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Efficient decoding of (binary) cyclic codes above the correction capacity of the code using Grobner bases
Daniel Augot, Magali Bardet, Jean-Charles Faugère
To cite this version:
Daniel Augot, Magali Bardet, Jean-Charles Faugère. Efficient decoding of (binary) cyclic codes above the correction capacity of the code using Grobner bases. IEEE International Symposium on Infor- mation Theory - ISIT’2003, Jun 2003, Yokohama, Japan. pp.362 - 362, �10.1109/ISIT.2003.1228378�.
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ISlT 2003, Yokohama, Japan, June 29 -July 4.2003
Efficient decoding of (binary) cyclic codes above the correction capacity off the code using Grobner bases
Daniel Augot Magali Bardet Jean- Charles Faug &re
INRIA-Rocquencourt, Bat. 10 Projet SPACES LIPG/LORIA Projet SPACES LIPG/LORIA
Domaine de Voluceau, B.P. 105 CNRS/UPMC/INRIA CNRS/UPMC/INRIA
F-78153 Le Chesnay Cedex 8, rue du capitaine Scott F-75015 Paris e-mail: Daniel . A u g o t Q i n r i a . f r e-mail: bardetQcalf o r . l i p 6 . f r
8, rue d u capitaine Scott F-75015 Paris e-mail: j cf Q c a l f o r . l i p 6 . f r Abstract - This paper revisits the topic of decoding
cyclic codes with Grobner bases. We introduce. new algebraic systems, for which the Grobner basis com- putation is easier. We show that formal decoding for- mulas are too huge to be useful, and that the most ef- ficient technique seems to be to recompute a Grobner basis for each word (online decoding). We use new Grobner basis algorithms and “trace preprocessing”
to gain in efficiency.
I. INTRODUCTION
Let C be a cyclic code of length n over Fz, with defining set Q c (1,. . . , n } and correction capacity t , and let CY E IF2- be a primitive n-th root of unity. For any error e of weight U ,
if Z; denote the locators of e , we can compute its syndroms S: = .(ai) = E,”,, Z;i V i E Q. As long as U 5 t , the system
SYN,, = { Si - E,”=, Zj, i E Q } specialized for Si = 5’:
has a unique solution (cf. 141). To use the symmetry of t h e problem, we introduce the symmetric functions of the locators:
SYM,, = { uj -E1l<...<l, Zil . . . z i j , J’ E ( l , ~ ] }. The S:’s
and the u;’s associated t o the.Zj*’s are also solutions of t h e following system (cf. [I])
A Grobner basis describes the set & ( I ) = {z E n“ : Vf 5
I , f(x) = 0) of solutions of an ideal I C JK[xl,. . . ,z,] where K is the algebraic closure of K To compute &(I) = Vz(I) nKS, we have t o add the field equations. We add a + t o an ideal to denote t h e ideal together with the field equations (Z?+l - Z~,S;”’ -si or a:“ - u j ) .
It has been shown that the problem of decoding cyclic codes up t o their true minimum distance can be solved by the use of Grobner bases 131, with the algebraic system SYN:.
11. NEW SYSTEMS AND THEIR PROPERTIES Starting from the system (l), we eliminate the unknowns syndroms Si, i 9 Q t o obtain t h e new system BIN = { S i - fi(a1,. . . ,a,) i E Q}, where the fi’s are the Waring functions. We show that this new system and the systems SYM,, and NEWTON, used in [3, 41 are closely related, and that for these systems the field equations are not necessary:
Proposition 1. The ideals and their variety are related by:
(BIN:) = (SYN,SYM$) n IF^ b,,~] = (NEWTON:) n [E,,, SJ (BIN) = ( S Y N , ~ Y M , ) n 4 k,, $1 = (NEWTON,,) n 4 [E,, a
V ~ ( S Y N , S Y M , ~ ) = V ~ ( S Y N , S Y M , ) V%(NEWTON,+) = V%(NEWTON,,)
Table 1: Decoding QR Codes
Proposition 2 (Uniqueness). Let s* c FZ- be the syn- drom of an e m r e of weight v 5 t , then the specialized sys- t em BIN(^) has a unique solution ( U : , . . . , U ; ) and L e ( Z ) =
Cy=, uj*Z”-j is the locator polynomial of e. I n practice, the Grobner basis of BIN($*) is always (01 + U : , . . . , U , + at}.
Proposition 3 (List Decoding). Ifv > t then the Grobner basis of BIN(^) gives all the possible errors of weight at most v that have as syndroms.
With these new systems, we are able t o do formal decoding as well as online decoding. But the size of the formal formu- las are so huge that the computation of t h e Grobner basis is intractable, and even if we could obtain these formulas, the cost of their evaluation would be much too large.
In practice we do online decoding with a subset of the system BIN (we take the minimal number of equations t o have a single solution, and choose the equations of minimal degree to speed the computation). This is a very general method, we only need the length and the defining set of.the cyclic code.
220), we use a general method for solving systems with parameters: the behavior of the Grobner basis computation is almost the same for all the possible values of the syndroms corresponding t o an error of a given weight. Hence as a preprocessing, we can compute a Grobner basis for BIN(S:,) for a random error eo of weight
U , and record the trace of the computation (we do it as a C program). Then for any error e, the C program executed on BIN(S,+) gives the values of the oj’s. This reduce drastically the complexity of t h e online computation (by a factor 1000).
This method is implemented in Maple, and call the C soft- ware FGb from t h e third author t o compute a Grobner basis.
FGb is an implementation of the algorithm F4 [2].
111. PRACTICAL DECODING
If the field is big enough (e.g.
REFERENCES
[l] D. Augot. Description of minimum weight codewords of cyclic codes by algebraic systems. Finite Fields Appl., vol. 2, pp.
[2] J.-C. FaugBre. A new efficient algorithm for computing Grobner bases (F4). J . Pure Appl. Algebra, 139(1-3):61-88, 1999.
(31 P. Loustaunau and E. Von York. On the decoding of cyclic codes using Grobner bases. A p p l . Algebra Eng. Commun. Comput., [4] I.S. Reed, T.K. Truong, X. Chen, and X. Yin. The algebraic decoding of the (41, 21, 9) quadratic residue code. ZEEE h n s . Inform. Theow, 38(3):974-986, 1992.
138-152, 1996.
8(6):469-483, 1997.
0-7803-7728-1 103/$17.00 02003 IEEE. 362
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