On the lifted Melas code (original) (raw)

On the Pless-construction and ML decoding of the (48, 24, 12) quadratic residue code

IEEE Transactions on Information Theory, 2003

We p r e s e n t a method for maximum likelihood decoding of the (48 24 12) quadratic residue code. This method is based on projecting the code onto a subcode with an acyclic Tanner graph, and representing the set of coset leaders by a trellis diagram. This results in a t wo level coset decoding which can be considered a systematic generalization of the Wagner rule. We s h o w that unlike the (24 12 8) Golay code, the (48 24 12) code does not have a Pless-construction which has been an open question in the literature. It is determined that the highest minimum distance of a (48 24) binary code having a Pless-construction is 10, and up to equivalence there are three such c o d e s .

On Decoding of the (73, 37, 13) Quadratic Residue Code

IEEE Transactions on Communications, 2014

In this paper, a method to search the set of syndromes' indices needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting index sets, one computes the unknown syndromes and thus finds the corresponding error-locator polynomial by using an inverse-free Berlekamp-Massey (BM) algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is new. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity. In fact, the algebraic hard-decision and soft-decision decoding of the (89, 45, 17) QR code outperforms that of the (73, 37, 13) QR code because the former has a larger minimal distance. However, experimental results indicate that the (73, 37, 13) QR code outperforms the (89, 45, 17) QR code with much fewer arithmetic operations when using the LP-based decoding algorithms. The pseudocodewords analysis partially explains this seemingly strange phenomenon.

Algebraic Decoding of the (31, 16, 7) Quadratic Residue Code by Using Berlekamp-Massey Algorithm

2010 International Conference on Communications and Mobile Computing, 2010

An analysis on the algebraic decoding of the (31, 16, 7) quadratic residue (QR) code with reducib le generator polynomial that uses the inverse-free Berlekamp-Massey (IFBM) algorith m to determine the error-locator polynomial is presented in this paper. The primary known syndrome S 1 will be equal to zero for some weight-3 error patterns. However, the zero S 1 does not cause a decoding failure while using the IFBM algorithm to determine the error-locator polynomial. Two examples with detailed step-by-step analysis show the decoding procedure.

On Decoding of Quadratic Residue Codes

2010

A binary Quadratic Residue(QR) code of length n is an (n, (n+1)/2) cyclic code over GF(2m) with generator polynomial g(x) where m is some integer. The length of this code is a prime number of the form n = 8l + 1 where l is some integer. The generator polynomial g(x) is defined by g(x)=∏_(i∈Q_n) (x-βi ) where β is a primitive nth root of unity in the finite field GF(2m) with m being the smallest positive integer such that n|2m-1 and Qn is the collection of all nonzero quadratic residues modulo n given by Qn={i│i≡j2 mod n for 1≤j≤n-1}. Algebraic approaches to the decoding of the quadratic residue (QR) codes were studied in [2], [3], [4], [5], [6] and [13]. Here, in this thesis, some new more general properties are found for the syndromes of the subclass of binary QR codes of length n = 8m + 1 or n = 8m - 1. A new algebraic decoding algorithm for the (41, 21, 9) binary QR code is presented by having the unknown syndrome S3 which is a necessary condition for decoding the (41, 21, 9) QR ...

The algebraic decoding of the (41, 21, 9) quadratic residue code

IEEE Transactions on Information Theory, 1992

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the code syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic and quartic equations over GF(2'9 in a new way which uses Zech's logarithms for the arithmetic. The algorithms developed here for Zech's logarithms save a substantial amount of computer memory by storing only a table of the fundamental parameters instead of a complete table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed here can apply generally to other codes such as the BCH and Reed-Solomon codes. Index Terms-Error-correction coding theory, cyclic code, quadratic residue code, finite field. ManuscriptreceivedJune3, 1991; revisedOctober4, 1991. This work was supported by the NSF under Grants NCR-8920060 and NCR-9016340.