On formulas for decoding binary cyclic codes

We adress the problem of the algebraic decoding of any cyclic code up to the true minimum distance. For this, we use the classical formulation of the problem, which is to find the error locator polynomial in terms of the syndroms of the received word. This is usually done with the Berlekamp-Massey algorithm in the case of BCH codes and related codes, but for the general case, there is no generic algorithm to decode cyclic codes. Even in the case of the quadratic residue codes, which are good codes with a very strong algebraic structure, there is no available general decoding algorithm. For this particular case of quadratic residue codes, several authors have worked out, by hand, formulas for the coefficients of the locator polynomial in terms of the syndroms, using the Newton identities. This work has to be done for each particular quadratic residue code, and is more and more difficult as the length is growing. Furthermore, it is error-prone. We propose to automate these computations, using elimination theory and Grbner bases. We prove that, by computing appropriate Grbner bases, one automatically recovers formulas for the coefficients of the locator polynomial, in terms of the syndroms

Functional diagnosability and recovery from massive faults in digital systems Quarterly progress reports, 17 May - 16 Nov. 1970 /final/

Diagnosability and recovery from massive faults in digital system

Cyclic Quantum Error-Correcting Codes and Quantum Shift Registers

We transfer the concept of linear feed-back shift registers to quantum circuits. It is shown how to use these quantum linear shift registers for encoding and decoding cyclic quantum error-correcting codes.Comment: 18 pages, 15 figures, submitted to Proc. R. Soc.

On the Shape of the General Error Locator Polynomial for Cyclic Codes

General error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes. We now present a conjecture on their sparsity, which would imply polynomial-time decoding for all cyclic codes. A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for specific code families, providing evidence to our conjecture. From these, a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with t <= 2 and n < 105, as well as for t = 3 and n < 63, except for some cases where the conjectured sparsity is proved by a computer check. Moreover, we summarize all related results, previously published, and we show how they provide further evidence to our conjecture. Finally, we discuss the link between our conjecture and the complexity of bounded-distance decoding of the cyclic codes

On Decoding of Quadratic Residue Codes

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 code