in software than the shift-search procedure.

The algebraic decoding of Goppa codes

An interesting class of linear error-correcting codes has been found by Goppa [3], [4]. This paper presents algebraic decoding algorithms for the Goppa codes. These algorithms are only a little more complex than Berlekamp's well-known algorithm for BCH codes and, in fact, make essential use of his procedure. Hence the cost of decoding a Goppa code is similar to the cost of decoding a BCH code of comparable block length.

Extended double-error-correcting binary Goppa codes are cyclic (Corresp.)

The class of codes introduced by Goppa [1]-[3] includes the BCH codes as a proper subset. It also includes a large subset of asymptotically good codes, each of which has an algebraic decoding algorithm for correcting some smaller number of errors. In Section 7 of [1], Goppa gives necessary and sufficient conditions for his codes to be isomorphic to cyclic codes under a certain correspondence. In this correspondence, we exhibit another correspondence which reveals that certain other Goppa codes (including the example of Goppa's Section 6) become cyclic when extended by an overall parity check. In particular, the extended Goppa codes with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,k,d) = (2^m + 1, 2^m - 2m, 6)</tex> are isomorphic to the reversible cyclic codes with check polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(x + 1)f(x)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x)</tex> is an irreducible polynomial of period <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^m + 1</tex> .

Goppa codes

Goppa described a new class of linear noncyclic error-correcting codes in [1] and [2]. This paper is a summary of Goppa's work, which is not yet available in English. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^1</tex> We prove the four most important properties of Goppa codes. 1) There exist <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> -ary Goppa codes with lengths and redundancies comparable to BCH codes. For the same redundancy, the Goppa code is typically one digit longer. 2) All Goppa codes have an algebraic decoding algorithm which will correct up to a certain number of errors, comparable to half the designed distance of BCH codes. 3) For binary Goppa codes, the algebraic decoding algorithm assumes a special form. 4) Unlike primitive BCH codes, which are known to have actual distances asymptotically equal to their designed distances, long Goppa codes have actual minimum distances much greater than twice the number of errors, which are guaranteed to be correctable by the algebraic decoding algorithm. In fact, long irreducible Goppa codes asymptotically meet the Gilbert bound.

Algebraic decoding of block codes over a q-ary input, Q-ary output channel, Q > q

A technique is presented for the algebraic decoding of block codes over a, q -ary input, Q -ary output channel ( Q > q ). It is assumed that an algebraic decoding algorithm is known for a simple channel such as a channel where the input alphabet is identical to the output alphabet. This decoding algorithm is then adapted for use over the actual channel. The technique can be used in conjunction with an arbitrary distance measure between input and output vectors. Thus, Hamming distance, Lee distance, or a burst distance can be assumed. Examples are presented for each of these distances.