Gui-Liang Feng and T. R. N. m e Center for Advanced Compuaer Studies, USL

Lafayette, LA. 70504, USA

In this paper, a new approach to determine a lower bound for the generalized Hamming weights of algebraic-geometric (AG) codes is discussed.

lh, be a well-behaving sequence of monomials based on LS. Let I { h,, , ... , h,Y 1 be a subset of LS called a maximal partially linearly dependent location set, on which Bp,, is c o n ~ i s ~ e ~ t ~ y and par- tially linearly dependent on its previous mon als. Define D { h,, , ... , h,p 1 = I I { h,, , ... , h,P 1 I .

ndent-degree of mono- define @) 6 max

{ ~ { h , , , ~ , , ,..., I 1 s * i l < i 2 a . - . c i , < r > .

T ~ e o r e ~ : For a linear code C, de [ h , , h,, ..., ~n, lT, if there is some d* such that Df?d*+h+l < d* - 1, then the generalized H m - ming weight d h is equal to or greater than d*.

Thus, the determination of a lower bound of the generalized Hamming weights reduces to the calculation of Df). Using an improved Bezout theorem, for the AG codes defined by a large class of plane curves, the value of Df) can be easily determined. In the following we show one exam- ple. Let the curve be a Hermitian curve over GF(24): x 5 + y 4 + y = 0 . We have the follow- ing well-behaving sequence H:

H = { 1, x, y , x 2 , xy, y 2 , x37 X 2 Y , xy2, Y 3 , x4, x3y, x2y2, xy3,x5, x4y, x3y2, x2y3? ... 1 = { xiyj I 0 < i 1 1 5 , 0 S j 1 3 >.

Let us consider c16, i.e., r = 16. The first 16 monomials are as follows: 1 1, x, y , x*, xy, y 2 , x 3 ,

x2y, xy2, y3 , x4, x3y, x2y2 , xy3,x5, x4y >. using the calculation of D f ) , we have the following values.

From these values and the above theorem, we Rave d l ( C 1 6 ) 2 12, d2(C16) 2 15, d3(c16) 2 16,

2 23, and d h ( C I 6 ) 2 h + 16, for h = 8, a D . , 48.

d4(C16) 2 19, dS(C16) 2 20, d6(C16) 2 21, d7(C16)

s new approach, some more odes with the minimum distances

4, 5, 6 and any lengths over G F ( P ) , and some more efficient AG codes have also been con-

. Wei, 'Generalized Hamming weights for linear codes," IEEE 7 " s . on Informa- tion Theory Vol. IT-37, pp. 1412-1428, Sept., 1991. K. Yang, P. W. Kumar, and H. ~ t i c h t e ~ o t h ~ "On the Weight Hierarchy of Geometric Goppa Codes," IEEE Trans. on Information Theory, Vol. IT-40, pp. 913-920, May 11994.

C. I,. Feng and T. R. N. Rao, "A Simple Approach for ~ o n s ~ c t i o ~ of Algebraic Geometric Codes from Affine Plane Curves," IEEE Trans. on Information Theory Vol. IT- 40, No.4., pp. 1003-1012, July 1994. 6. I,. Feng and T. R. N. Rao, "Improved Geometric Goppa Codes, Part I: Basic Theory" to appear in IEEE Trans. on Infor- mation Theory.

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