AGCT-8, Luminy, Marseille, France, Mai 14 { Mai 18, 2001
On Hyperbolic Codes O. Geil
T. H holdt
Aalborg University Technical University of Denmark Department of Mathematics Department of Mathematics Fr. Bajersvej 7G, 9220 Aalborg Bldg. 303 DK-2800 Lyngby Denmark Denmark Email:
[email protected] Email:
[email protected] Abstract | We give a new description of the so- De ne the map called hyperbolic codes from which the minimum dis M ! N0 tance and the generator matrix are easily determined. D: M 7! # (hM; X1 ; : : : ; X i) We also give a method for the determination of the dimension of the codes and nally some results on the and let E (s) := SpanFq fev(M ) j M 2 M ; D(M ) sg. weight hierarchy are presented. m
q
q m
m
Note that the value D(M ) is easily calculated. It is simply the number of monomials in M that are not divisible by any of the monomials M; X ; : : : ; X . De nition 3 We de ne M (s) := fM 2 M j deg < q for i = 1; : : : ; m; and D(M ) sg : We have E (s) = SpanF fev(M ) j M 2 M (s)g : In order to estimate/ nd the minimum distance of the codes given in De nition 2 we will need the following result known as the footprint bound.
I. Introduction
m q m
In [5] Saints and Heegard considered a class of codes called hyperbolic cascaded Reed-Solomon codes which can be seen as an improvement of the generalized Reed-Muller codes RM (r; 2). The construction was further generalized by Feng and Rao in [1] to an improvement of the generalized ReedMuller codes RM (r; m) for arbitrary m. Feng et al. also estimated the minimum distance of the new codes. The codes were further studied in [4] and [2] where the minimum distance was estimated by means of order functions and it was shown using the theory of order domains that the codes are asymptot- Theorem 4 Assume we are given an ideal I and a monoically bad with respect to the order bound and the codes were mial ordering such that (I ) is a nite set. Then the renamed hyperbolic codes. By use of the so-called footprint of (I ) is independent of the actual choice of . The from Grobner basis theory we construct a class of codes where size number of common solutions in F of F (X ; : : : ; X );: : : ; the minimum distance is easy to determine. We then show F (X ; : : : ; X ) is at most equal to # (I ). that these codes are actually the hyperbolic codes, thereby obtaining generator matrices of these, and give a method for We get. the determination of the dimension. It follows that the estimation in [4] of the minimum distance of the hyperbolic codes Proposition 5 The code E (s) is of length n = q and minactually gives the correct minimum distance. We show how to imum distance d q s. estimate, and in certain cases nd, the generalized Hamming Whenever s is chosen properly we can say even more. weights of the codes. q
1
(q ) m
q
m
Xi
(q ) m
q
q
m q
1
l
1
1
m
m
m
m
De nition 6 De ne
II. A class of codes with known minimum distance
We give a new description of a class of codes related to F [X ; : : : ; X ], m 1. The presentation of the codes relies on the Grobner basis theoretical concept of a footprint. q
1
m
De nition 1 Assume we are given an ideal I
= hF (X ; : : : ; X ); : : : ; F (X ; : : : ; X )i F [X ; : : : ; X ] 1
1
m
l
1
m
1
q
S
:= fD(M ) j M 2 M ; deg m
< q; i
= 1; : : : ; mg :
Theorem 7 For any s0 2 N0 there exists a unique s 2 S such that E (s0 ) = E (s). The minimum distance of E (s) is given by d=q s. m
III. Hyperbolic codes
m
and a monomial ordering on the set Mm of monomials in the variables X1 ; : : : ; Xm . The footprint (I ) of I with respect to is the set of monomials in Mm that can not be found as a leading monomial of any polynomial in I .
Xi
In [4, p. 922] the so-called hyperbolic codes are considered. De nition 8 LetQN (s) := fX 1 X 2 M j a < q; for i = 1; : : : ; m; (a + 1) < q sg: The hyperbolic codes are now de ned as follows. (q ) m m i=1
a
1
i
m
am m
m
i
De nition 2 Given a polynomial ring F [X1 ; : : : ; X ] and an indexing F = fP1 ; P2 ; : : : ; P g ; where n = q . Consider De nition 9 the evaluation map Hyp (s; m) := fc 2 F j hc ; ev(M )i = 0 for all M 2 N ( ) (s)g: F ev : F [X1 ; F: : : ; X ] ! Here n = q and h ; i is the standard inner product in F . 7! (F (P1 ); : : : ; F (P )) : q
m q
q
m
m
m
n
n q
q
n q
m
n
q m
n q
GF(64)
1
IV. The generalized Hamming weights
As demonstrated below the hth generalized Hamming weight of the hyperbolic code Hyp (m; s) is related to the following number. q
0.8
De nition 12
(
) := maxf# (hM ; : : : ; M ; X ; : : : ; X i) j M 6= M for i 6= j; M 2 M (s) for i = 1; : : : ; hg : Note that the number # (hM ; : : : ; M ; X ; : : : ; X i) is easily calculated. It is simply the number of monomials in M that are not divisible by any of the monomials M ; : : : ; M ; X ; : : : ; X . To establish the correspondence between (q; s; m) and the hth generalized Hamming weight we will need the following de nition. De nition 13 For M ; : : : ; M 2 M where h 2, let gcd(M ; : : : ; M ) denote the greatest common divisor of M ; : : : ; M . For a single element M 2 M we write gcd(M ) := M . The set D = fM ; : : : ; M g M (s) is said to be a dense set related to Hyp (s; m) if fX 1 X 2 (hM : : : : ; M ; X ; : : : ; X i) j a b ; i = 1; : : : ; mg M (s) ; where X 1 X = gcd(M ; : : : ; M ). A set D = fM ; : : : ; M g M (s) is said to be an optimal set of size h related to Hyp (s; m) if M 6= M for i 6= j and # (hM ; : : : ; M ; X ; : : : ; X i) = (q; s; m) : We can show the following theorem concerning the hth generalized Hamming weight. This theorem is a generalization of Theorem 7. h q; s; m
0.6
1
i
j
i
0.4
m
1
q
h
q m
1
m
1
d/n
q
h
(q )
q
h
q m
1
q m
1
h
0.2
1
1
1
0
0.2
0.4
k/n
0.6
0.8
1
m
h
h
1
h
1
1
m
1
(q )
h
m
q
b
In [4] the minimum distance of these codes is estimated using the order bound. One gets d(Hyp (s; m)) q s. By Theorem 7 and the following result this estimate is actually equal to the true minimum distance of the hyperbolic code. Theorem 10 Consider F [X ; : : : ; X ] and s 2 S , then E (s) = Hyp (s; m). It follows from Theorem 10 that we now have the generator matrices of the hyperbolic codes. For a 2 N we de ne V (m; a) := #f(x ; : : : ; x ) j x 2 N; 1 x q; Q x ag i = 1; : : : ; m; then it follows from above that dim(Hyp (s; m)) = q V (m; q s 1): It is not obvious how to get a closed form expression for since V (1; a) = minfa; qg and P V (Vm(m;1;ab) but V (m; a) = c) we can easily calculate V (m; a) recursively. One can verify that V (2; a) = bq + P f gb c where b := minfb c; qg and the last sum is zero if b q. The description in [4] of the hyperbolic codes is based on order domain theory. From the theory in [4] it is clear that the hyperbolic code construction is an improvement of the generalized Reed-Muller code construction. Example 11 There are 190 di erent generalized Reed-Muller codes RM (r; 3) and 14 224 di erent hyperbolic codes Hyp (s; 3). These codes are of length n = 262144. In the gure every + corresponds to a generalized Reed-Muller code of m
q
q
1
m
q
1
m
i
i
m
i=1
m
i
q
m
q
a j
j =1
a q
min a;q j =b+1
a j
64
64
the given parameters. The graph marked with a Æ corresponds to the hyperbolic codes. It appears that given a generalized Reed-Muller code, then in almost all cases there are hyperbolic codes that are of larger minimum distance and are of larger dimension.
It is well-known that generalized Reed-Muller codes are asymptotically bad and it follows from [2, Corollary 2] that the hyperbolic codes are also asymptotically bad since their minimum distance as we have shown equals the order bound.
bm m
1
1
i
a
am m
1
1
q m
1
(q )
i
m
1
(q )
q
h
h
m
h
i
q
1
j
q
h
q m
1
h
Theorem 14 The hth generalized Hamming weight of Hyp (s; m) satis es q
dh
q
m
(
)
h q; s; m :
(1)
If a dense optimal set of size h related to Hypq (s; m) exists then equality holds in (1).
We can show that for any hyperbolic code of the form Hyp (s; 2) and of dimension at least two there exists a related dense and optimal set of size two. Therefore we have the following proposition. q
Proposition 15 The second generalized Hamming weight of a hyperbolic code Hyp (s; 2) of dimension at least 2 is given by d2 = q 2 2 (q; s; 2). q
References
[1] G.-L. Feng and T.R.N. Rao, \Improved Geometric Goppa Codes, Part I:Basic theory," IEEE Trans. Inform. Theory, vol. 41, pp. 1678-1693, Nov. 1995. [2] O. Geil, \On the Construction of Codes from Order Domains", submitted to IEEE Trans. Inform. Theory, Sep. 2000. [3] O. Geil, and T. H holdt, \Footprints or Generalized Bezout's Theorem", IEEE Trans. Inform. Theory, vol. 46, pp. 635-641, Mar. 2000. [4] T. H holdt, J. H. van Lint, and R. Pellikaan, \Algebraic Geometry Godes", in Handbook of Coding Theory, (V. S. Pless, and W. C. Hufman Eds.), vol 1, pp. 871-961, Elsevier, Amsterdam 1998. [5] K. Saints, and C. Heegard, \On Hyperbolic Cascaded ReedSolomon codes", Proc. AAECC-10, Lecture Notes in Comput. Sci. Vol. 673, pp. 291-303, Springer, Berlin 1993.
