Generalized Hamming weights of q-ary Reed-Muller codes

41 Pages·2004·0.27 MB·English

Generalized Hamming weights of q-ary
Reed-Muller codes
Petra Heijnen and Ruud Pellikaan ∗
Appeared in IEEE Trans. Inform. Theory,
vol. 44, pp. 181-196, Jan. 1998.
Abstract The order bound on generalized Hamming weights is introduced
in a general setting of codes on varieties which comprises both the one point
geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes
it is shown that this bound is sharp and that they satisfy the double chain
condition.
1 Introduction
In terms of polynomials the problem of finding the generalized Hamming
weights of q-ary Reed-Muller codes is equivalent to Problem 7.1 of [39]:
Let f ,...,f be linearly independent polynomials in m variables of degree
1 r
u or less with coefficients in F , the finite field of q elements.
q
What is the maximum possible number of solutions in Fm of the system
q
f = ··· = f = 0 ?
1 r
Let n = qm. Let P ,...,P be an enumeration of the points of Fm. The
1 n q
ring of polynomials in m variables with coefficients in F is denoted by
q
F [X ,...,X ]. Consider the evaluation map
q 1 m
ev : F [X ,...,X ] −→ Fn
q 1 m q
∗BothauthorsarefromtheDepartmentofMathematicsandComputingScience,Eind-
hoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
1
defined by ev(f) = (f(P ),...,f(P )). The q-ary Reed-Muller (RM) code
1 n
RM (u,m) of order u in m variables is the image of this evaluation map
q
of all polynomials of degree u or less. The polynomials which evaluate to
the zero word form the ideal generated by Xq −X for i = 1,...,m. If the
i i
polynomial f is evaluated to the word c, then the number of zeros of f in Fm
q
is equal to n−wt(c). If the polynomials f ,...,f are evaluated to the words
1 r
c ,...,c , then the number of common zeros of these polynomials is equal
1 r
to n minus the size of the support of the linear code generated by c ,...,c .
1 r
The support of a code C is defined by
supp(C) = {i : c 6= 0 for some c ∈ C}.
i
The support weight or effective length of C is the number of elements of its
support.
Therthgeneralized Hamming weight(GHW)orrthminimum support weight
of a linear code C is defined by
d (C) = min{|supp(D)| : D is a linear subcode of C and dim(D) = r}.
r
The RM codes were defined in [32, 28] for the binary case and this was
generalized to arbitrary q in [7, 18, 43].
Generalized Hamming weights were introduced in the study [15, 24] on the
weight distribution of codes over extensions of F . The concept was redis-
q
covered in [41] where the problem of the GHW’s of the binary RM codes
was solved. In [1] the GHW’s of the ternary RM codes was computed. An
equivalent definition was found for projective systems in projective space
in [38, 39]. The generalized weight enumerator was defined in [25] and a
Macwilliams identity was proved in [34].
The connection between the trellis or state complexity of a code and its
GHW’swasmadein[20,21]. ThesequenceofGHW’siscalledthelength/dimension
profile(LDP)in[13]. Theinversefunctioniscalledthedimension/lengthpro-
file (DLP). The DLP bound on the state complexity becomes an equality if
the code satisfies the double chain condition [8, 14, 20, 42].
A code C satisfies the chain condition if there is an increasing sequence (D )
r
of subcodes of C such that D ⊂ D , dim(D ) = r and the effective length
r r+1 r
of D is d (C) for all r. Or equivalently, there exists a permutation of the
r r
coordinates and there is an increasing sequence (D ) of subcodes of C such
r
that D ⊂ D , dim(D ) = r and supp(D ) = {1,2,...,d } for all r.
r r+1 r r r
2
A code C satisfies the two way or double chain condition if there exists
a permutation of the coordinates and there are increasing sequences (DL)
r
and (DR) of subcodes of C such that DL ⊂ DL , dim(DL) = r and
r r r+1 r
such that DR ⊂ DR , dim(DR) = r and supp(DL) = {1,2,...,d } and
r r+1 r r r
supp(DR) = {n+1−d ,n+2−d ,...,n} for all r.
r r r
A general setting will be given where our method applies. This is taken from
[16]andoriginatesfrom[11,12]. ConsiderageometricobjectX withasubset
P consisting of n points which are enumerated by P ,...,P . Suppose that
1 n
we have a vector space L over F of functions on X such that f(P ) ∈ F for
q i q
all i and f ∈ L. In this way one has an evaluation map
ev : L −→ Fn
P q
which is defined by ev (f) = (f(P ),...,f(P )). If this evaluation map is
P 1 n
linear, then its image E and its dual C are linear codes. The notion of an
orderfunctiononthespaceoffunctionsonX givessequencesofvectorspaces
L(l), evaluation codes E(l) and their duals C(l), where l is a parameter de-
noting the dimension of L(l). These codes comprise the one point geometric
Goppa codes and the RM codes. In the first case X is an algebraic curve
defined over F , P is a set of rational points on X and L is a vector space
q
of rational functions which have only poles at a fixed point P with an upper
bound on the pole order at P. In the second case X is the affine space of
dimension m, P is the set of all qm rational points of X and L is the vector
space of all polynomials in m variables with coefficients in F of degree at
q
most u.
The decoding of these codes can be done by majority voting of unknown
syndromes [10, 12, 16, 17, 30].
In this paper the order bound on the generalized Hamming weights is defined
and in detail studied for the RM codes over all finite fields. It turns out that
the order bound is sharp in this case. The shift bound as defined in [30]
is tight for the minimum distance of RM codes but does not give the right
answer for the higher Hamming weights.
A direct consequence of our methods is that the RM codes satisfy the double
chain condition. This gives the state complexity of these codes as done in
the binary case by [20, 21].
3
In Section 2 the definition of an order function is given. The order bound
on the generalized Hamming weights is given in Section 3. The theory is
applied to Reed-Muller codes in Section 4. Some combinatorics of extremal
poset theory is applied to get a formula for the higher weights of RM codes
in Section 5. Another formula is derived in Section 6. We conclude with
some remarks and open problems.
2 Order functions and higher weights
In this section the notion of an order function will be defined. It has its
origin in valuation theory [9] and the theory of Gr¨obner bases [4, 6].
In this paper F denotes a field, F the finite field with q elements. An F-
q
algebra will be a commutative ring with a unit that contains F as a unitary
subring. N denotes the positive integers and N the nonnegative integers.
0
The dimension of polynomials in one variable of degree at most d is equal
to d + 1. This is no longer the case for polynomials in several variables.
Decreasing sequences of codes (C(l) | l ∈ N) will be introduced. The code
C(l) is defined by parity checks with words that are evaluations of functions
in a certain F -algebra of a given order. For inductive arguments for bounds
q
on the generalized Hamming weights it is necessary that the dimension of
C(l) is at most one more than the dimension of C(l+1).
A way to obtain this objective for the ring of polynomials in two variables
is to define the order by looking at the degree firstly and the lexicographic
order on the exponents (i,j) of the monomial XiYj secondly. This order ≺
looks like
1 ≺ Y ≺ X ≺ Y2 ≺ XY ≺ X2 ≺ Y3 ≺ XY2 ···
Notice that this order respects multiplication of monomials. This is formal-
ized as follows.
Definition 2.1 Let R = F[X ,...,X ]. Suppose that ≺ is a total order on
1 m
the set of monomials in the variables X ,...,X such that for all monomials
1 m
M ,M , and M the following holds
1 2
(R.1) If M 6= 1, then 1 ≺ M,
(R.2) If M ≺ M , then MM ≺ MM .
1 2 1 2
4
Then≺iscalledareduction,termoradmissibleorderonthemonomials[4,6].
Themulti-indexnotationisusedformonomials. ThatmeansXα = Qm Xαs
s=1 s
if α = (α ,...,α ). The degree of Xα and α is defined by
1 m
m
X
deg(Xα) = deg(α) = α .
s
s=1
Giving a reduction order on monomials in m variables is the same as giving a
total order on Nm such that, for all α ,α , and α in Nm, the following holds
0 1 2 0
(E.1) If α 6= 0, then 0 ≺ α,
(E.2) If α ≺ α , then α+α ≺ α+α .
1 2 1 2
We use ≺ both for monomials and exponents.
Example 2.2 The lexicographic order ≺ is defined by
L
Xα ≺ Xβ ⇔ α = β ,...,α = β and α < β for some l.
L 1 1 l−1 l−1 l l
Thelexicographicorderisareductionorder. See[6]. Form = 2withX = X
1
and Y = X the start of the lexicographic order looks like
2
1 ≺ Y ≺ Y2 ≺ ··· ≺ Yj ≺ Yj+1 ≺ ···
X ≺ XY ≺ XY2 ≺ ··· ≺ XYj ≺ XYj+1 ≺ ···
X2 ≺ ···
Hence Xi+1 is the supremum of the set {XiYj | j ∈ N }. If m ≥ 2, then the
0
lexicographic order is not isomorphic with the positive integers.
Example 2.3 The graded lexicographic order ≺ is defined by
D
Xα ≺ Xβ ⇔ deg(Xα) < deg(Xβ) or deg(Xα) = deg(Xβ),Xα ≺ Xβ.
D L
The graded lexicographic order is a reduction order [6] which is isomorphic
with the positive integers. This section started with the two variable case as
an example.
The order is extended to a function on all polynomials in the following way.
Let ≺ be a reduction order which is isomorphic with the positive integers.
5
Let f ,f ,... be an enumeration of the set of monomials such that f ≺ f
1 2 i i+1
for all i. The monomials are a basis of F[X ,...,X ] over F. Hence every
1 m
nonzero polynomial f can be written in a unique way as
j
X
f = λ f ,
i i
i=1
where λ ∈ F for all i, and λ 6= 0. Then f is called the leading monomial
i j j
of f. Define a function
ρ : F[X ,...,X ] −→ N ∪{−∞},
1 m 0
by ρ(0) = −∞ and ρ(f) = j−1 where j is the smallest positive integer such
that f can be written as a linear combination of the first j monomials. It is
not difficult to show that ρ satisfies the following conditions
(O.0) ρ(f) = −∞ if and only if f = 0
(O.1) ρ(λf) = ρ(f) for all nonzero λ ∈ F
(O.2) ρ(f +g) ≤ max{ρ(f),ρ(g)}
and equality holds when ρ(f) < ρ(g).
(O.3) If ρ(f) < ρ(g) and h 6= 0, then ρ(fh) < ρ(gh)
(O.4) If ρ(f) = ρ(g), then there exists a nonzero λ ∈ F such that
ρ(f −λg) < ρ(g).
for all f,g,h ∈ R. Here −∞ < n for all n ∈ N . The properties of the
0
function ρ are captured in the following definition.
Definition 2.4 Let R be an F-algebra. An order function on R is a map
ρ : R −→ N ∪{−∞},
0
that satisfies conditions (O.0),...,(O.4).
3 The order bound on the generalized weights
In this section a sequence (C(l) : l ∈ N) of codes is defined by means of
an F -algebra R together with an order function ρ on R and a morphism of
q
F -algebras ϕ : R → Fn. The order bound d (l) is defined and it is shown
q q r
6
that it is a lower bound on the rth generalized Hamming weight of C(l).
Let R be an F -algebra with an order function ρ. So let {f : i ∈ N} be a
q i
basis of R over F such that ρ(f ) < ρ(f ) for all i ∈ N, and for all nonzero
q i i+1
f ∈ R there exists a j with ρ(f) = ρ(f ). The existence of such a basis can
j
be shown in general [16, 30], but this fact is not needed in this paper since
we specialize to the ring R = F [X ,...,X ] which has a basis of monomials
q 1 m
with the above mentioned property. Let L(l) be the vector space generated
by f ,...,f . Hence for all nonzero f ∈ R we have that ρ(f) = ρ(f ) if and
1 l l
only if l is the smallest integer such that f ∈ L(l). Let l(i,j) be the smallest
positive integer l such that f f ∈ L(l). So l(i,j) < l(i+1,j) for all i,j ∈ N.
i j
P
The standard innerproduct is defined by a·b = a b for a = (a ,...,a )
i i 1 n
and b = (b ,...,b ). The coordinatewise multiplication ∗ on Fn is defined
1 n q
by a∗b = (a b ,...,a b ). The vector space Fn becomes an F -algebra with
1 1 n n q q
this multiplication.
Let
ϕ : R −→ Fn,
q
be a morphism of F -algebras, that means ϕ is F-linear and
q
ϕ(fg) = ϕ(f)∗ϕ(g).
Let h = ϕ(f ). Define
i i
C(l) = {c ∈ Fn : c·h = 0 for all i ≤ l}.
q i
In this paper only those algebra morphisms ϕ will be considered that are
surjective. Then there exists an N such that h ,...,h generate Fn. Hence
1 N q
C(l) = 0 for all l ≥ N.
Definition 3.1 Let y ∈ Fn. Consider the syndromes
q
s (y) = y·h and s (y) = y·(h ∗h ).
i i ij i j
Then S(y) = (s (y) : 1 ≤ i,j ≤ N) is the matrix of syndromes of y.
ij
Lemma 3.2 Let y ∈ Fn. Let D(y) be the n×n diagonal matrix with y on
q
the diagonal. Let H be the N ×n matrix with h on the ith row. Then
i
S(y) = HD(y)HT,
7
Proof. The matrix of syndromes S(y) is equal to HD(y)HT, since
n
X
s (y) = y·(h ∗h ) = y h h ,
ij i j l il jl
l=1
where h is the lth entry of h . (cid:3)
il i
Definition 3.3 We define the image of S with respect to U as
Im(S,U) = {y ∈ FN : there exist u ∈ U and x ∈ FN such that xS(u) = y}
q q
and its linear span is denoted by Im∗(S,U). The image of D with respect
to U is defined by
Im(D,U) = {w ∈ Fn : there exist u ∈ U and v ∈ Fn such that vD(u) = w}
q q
and its linear span is denoted by Im∗(D,U).
Remark 3.4 The images Im(S,U) and Im(D,U) are in general not lin-
ear spaces. Take for instance q = 2 and U = h(1,1,0),(0,1,1)i. Then
Im(D,U) = F3 \{(1,1,1)}.
2
Lemma 3.5
1) Im∗(S,U) ∼= Im∗(D,U),
2) Im∗(D,U) = he : j ∈ supp(U)i ,
j
3) dim(Im∗(S,U)) = |supp(U)|.
Proof.
1) Consider the linear map η : Fn → FN with matrix HT, so η(w) = wHT.
q q
We claim that η(Im(D,U)) = Im(S,U).
Let w ∈ Im(D,U), then there exists a v ∈ Fn such that w = vD(u). The
q
rows of H generate Fn, since ϕ is surjective. So v = xH for some x ∈ FN.
q q
Hence η(w) = wHT = vD(u)HT = xHD(u)HT = xS(u) ∈ Im(S,U),
where the last equality is stated in Lemma 3.2.
Suppose y ∈ Im(S,U). Then there exists an x ∈ FN and a u ∈ U such
q
that y = xS(u). We can write S(u) = HD(u)HT, so y = xHD(u)HT.
Let v = xH, then v ∈ Fn. Let w = vD(u), then w ∈ Im(D,U) and
q
η(w) = wHT = y. This proves the claim.
Themapη isinjective,sinceH hasrankn. ThereforeIm∗(S,U) ∼= Im∗(D,U)
under η.
8
2) Let y ∈ Im(D,U). Then there is a u ∈ U and an x ∈ Fn, such that
q
y = D(u)x = u ∗ x. So y = u x , and y = 0 if j 6∈ supp(U). Hence
j j j j
he : j ∈ supp(U)i is a basis of Im∗(D,U), where e is the jth standard basis
j j
vector of length n, that is to say the ith coordinate of e is one if i = j and
j
zero otherwise.
3) This follows directly from 1) and 2). (cid:3)
Remark 3.6 One can show that Im(D,U) = Im∗(D,U) and Im(S,U) =
Im∗(S,U) if q > |supp(U)|.
Lemma 3.7 Let U be a linear subspace of C(l) of dimension r with r > 0.
Then there exists a unique r-tuple (l ,...,l ) such that l ≤ l < ··· < l < N
1 r 1 r
and dim (U ∩C(l +1)) = dim (U ∩C(l ))−1 for i = 1,...,r.
i i
Proof. Let U(m) = U ∩C(m). Then (U(m) : m ∈ N) is a decreasing chain
of subspaces of U. Furthermore U(l) = U has dimension r > 0, U(m) = 0
for m ≥ N and
dim U(m)−1 ≤ dim U(m+1) ≤ dim U(m).
So there are exactly r values for m such that l ≤ m < N and dim U(m+1) =
dim U(m)−1, which are denoted by l < ··· < l . (cid:3)
1 r
Definition 3.8 The (l ,...,l ) is called the associated r-tuple of the sub-
1 r
space U of C(l) of dimension r.
Definition 3.9 Define the sets N(l) and A(l) by
N(l) = {(i,j) ∈ N2 : l(i,j) = l+1},
A(l) = {i ∈ N : l(i,j) = l+1 for some j}.
Define for l < ··· < l the set A(l ,...,l ) by
1 r 1 r
A(l ,...,l ) = ∪r A(l ).
1 r t=1 t
Let a(l ,...,l ) denote the number of elements of A(l ,...,l ).
1 r 1 r
Lemma 3.10
1) If y ∈ C(l) and l(i,j) ≤ l, then s (y) = 0.
ij
2) If y ∈ C(l)\C(l+1) and l(i,j) = l+1, then s (y) 6= 0.
ij
9
Proof.
1) Let y ∈ C(l). If l(i,j) ≤ l, then f f ∈ L(l). So h ∗h = ϕ(f f ) is an
i j i j i j
element of ϕ(L(l)), which is the dual of C(l). Hence s (y) = y·(h ∗h ) = 0.
ij i j
2) Let y ∈ C(l)\C(l +1). If l(i,j) = l +1, then f f ∈ L(l +1)\L(l). So
i j
f f ≡ µf modulo L(l) for some nonzero µ ∈ F . Hence h ∗h ≡ µh
i j l+1 q i j l+1
modulo ϕ(L(l)). Now y 6∈ C(l+1), so s (y) 6= 0. Therefore s (y) 6= 0 (cid:3)
l+1 ij
Lemma 3.11 If t = |N(l)| and (i ,j ),...,(i ,j ) is an enumeration of the
1 1 t t
elements of N(l) in increasing order with respect to the lexicographic order
on N2, then i < ··· < i and j < ··· < j . In particular N(l) and A(l) have
1 t t 1
the same number of elements. If moreover y ∈ C(l)\C(l+1), then
(cid:26)
0 if r < s
s (y) =
irjs not zero if r = s.
Proof. The sequence (i ,j ),...,(i ,j ) is ordered in such a way that i ≤
1 1 t t 1
... ≤ i and j < j if i = i . If i = i , then j < j . So
t r r+1 r r+1 r r+1 r r+1
l+1 = l(i ,j ) < l(i ,j ) = l(i ,j ) = l+1,
r r r r+1 r+1 r+1
which is a contradiction. Hence the sequence i ,...,i is strictly increasing.
1 t
A similar argument shows that j < j for all s < t.
s+1 s
Let y ∈ C(l). If r < s, then l(i ,j ) < l(i ,j ) = l +1. Lemma 3.10 implies
r s s s
that s (y) = 0.
irjs
Moreover, let y 6∈ C(l + 1). If r = s, then l(i ,j ) = l + 1. Lemma 3.10
r s
implies that s (y) 6= 0. (cid:3)
irjs
Proposition 3.12 Let U be a linear subspace of C(l) of dimension r with
associated r-tuple (l ,...,l ). Then
1 r
|supp(U)| ≥ a(l ,...,l ).
1 r
Proof. Let U be a linear subspace of C(l) of dimension r. Let u ,...,u
1 r
be a basis of U with indices l ≤ l < ··· < l ≤ N such that u ∈ C(l ) and
1 r i i
u 6∈ C(l +1) for all i = 1,...,r.
i i
Let M be the a(l )×N submatrix of S(u ) with the a(l ) rows correspond-
k k k k
ing to the indices of A(l ). The matrix M has rank a(l ), since it has an
k k k
a(l )×a(l ) submatrix with nonzeros on the backdiagonal and zeros above
k k
the backdiagonal, according to Lemma 3.11.
10