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ON THE MINIMUM WEIGHT CODEWORDS OF SOME BINARY BCH CODES
Daniel Augot, Pascale Charpin, Nicolas Sendrier
To cite this version:
Daniel Augot, Pascale Charpin, Nicolas Sendrier. ON THE MINIMUM WEIGHT CODEWORDS OF SOME BINARY BCH CODES. ISIT 2991, Jun 1991, Budapest, Hungary. pp.7-7. �hal-01216856�
ON THE MINIMUM WEIGHT CODEWORDS OF SOME BINARY BCH CODES
D. Augot P. Charpin N. Sendrier INRIA, Domaine de Voluceau, R.ocquericoiu-t.
BP 105, 78153 Le Chesnay Cedex, FRANCE
Abstra.ct: We give the true minimum distances 0 1 some RCH codes of length 255 and 511, which wcre iiot Itnown.
We describe the set of the minimum weight, codewords of tlie BCH codes with designed distancc 2"-' - 1.
Notation
We consider binary cyclic codes of length n = 2" - 1. A 13CH code C is always a narrow-sense BCH code, and is cliaractcr- ized by its defining-set:
I ( C ) = { s E [O,n - 11 I as is a zero of C}.
where a is a primitive n-th root of unity.
with its locator polynomial [ 3 p. 3431,
We identify a codeword z with its locators (X,),,,,,, aiitl
w UI
o ( z ) = U(1 - S , Z ) = C O j Z f ,
1=1 i=O
where w is the Hamming weight of I.
T h e power sum symmetric fiinctions of T ,
Ak =
CAY,!,
, = I
are related to the 0, by the Newton's identitics [3 p. 2451:
r 5 w , I , : A,
+
A r - i c 7 j+
r o , = 0;{
T > '10, I , : A ,+
Cy=i A T - i 0 , = 0.Remark that for each codeword 5 E C , we have:
VS E I ( C ) , A, = 0.
We will denote by B ( n , 5 ) tlic BCH cotlr of length 11 a n d designed distance 6.
BCH codes of length 2"l - 1, m = S , 9 , 1 0 Mac-Williams and Sloane give A table of BCH codes [ 3 p. 2671, in which the true minimum distance of some BCH codes of length 255 is not known. This table is improved by Cohen [2]; however some ca.ses remain unsolved.
We write the Newton's identities for a given BCII code C and a given weight U , and we check the consistency of thc resulting polynomial system;
- if we find a contradiction, then C has no codeword of weight w ,
- if we are able to find a solution, we have a codeword of weight w.
This method enables us to complete thc table of the miiii- mum distance of the BCH codes of length 255, and to cstcrid
0111' knowledge of BCII codes of length 511. For the latter
length we obtain minimum weight codewords which are ideiii- potents of the code. We present some proofs; for instance, thc BCH code of length 255 and designed distance 59 (resp. 61) has minimum distance 61 (resp. 6 3 ) .
Minimum weight codewords of B(2" - 1,2"-' - 1) n = 2 m - 1 and 6 = 2"-' - 1. Since the code B ( n , S ) contains the Reed-Muller code R ( 2 , m ) , its true minimum dis- tance is 6.
Using the Newton's idcntities, we prove the following the- Theorem 1: The minimum weiglit codewords of the BCH code of length 2'" - 1 and desigiietl distance am-' - 1 are those of the punctured RM code of sa.me length and order 2.
Corollary 1: Let I E B( n., S) , 6 = 2m-2 - 1, such that
&(x) = 2"-'. Then z is a cotlcword of the punctured RM code R(2,m) - i.e. t h e set of the locators of I is an m - 2- tliinensional affine subspace of GJ'(2m).
Corollary 2: m > 5. The autoniorphism group of the BCII code R ( n , 6 ) is contained i i i G L ( 2 , m). The code generated by tlie set of the minimum weight codewords of B ( n , 6) is strictly cont,ained in B ( n , 6 ) .
ore in :
Referencrs
(11 D. Augot, P. Charpin arid N. Scndrier. The minimum dis- tance of some binary codes via the Newton's identities.
EURO C 0 D E'90, LN C S , to appear.
[3] G. Cohen. On the minimuiii distance of some BCII codes.
IEEE Transaction on Information Theory, vol. 26, 1980.
[3] F.J. Macwilliams and N.J.A. Sloanc.
The Theory of Error Corrccting Codes. North-Holland, 1986.
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