On the next-to-minimal weight of projective Reed-Muller codes

0.12 MB·

by  Cícero Carvalho

On the next-to-minimal weight of projective
Reed-Muller codes
C´ıcero Carvalho and Victor G.L. Neumann1
FaculdadedeMatem´atica, UniversidadeFederaldeUberlaˆndia,Av.J.N.A´vila2121, 38.408-902
Uberlaˆndia - MG, Brazil
7
1
0 Abstract. In this paper we present several values for the next-to-minimal weights of projective Reed-
2 Muller codes. We work over F with q ≥ 3 since in [3] we have determined the complete values for the
q
n
next-to-minimal weights of binary projective Reed-Muller codes. As in [3] here we also find examples of
a
J codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement.

6

]
T
1 Introduction
I
.
s
c
F
[ Reed-Muller codes were introduced in 1954 by D.E. Muller ([11]) as codes defined over ,
2

1 and a decoding algorithm for them was devised by I.S. Reed ([12]). In 1968 Kasami, Lin,
v
3 and Peterson ([7]) extended the original definition to a finite field Fq, where q is any prime
6
power, and named these codes “generalized Reed-Muller codes”. They also presented
6
1 some results on the weight distribution, the dimension of the codes being determined
0
in later works. In coding theory one is always interested in the values of the higher
.
1
Hamming weights of a code because of their relationship with the code performance,
0
7 but usually this is not a simple problem. For the generalized Reed-Muller codes, the
1
completedeterminationofthesecondlowest Hammingweight, alsocallednext-to-minimal
:
v
i weight, was only completed in 2010, when Bruen ([2]) observed that the value of these
X
weights could be obtained from unpublished results in the Ph.D. thesis of D. Erickson
r
a
([4]) and Bruen’s own results from 1992 and 2006. Now that Bruen called the attention
the Erickson’s thesis we know that the complete list of the next-to-minimal weights for
the generalized Reed-Muller codes may be obtained by combining results from Erickson’s
thesis with results by Geil (see [6]) or with results by Rolland (see [16]).
In 1990 Lachaud introduced the class of projective Reed-Muller codes (see [8]). The
parameters of this codes were determined by Serre ([19]), for some cases, and by Sørensen
([20]) for the general case. As for the determination of the next-to-minimal weight for
these codes, there are some results (also about higher Hamming weights) on this subject
by Rodier and Sboui ([14], [15], [18]) and also by Ballet and Rolland ([1]). Recently
1
Authors’ emails: [email protected] and [email protected]. Both authors were partially supported
by grants from CNPq and FAPEMIG.
1
the authors of this note completely determined the next-to-minimal weight of projective
F
Reed-Muller codes defined over (see [3]) In this paper we present several results on
2
the next-to-minimal Hamming weights for projective Reed-Muller codes, including the
complete determination of the next-to-minimal weights for the case of projective Reed-
F
Mullercodesdefinedover . Inthenextsectionwerecallthedefinitionsofthegeneralized
3
and projective Reed-Muller codes, and some results of geometrical nature that will allow
us to determine many cases of higher Hamming weights for the projective Reed-Muller
codes, which we do in the last section.
2 Preliminary results
Let F be a finite field and let I = (Xq − X ,...,Xq − X ) ⊂ F [X ,...,X ] be the
q q 1 1 n n q 1 n
ideal of polynomials which vanish at all points P1,...,Pqn of the affine space An(Fq).
Let ϕ : F [X ,...,X ]/I → Fqn be the F -linear transformation given by ϕ(g + I ) =
q 1 n q q q q
(g(P1),...,g(Pqn)).
Definition 2.1 Let d be a nonnegative integer. The generalized Reed-Muller code of
order d is defined as RM(n,d) = {ϕ(g +I )|g = 0 or deg(g) ≤ d}.
q
It is not difficult to prove that RM(n,d) = Fqn if d ≥ n(q − 1), so in this case the
q
minimum distance is 1. Let d ≤ n(q −1) and write d = a(q −1)+b with 0 < b ≤ q −1,
then the minimum distance of RM(n,d) is
W(1)(n,d) = (q −b)qn−a−1.
RM
(2)
According to [4] and [2] (see also [1, Thm. 9]) the next-to-minimal weight W (n,d) of
RM
RM(n,d) is equal to
c
W(2)(n,d) = W(1)(n,d)+cqn−a−2 = 1+ (q −b)qn−a−1 (2.1)
RM RM (q −b)q
(cid:18) (cid:19)
where
q if a = n−1;
b−1 if a < n−1 and 1 < b ≤ (q +1)/2;

or a < n−1 and b = q −1 6= 1;


 q if a = 0 and b = 1;


c =  q −1 if q < 4,0 < a < n−2, and b = 1;


 q −1 if q = 3,0 < a = n−2 and b = 1;

q if q = 2,a = n−2 and b = 1;

 q if q ≥ 4,0 < a ≤ n−2 and b = 1;



 b−1 if q ≥ 4,a ≤ n−2 and (q +1)/2 < b.






 2
(2)
We will need some specific values of W (n,d) in the next section.
RM
Let Q ,...,Q be the points of Pn(F ), where N = qn + ...+q +1. From e.g. [13]
1 N q
or [10] we get that the homogeneous ideal J ⊂ F [X ,...,X ] of the polynomials which
q q 0 n
vanish in all points of Pn(F ) is generated by {XqX − XqX |0 ≤ i < j ≤ n}. We
q j i i j
denote by F [X ,...,X ] (respectively, (J ) ) the F -vector subspace formed by the ho-
q 0 n d q d q
mogeneous polynomials of degree d (together with the zero polynomial) in F [X ,...,X ]
q 0 n
(respectively, J ).
q
Definition 2.2 Let d be a positive integers and let ψ : F [X ,...,X ] /(J ) → FN be
q 0 n d q d q
the F -linear transformation given by ψ(f +(J ) ) = (f(Q )...,f(Q )), where we write
q q d 1 N
the points of Pn(F ) in the standard notation, i.e. the first nonzero entry from the left
q
is equal to 1. The projective Reed-Muller code of order d, denoted by PRM(n,d), is the
image of ψ.
(1)
In[20] Sørensen determined all values forthe minimum distance W (n,d)ofPRM(n,d)
PRM
and proved that
(1) (1)
W (n,d) = W (n,d−1).
PRM RM
One may wonder if there is a similar relation between the next-to-minimal weight of
projective Reed-Muller codes and the next-to-minimal weight of generalized Reed-Muller
codes. A hint that this might be true comes from the following reasoning. Let ω be the
Hamming weight of ϕ(g +I ), where g ∈ F [X ,...,X ] is a polynomial of degree d−1,
q q 1 n
and let g(h) be the homogenization of g with respect to X . Then the degree of g(h) is
0
d−1 and the weight of ψ(X g(h)+(J ) ) is ω. This shows that, denoting by W(2) (n,d)
0 q d PRM
the next-to-minimal weight of PRM(n,d), we have
(2) (2)
W (n,d) ≤ W (n,d−1). (2.2)
PRM RM
In [3] we determined all values for the next-to-minimal weights of projective Reed-
F
Muller codes defined over , and from those results we get that there are cases for which
2
equality does not hold in (2.2). In the next section we will determine several values for
the next-to-minimal weights of projective Reed-Muller codes defined over F , with q ≥ 3,
q
and we will also find some cases where equality does not hold in (2.2). We recall from [3]
a definition and some results that we will need to prove the main results.
Definition 2.3 Let f ∈ F [X ,...,X ] . The set of points of Pn(F ) which are not zeros
q 0 n d q
of f is called the support of f, and we denote its cardinality by |f| (hence |f| is the weight
of the codeword ψ(f +(J ) )).
q d
3
In what follows the integers k and ℓ will always be the ones uniquely defined by the
equality
d−1 = k(q −1)+ℓ
with 0 ≤ k ≤ n−1 and 0 < ℓ ≤ q −1.
Theorem 2.4 ([3, Lemma2.4, Lemma2.5, Prop. 2.6 andProp. 2.7]) Letf ∈ F [X ,...,X ]
q 0 n d
be a nonzero polynomial, and let S be it support, which we assume to be nonempty. Then:
i) if there exists a hyperplane H ⊂ Pn(F ) such that S ∩ H = ∅ and |f| > W(1) (n,d)
q PRM
(2)
then |f| ≥ W (n,d−1);
RM
1
ii) if |S| < 1+ (q −ℓ)qn−k−1 then there exists a hyperplane H ⊂ Pn(F ) such that
q
q
(cid:18) (cid:19)
S ∩H = ∅;
1
iii) if |S| ≤ 1+ (q −ℓ)qn−k−1 = (q −ℓ+1)qn−k−1 then there exists r ≥ k and
(q −ℓ)
(cid:18) (cid:19)
a linear subspace H ⊂ Pn(F ) of dimension r such that S ∩H = ∅.
r q r
3 Main results
In this section we determine thenext-to-minimal weight for most cases of projective Reed-
Muller codes. Recall that we are assuming q ≥ 3. We start by treating the case where
k = n−1.
Proposition 3.1 If d−1 = (n−1)(q −1)+ℓ, with 0 < ℓ ≤ q −1 then
(2) (2)
W (n,d) = W (n,d−1) = q −ℓ+1.
PRM RM
Proof: Let f ∈ F [X ,...,X ] be a nonzero polynomial such that
q 0 n d
(2)
0 6= |f| ≤ W (n,d−1) = q −ℓ+1
RM
(1) (1)
(such a polynomial exists because W (n,d) = W (n,d− 1) = q − ℓ). Let S be the
PRM RM
support of f, from Theorem 2.4 (iii) there exists a hyperplane H such that S ∩H = ∅.
(1) (2)
From Theorem 2.4 (i) we get that |f| = W (n,d) or |f| ≥ W (n,d − 1), so that
PRM RM
(2) (2) (2) (2)
W (n,d) ≥ W (n,d−1), and from inequality (2.2) we get W (n,d) = W (n,d−
PRM RM PRM RM
(cid:3)
1).
Now we start to study the case where k < n−1.
4
q +1
Proposition 3.2 Let d−1 = k(q−1)+ℓ be such that k < n−1. If either 1 < ℓ ≤ ,
2
or q = 3, k > 0 and ℓ = 1, we get
q +1
(q −1)(q −ℓ+1)qn−k−2 if 1 < ℓ ≤ ,
(2) (2)
WPRM(n,d) = WRM(n,d−1) = 2
( 8·3n−k−2 if q = 3,k > 0 and ℓ = 1.
Proof: From (2.1) we get that
c
W(2)(n,d−1) = 1+ (q −ℓ)qn−k−1,
RM (q −ℓ)q
(cid:18) (cid:19)
where, when q = 3, 0 < k ≤ n − 2 and ℓ = 1 we have c = q − 1 = 2 (and a fortiori
c 1
= ), and when q ≥ 3 and 1 < ℓ ≤ (q +1)/2 we have c = ℓ−1 (and a fortiori
(q −ℓ)q q
c 1
≤ ). Assume, by means of absurd, that there exists f ∈ F [X ,...,X ] such
q 0 n d
(q −ℓ)q q
that
1
W(1) (n,d) < |f| < W(2)(n,d−1) ≤ 1+ (q −ℓ)qn−k−1
PRM RM q
(cid:18) (cid:19)
(2)
and let S be the support of f. From Theorem 2.4 (ii) and (i) we get |f| ≥ W (n,d−
RM
(2) (2) (2)
1), a contradiction. Hence W (n,d) ≥ W (n,d − 1) and a fortiori W (n,d) =
PRM RM PRM
W(2)(n,d−1). (cid:3)
RM
Next we treat the case where n ≥ 3, 0 ≤ k < n−2 and ℓ = 1. We observe from (2.1)
that (q2−1)qn−k−2 ≤ W(2)(n,d−1) (actually we have equality when q = 3 and an strict
RM
inequality when q > 3).
Proposition 3.3 Let n ≥ 3, 0 ≤ k < n−2 and ℓ = 1, then
1
W(2) (n,d) = 1+ (q −1)qn−k−1 = (q2 −1)qn−k−2.
PRM
q
(cid:18) (cid:19)
Proof: In case 1 ≤ k < n−2 let f = X X g +X X h, where
1 k+3 0 k+2
k+1 k+1
g = Xq−1 −Xq−1 and h = Xq−1 −Xq−1 ,
i 1 i 0
i=2 i=2
Y(cid:0) (cid:1) Y(cid:0) (cid:1)
or let f = X X + X X in case k = 0. We claim that |f| = (q2 − 1)qn−k−2, and let
1 3 0 2
α = (α : ... : α ) be a point in the support of f. If α = 0 then we may take α = 1,
0 n 0 1
5
and we have f(α) = α k+1(αq−1 −1) (or f(α) = α in case k = 0). Thus we must
k+3 i=2 i 3
have α ∈ F∗ and α = ··· = α = 0, so we get (q −1)qn−k−2 points of this form in
k+3 q 2 Q k+1
the support. If α = 1 then
0
k+1 k+1
f(α) = α α (αq−1 −αq−1)+α (αq−1 −1)
1 k+3 i 1 k+2 i
i=2 i=2
Y Y
in case 1 ≤ k < n−2, or f(α) = α α +α in case k = 0. We see that for any α ∈ F
1 3 2 1 q
we get f(α) = (α α +α ) k+1(αq−1−1), thus we must have α 6= −α α and
1 k+3 k+2 i=2 i k+2 1 k+3
α = ··· = α = 0, so we get (q −1)qn−k−1 points of this form in the support, and we
2 k+1 Q
get
|f| = (q−1)qn−k−2+(q −1)qn−k−1 = (q2 −1)qn−k−2,
this proves that W(2) (n,d) ≤ (q2 −1)qn−k−2. Assume, by means of absurd, that there
PRM
exists a ∈ F [X ,...,X ] such that
q 0 n d
1
W(1) (n,d) < |a| < (q2 −1)qn−k−2 = 1+ (q −1)qn−k−1 ≤ W(2)(n,d−1)
PRM q RM
(cid:18) (cid:19)
(2)
and let S be the support of a. From Theorem 2.4 (ii) and (i) we get |a| ≥ W (n,d−1),
RM
a contradiction, and we conclude that W(2) (n,d) = (q2 −1)qn−k−2. (cid:3)
PRM
The above proof shows that the next-to-minimal weight of projective Reed-Muller
codes is not, in all cases, attained only by evaluating polynomials that split completely
as a product of degree one factors. For example, in the case k = 0 of the above result
(hence n = 2 and d = 2) the next-to-minimal weight is attained from the evaluation of an
irreducible quadric. This is in contrast with what happens with polynomials that attain
the minimum distance of RM(n,d) and PRM(n,d) which must be product of degree one
polynomials (see [5] and [17] respectively). Also, in [9], the author proves that codewords
of next-to-minimal weight in RM(n,d), when q ≥ 3, always come from the evaluation of
polynomials which are products of degree one polynomials.
The following result deals with the case where n = 2 and d = 2 (so that k = 0 and
ℓ = 1).
Proposition 3.4 Let n = 2 and d = 2 then
W(2) (2,2) = q2.
PRM
Proof: Let g ∈ F [X ,X ,X ] be such that
q 0 1 2 2
1
W(1) (2,2) < |g| < (1+ )W(1) (2,2) = q2 −1.
PRM q PRM
6
From Theorem 2.4 (ii) and (i) we get |g| ≥ W(2)(2,1) = q2, a contradiction, hence
RM
from (2.2) we get W(2) (2,2) ∈ {q2 − 1,q2}. Let f ∈ F [X ,X ,X ] be such that
PRM q 0 1 2 2
|f| ∈ {q2 − 1,q2} and let S be its support. From Theorem 2.4 (i) if there exists a line
in P2(F ) not intersecting S then |f| = q2, so let’s assume that there is no such line.
q
After a projective transformation we may assume that P = (0 : 0 : 1) is not in S, and
let L ,...,L be the lines that contain P. Let i ∈ {0,...,q}, since deg(f) = 2 we have
0 q
|S ∩L | ≥ q −1 and from |S| ∈ {q2 −1,q2} we must have at most one line intersecting
i
S in q points. After a projective transformation that fixes P we may assume that the
line X = 0 intersects S in q − 1 points, and that P and Q = (0 : 1 : 0) are the points
0
of the line missing S, so we may write f = X (a X + a X + a X ) + X X . Observe
0 0 0 1 1 2 2 1 2
that there are q−1 points of the form (0 : 1 : α) in the support of f. Let α,β ∈ F , then
q
f(1 : α : β) = a + a α + (a + α)β and for each α 6= −a we have q − 1 values for β
0 1 2 2
such that f(1 : α : β) 6= 0. On the other hand, if α = −a then either f(1 : −a : β) = 0
2 2
or f(1 : −a : β) 6= 0 for β ∈ F , so that |f| = (q − 1) + (q − 1)2 = (q − 1)q or
2 q
|f| = (q −1)+(q −1)2 +q = q2, which proves |f| = q2. (cid:3)
(2)
We summarize the results for W (n,d) obtained in [3] and in this paper in the fol-
PRM
(2)
lowing tables, where we also list the corresponding valuesof W (n,d−1)for comparison.
RM
(2) (2)
n k ℓ W (n,d−1) W (n,d)
RM PRM
n ≥ 3 k = 0 ℓ = 1 2n 3·2n−2
n ≥ 4 1 ≤ k < n−2 ℓ = 1 3·2n−k−2 3·2n−k−2
n ≥ 2 k = n−2 ℓ = 1 4 4
n ≥ 2 k = n−1 ℓ = 1 2 2
Table 1: Next-to-minimal weights for RM(n,d) and PRM(n,d) when n ≥ 2 and q = 2
(2) (2)
n k ℓ W (n,d−1) W (n,d)
RM PRM
n = 2 k = 0 ℓ = 1 32 32
n ≥ 3 k = 0 ℓ = 1 3n 8·3n−2
n ≥ 3 1 ≤ k ≤ n−2 ℓ = 1 8·3n−k−2 8·3n−k−2
n ≥ 2 0 ≤ k ≤ n−2 ℓ = 2 4·3n−k−2 4·3n−k−2
n ≥ 1 k = n−1 ℓ = 1,2 4−ℓ 4−ℓ
Table 2: Next-to-minimal weights for RM(n,d) and PRM(n,d) when n ≥ 1 and q = 3
7
(2) (2)
n k ℓ W (n,d−1) W (n,d)
RM PRM
n = 2 k = 0 ℓ = 1 q2 q2
n ≥ 3 k < n−2 ℓ = 1 qn−k qn−k −qn−k−2
n ≥ 3 k = n−2 ℓ = 1 q2 ???
n ≥ 2 k ≤ n−2 1 < ℓ ≤ q+1 (q−1)(q −ℓ+1)qn−k−2 (q −1)(q−ℓ+1)qn−k−2
2
n ≥ 2 k ≤ n−2 q+1 < ℓ ≤ q −1 (q−1)(q −ℓ+1)qn−k−2 ???
2
n ≥ 1 k = n−1 1 ≤ ℓ ≤ q −1 q −ℓ+1 q −ℓ+1
Table 3: Next-to-minimal weights for RM(n,d) and PRM(n,d) when n ≥ 1 and q ≥ 4
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