A criterion for detecting unnecessary reductions in the construction of Groebner bases
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Properties of Gröbner bases under specializations
EUROCAL '87 Proceedings of the European Conference on Computer Algebra
Solving systems of algebraic equations by using Gröbner bases
EUROCAL '87 Proceedings of the European Conference on Computer Algebra
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
IEEE Transactions on Information Theory
Algebraic decoding of the (32, 16, 8) quadratic residue code
IEEE Transactions on Information Theory
Decoding the (47,24,11) quadratic residue code
IEEE Transactions on Information Theory
Cyclic decoding procedures for Bose- Chaudhuri-Hocquenghem codes
IEEE Transactions on Information Theory
Studying the locator polynomials of minimum weight codewords of BCH codes
IEEE Transactions on Information Theory
General Error Locator Polynomials for Binary Cyclic Codes With and
IEEE Transactions on Information Theory
Use of Grobner bases to decode binary cyclic codes up to the true minimum distance
IEEE Transactions on Information Theory
General principles for the algebraic decoding of cyclic codes
IEEE Transactions on Information Theory
Unusual general error locator polynomial for the (23, 12, 7) golay Code
IEEE Communications Letters
Decoding binary cyclic codes with irreducible generator polynomials up to actual minimum distance
IEEE Communications Letters
CT-RSA'11 Proceedings of the 11th international conference on Topics in cryptology: CT-RSA 2011
| Hi-index | 0.00 |
We revisit in this paper the concept of decoding binary cyclic codes with Grobner bases. These ideas were first introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Grobner bases. Thus, we study in this paper the properties of the system defined by the Newton identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated with this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas. Then we consider the so-called online Grobner basis decoding, where the work of computing a Grobner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred).