ITW 1998. Killarney, Ireland. June 22 - 26

On the missing functions of Hermitian-like codes over GF( a2) Gui-Liang Feng, Xin-Wen Wu, a n d T. R. N. Rao

The Center for Advanced Computer Studies University of Southwestern Louisiana

Lafayette, LA. 70504 Email: glf@cacs.usl.edu, wxwQcacs.usl.edu, and rao@cacs.usl.edu

I. INTRODUCTION

For the past thirty-five years, finding good codes, especially a sequence of linear codes which meet or exceed the Gilbert- Varshamov bound, has been a very active area of research. Unfortunately, many known linear codes are bad codes.

The notation [n, k , d] denotes a code of length n with di- mension k and minimum distance d. For such a code, the information rate and relative minimum distance rate are de- fined as R = ! and d = $, respectively. A sequence of linear codes {Cm}g=l = {[nm, k m , lover a finite field GF(q) such that the code length n, + co when m + co, is said to be asymptotically good, if the information rates R,,, and the relative minimum distance rates 6, a.re bounded away from zero. Let the q-ary entropy function be

H q ( z ) = z.log,(q - 1) - z.log,z - (1 - z).log,(l - z). (1.1)

The well-known Gilbert-Varshamov bound can be simply de- scribed as follows [5].

Gi lber t -Varshamov bound If 0 :I 6 < y, then there exists an infinite sequence of [n, k,d] q-ary linear codes with R = and $ = 6 satisfying the equation:

R 2 1 - Hq(6) for all n. (1.2) It is known that there exists long Goppa codes which meet

the Gilbert-Varshamov bound [5]. However, a solution for finding these long Goppa codes has not been found. In the past, one of the most difficult problems of coding theory has been an attempt to construct asymptotically good codes.

A very sensational development in the study of the prob- lem of asymptotic bounds and constiructing the asymptoti- cally good codes was presented in a paper [9]. In this paper, it is shown that there exist asymptotic good algebraic geo- metric codes exceeding the Gilbert-Varshamov bound when q2 2 49. Based on this work, Katsman,, Tsfasman, and Vladut [4] presented a polynomial construction of q-ary codes arising from modular curves &( l l l ) over GF(q2), which are better than the Gilbert-Varshamov bound, when q 2 7. But their proposed construction has very high complexity. Therefore, a solution for the explicit construction of sequences of good AG codes which exceed the Gilbert-Varshamov bound has not been found.

In this paper, we study the explicit construction of asymp- totically good improved AG codes that exceed the Gilbert- Varshamov bound. We will find missing functions in improved well-behaving sequences.

II. ASYhlPTOTICALLY GOOD CODES OVER HERMITIAN-LIKE CURVES

Consider the curve over GF(q2) defined by

y9 + 1:QyQ-l + 1: = 0. (2.1)

We call the curve Hermitian-like curve. For each y # 0, there are q distinct nonzero z's such that (z,y) are points of this curve. Obviously, (0,O) is a point. Therefore, this curve has n = (q2 - l )q + 1 points. The Hermitian-like curve Xm in m-dimensional affine space over GF(q2) is defined by the fol- lowing system of equations:

z': + x;x;-' + 2 2 = 0, x; + z;.;-' + z3 = 0,

...... z:-l + zLz:-:, + zm = 0.

It has (4' - l)qm-' + 1 points. In [2], a tower of function fields over GF(q') was con-

structed, they correspond to the sequence of the Hermitian- like curves over GF(q2) . The genus of X, is given by

odd m, + 1, even m,

m+2 7 7 - 2 gm + p - 1 - i q T - - $qY - q2

(2.3)

(2.4) and it satisfies nm lim - = q - 1,

m+m 9.. where n, is the number of points of X,.

It is well known that if C is an q-ary [n, I C , d] AG code over an algebraic curve (or equivalently, a function field) XI then its parameters satisfy the condition k + d 2 n - g 4- 1, where g is the genus of the curve. Thus, we have

R + 6 2 1-2+-. 1 n n (2.5)

By selecting good curves over GF(qZ) , it is shown that there exist asymptotically good AG codes attaining the Tsfasman- Vladut-Zink bound [9]:

1 R L 1 - 6 - - q - 1'

When q2 2 49, the Tsfasman-Vladut-Zink bound is better than the Gilbert-Varshamov bound. However, no explicit con- struction of sequences of codes satisfying (2.6) was given.

In [lo], a sequence of codes attaining the Tsfasman-Vladut- Zink bound was presented, the construction was based on the tower of Artin-Schreier extensions of function fields found by Garcia and Stichtenoth [2]. However, this is not the explicit construction of a sequence of asymptotically good codes that exceed the Gilbert-Varshamov bound, since many functions in the well-behaving sequences are missing.

Let LS be the set of points of an algebraic curve X in an m dimensional space over GF(q), and let H = {hi, h2,. . . , h n } be a well-behaving sequence a t the location set LS. The ma- trix H, = [hl,. . . , hpIT can be used as a parity check matrix

29 0-7803-4408-1/98/$10.00 0 1 9 9 8 IEEE

which defines an AG code. For each element h, in H, let N, be the number of terms which are consistent with h, and are well-behaving terms. Then the minimum distance of the code defined by H, is a t least min{Nklk > r } . However, the bound obtained by the well-behaving sequence is not tight enough. Some of the conditions are too strong and can be weakened, which may increase the estimated lower bound.

D. GOOD SEQUENCES AND MISSING FUNCTIONS

In this section we further improve the well-behaving se- quences. Using the improved sequence, the missing functions can be found.

Let LS = {PI, . ’ , P,} be the set of points of the Hermitian-like curve over GF(22), if h = (h (P l ) , . . . , h (P , ) ) = PI),..., Ii‘(P,)) = h‘, we say two polynomials h and h’ are equivalent at LS , and denote h - h’. From [l], by the equations of the Hermitian-like curve and the properties of the finite field GF(2’), we have the following relations of polynomials (let a,a,-1 . . . a2a1 denote z&mzky;l . . . zg’zp’):

At first we give some definitions and notations.

2 1 . . . = . . . 0 2 . . . + . . .IO.. . ,

. . . (i + 2 ) ( j + 1). . f = . . . i(j + 2) . . . + f . . (i + 1)j.. . , . . , 2 2 . . . = . . . o o . . . + ... 11 .. . . . . 2 0 . . . = . . . 1 2 . . . + ...()I.. . .

For a monomial h = ama,-l . . . a2a1, we define three equiva- lent transformations as follows:

h -* h‘ means: a,a,-1 , . . a z a l is transformed t o . . . aiai by 21 -+ 02 + 10 and 02 -+ 21 + 10.

h wGF h’ means: ama,-1...a2 is transformed to U ~ U ~ - ~ . . . U ; by 21 -+ 02+10,20 -+ 12+01, and 22 -+ 11+00, while a2u1 is transformed by 21 -+ 02 + 10 and 20 -+ 12 + 01.

h -+GF h’ means: a; = a1 - 3k or a1 = a: - 3k (in general, k = 1).

Definit ion 4.1 let CH = (hl , h2,. . . , h,} be a sequence of polynomials, we call C H a candidate well-behaving sequence if the following conditions are satisfied:

(1). hl, hZ,. . . , h, are linearly independent. And w(h1) < w(h2) < . . . < w(h,), w(hn) is denoted as CW,,,.

(2). For every hi, hj E C H , hi) + w ( h j ) 5 CWmax, let hi . hj -* h,* + h,*, where w(h,*) < w(h,*). Then,

(i). If w(hi) + w(h j ) = w(h,*), then h,., h,* E CH.

(ii). If w(hi) + w(h j ) > w(h,*). Let h,* -+GF h, and h,. -+GF h,, then h, and h, are in CH.

(iii). If w(hi) + w(h j ) < w(hr*). Then h,* and h,* are not in CH. Let h,* +GF h, and h,. jGF h,, then hr and h, are in C H .

Obviously, a well-behaving sequence is a candidate well-

Consider the Hermitian-like curve in 4-dimensional space behaving sequence but the converse is not true.

over GF(2’) 2 2 + 2 ; 2 1 + 2 2 = 0, z; + 2g22 + 2 3 = 0, zg + 2 : 2 3 + 2 4 = 0.

It has 25 points. We have ~ ( 2 % ) = 24-’, for i = l , . . . ,4 . Let LS be the set of the points of this curve. We have a well- behaving sequence H = (0000, 0001, 0011, 0002, 0102, 0012, 1012, 0112, 1112, 0003, 1003, 0103, 1103, 0013, 1013, 0113, 1113, 1004, 0104, 1104, 1014, 0114, 1114, 1005, 1105). The corresponding weight sequence is W = ( 0 , 8, 12, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43). By (2.3), the genus is g4 = 13, but in W , 16 weights are missing. So there are three missing functions.

Based on the sequence H we find a candidate well-behaving sequence C H . In the corresponding weight sequence CW the weights 14, 15 and 19 are not missing. And we find the miss- ing functions. In [6 ] , the authors also found the three missing functions with weights 14, 15 and 19 by a different method. But our method has two advantages. Firstly, i t can be eas- ily generalized to the general cases. We will find a recursion method to find missing functions for large m. Secondly, in the process of finding the missing functions, we do not need to check all the weights. For example, in the case of m = 4, the function with weight 17 is not a missing function, so we do not need to check the weight 17. In fact, the monomial with weight 17 is 1002, if 1002 E C H , then (1002)’ = 2004 is in C H . Now 2004 - 0104 + 1204, 0104 E C H ; 1204 - 1014 + 1124, 1014 GGF 1011, 1124 - 1105 + 1113, 1105 -+GF 1102 E C H , and 1113 E C H . So if 2004 E C H , then 1011 is in C H , so

1213 1103 + 1023, 1103 E C H ; 1023 - 1012 + 1004, 1012 E C H , 1004=1001, so 1001 must be in C H . (1001)2 = 2002, 2002 - 0102 + 1202, 0102 E C H ; 1202 - 1012 + 1122, 1122 - 1111 + 1103, 1103=1100, but 1012 and 1100 are not in C H , this is a contradition. Thus i t implies that 1002 is not in C H .

2013 = 1011 * 1002 E CH. 2013 0113 + 1213, 0113 E C H ;

REFERENCES [l] G. L. Feng and T. R. N. Rao, “Improved Hermitian-like Codes

over GF(4)”, preprint, 1997. [2] A. Garcia and H. Stichtenoth, “A tower of Artin-Schreier ex-

tensions of function fields attaining the Drinfeld-Vladut bound”, Invent. Math., Vol. 121, pp.211-222, 1995.

[3] A. Garcia and H. Stichtenoth, “On the asymptotic behavior of some towers of function fields over finite fields”, J . Number Theory, Vol. 61, pp.284-273, 1996.

[4] G. L. Katsman, M. A. Tsfasman, and S. G. Vladut, “Modular curves and codes with a polynomial construction”, IEEE Trans. on Infor. Theory, Vol. IT-30, pp.353-355, 1984.

[5] F. J. MacWilliams and N. 3. A. Sloane, The Theory of Error- Correcting Codes, Amsterdam: North-Holland, 1977.

[6] R. Pellikaan, “Asymptotically good sequences of curves and codes”, in Proceeding of 34th Annual Allerton Conference on Communication, Control, and Computing, pp.276-285, Oct. 1996.

[7] R. Pellikaan, “On the missingfunctions of a pyramid of curves”, to appear in Proceeding of 35th Annual Allerton Conference on Communication, Control, and Computing, Sept. 1997.

[E] R. Pellikaan, H. Stichtenoth and F. Torres, “Weierstrass semi- groups in an asymptotically good tower of function fields”, to appear in Finite Fields and their Applications.

[9] M. A. Tsfasman, S. G. Vladut, and Zink, “Modular curves, Shimura curves and Goppa codes, better than Varshamov- Gilbert bound”, Math. Nachr., 104(1982), 13-28.

[lo] C. Voss and T. Hoholdt, “An explicit construction of a se- quence of codes attaining the Tsfasman-Vladut-Zink bound. The first steps”, IEEE Trans. Infor. Theory, Vol. IT-43, pp.128- 135, Jan, 1997.

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