New descriptions of the weighted Reed-Muller codes and the homogeneous Reed-Muller codes

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by  Harinaivo Andriatahiny

New descriptions of the weighted Reed-Muller
codes and the homogeneous Reed-Muller codes
7
1 Harinaivo ANDRIATAHINY
0 Mention: Mathematics and Computer Science,
2
Domain: Sciences and Technologies,
n
University of Antananarivo, Madagascar
a
J e-mail: [email protected]

4

Vololona Harinoro RAKOTOMALALA
]
T
Mention: Meteorology,
I
. Domain: Sciences of the Engineer,
s
c Higher Polytechnic School of Antananarivo (ESPA),
[
University of Antananarivo, Madagascar

1 e-mail: [email protected]
v
9
January 5, 2017
5
0
1
0 Abstract
1. Wegiveadescription oftheweightedReed-Mullercodesoveraprime
0 fieldinamodularalgebra. AdescriptionofthehomogeneousReed-Muller
7 codes in the same ambient space is presented for the binary case. A
1 decoding procedure using theLandrock-Manz method is developed.
:
v Keywords: weightedReed-Mullercodes,homogeneousReed-Mullercodes,mod-
Xi ular algebra,Jennings basis, decoding.
MSC 2010: 94B05, 94B35, 12E05.
r
a
1 Introduction
It is well known that the Generalized Reed-Muller (GRM) codes of length pm
overtheprimefieldF canbe viewedastheradicalpowersofthemodularalge-
p
braA=F [X ,...,X ]/(Xp−1,...,Xp −1)([1],[4],[5]). A isisomorphic
p 0 m−1 0 m−1
to the group algebra Fp[Fpm].
The weighted Reed-Muller codes and the homogeneous Reed-Muller codes are
classes of codes in the Reed-Muller family. The Jennings basis are used to de-
scribetheGRMcodesoverF . WeutilizetheelementsoftheJenningsbasisfor
p
the description of the weighted Reed-Muller codes and the homogeneous Reed-
Muller codes in A. P. Landrock and O. Manz developed a decoding algorithm
1
for the binary Reed-Muller codes in [9]. We use here the same method for the
binary homogeneous Reed-Muller codes.
The weighted Reed-Muller codes can be considered as a generalization of the
GRM codes. Some classes of the weighted Reed-Muller codes are algebraic-
geometric codes.
The homogeneous Reed-Muller codes are subcodes of the GRM codes. In gen-
eral, they have a much better minimum distance than the GRM codes.
We give, in section 2, the definition and some properties of the weighted Reed-
Muller codes. We consider here the affine case. In section 3, a description of
the weighted Reed-Muller codes over F in the quotient ring A is presented.
p
In section 4, we recall the definition and the parameters of the homogeneous
Reed-Muller codes. In section 5, we describe the homogeneous Reed-Muller
codes over the two elements field F in A (with p=2). In section 6, we use the
2
Landrock-Manzmethodtoconstructadecodingprocedureforthehomogeneous
Reed-Muller codes in the binary case. In section 7, an example is given.
2 Weighted Reed-Muller codes
The definition and the properties of the weighted Reed-Muller codes presented
in this section are from [11]. Let F the field of q = pr elements where p is
q
a prime number and r ≥ 1 is an integer. Let (F )m be the m-dimensional
q
affine space defined over F . F [Y ,Y ,...,Y ] is the ring of polynomials in
q q 0 1 m−1
m variables with coefficients in F . If we attach to each variables Y a natural
q i
numberw ,calledweightofY ,wespeakabouttheringofweightedpolynomials,
i i
WF [Y ,Y ,...,Y ]. The weighted degree of F ∈ WF [Y ,Y ,...,Y ], is
q 0 1 m−1 q 0 1 m−1
defined as
deg (F)=deg (F(Y ,...,Y ))=deg(F(Yw0,...,Ywm−1)),
̟ ̟ 0 m−1 0 m−1
where deg is the usual degree.
We will, without loss of generality, always assume that the weights are ordered
w ≤w ≤...≤w . Consider the evaluation map
0 1 m−1
φ: WF [Y ,...,Y ]−→(F )qm
q 0 m−1 q
(1)
F 7−→φ(F)=(F(P ),...,F(P ))
1 n
where P ,...,P (n=qm) is an arbitrary ordering of the elements of (F )m.
1 n q
For w =w =...=w =1 we have the following definition.
0 1 m−1
The Generalized Reed-Muller codes of order ν (1 ≤ ν ≤ m(q−1)) and length
n=qm is defined by
C (m,q)=φ(V(ν)),
ν
where
V(ν)={F ∈F [Y ,...,Y ]|deg(F)≤ν}.
q 0 m−1
Letω beanaturalnumberand{w ,...,w }beweightscorrespondingtothe
0 m−1
ring of weighted polynomials WF [Y ,...,Y ]. The weighted Reed-Muller
q 0 m−1
2
codesWRMC (m,q) ofweightedorderω andlengthn=qm, correspondingto
ω
the weights {w ,...,w } is defined by
0 m−1
WRMC (m,q)=φ(V (ω)), (2)
ω ̟
where
V (ω)={F ∈WF [Y ,...,Y ]|deg (F)≤ω}. (3)
̟ q 0 m−1 ̟
For a polynomial F ∈ F [Y ,...,Y ], F denotes the reduced form of F, i.e.
q 0 m−1
the polynomial of lowest degree equivalent to F modulo the ideal (Yq−Y ,i=
i i
0,...m−1). For any subset M of F [Y ,...,Y ], the set M denotes the set
q 0 m−1
of reduced elements of M.
2.1 Remark. For F ∈F [Y ,...,Y ], we have
q 0 m−1
1. for every P ∈(F )m :F(P)=F(P).
q
2. if F(P)=0 for all P ∈(F )m, then F =0.
q
Given natural numbers ω,ν, and a set of weights {w ,...,w } such that
0 m−1
m
1≤ν ≤m(q−1) and 1≤ω ≤(q−1) w .
i=1 i
Let
ν (ω)=QP′(q−1)+R′
max
where
Q
Q′ =max{Q|ω ≥ (q−1)w }
i
i=0
X
and
Q′
R′ =max{R|ω ≥ (q−1)wi+RwQ′+1}.
i=0
X
2.2 Theorem. Given a natural number ω and a set of ordered weights
{w ,...,w } such that ω ≤ (q −1) m−1w . The code WRMC (m,q) is
0 m−1 i=0 i ω
an F -linear [qm,k,d] code with
q
P
m−1
k=card({(e ,...,e )| w e ≤ω,0≤e <q})
0 m−1 i i i
i=0
X
and
d=qm−Q−1(q−R)
where Q and R are given by
ν (ω)=Q(q−1)+R,
max
with 0≤R<q−1.
2.3 Remark. The set of monomials
m−1 m−1
{ Yei | w e ≤ω,0≤e <q}
i i i i
i=0 i=0
Y X
is a basis of V (ω).
̟
3
3 Description of the weighted Reed-Muller codes
in A
Consider the modular algebra
A=F [X ,X ,...,X ]/(Xp−1,...,Xp −1)
p 0 1 m−1 0 m−1
and the ideal
I = Xp−1,...,Xp −1
0 m−1
of the polynomial ring Fp[X0(cid:0),...,Xm−1], where Fp(cid:1)is the prime field of p (a
prime number) elements.
Set x = X + I,...,x = X +I. Let us fix an order on the set of
0 0 m−1 m−1
monomials
xi0...xim−1 |0≤i ,...,i ≤p−1 .
0 m−1 0 m−1
n o
Then
p−1 p−1
A= ··· a xi0...xim−1 |a ∈F . (4)
 i0...im−1 0 m−1 i0...im−1 p
iX0=0 imX−1=0 
And we have the following identification: 
A ∋ p−1 ··· p−1 a xi0...xim−1 ←→ (a ) ∈
i0=0 im−1=0 i0...im−1 0 m−1 i0...im−1 0≤i0,...,im−1≤p−1
(F )pm.
p P P
Hence the modular algebra A is identified with (F )pm.
p
P(m,p)denotesthevectorspaceofthereducedpolynomialsinmvariablesover
F :
p
p−1 p−1
P(Y ,...,Y )= ··· u Yi0...Yim−1 |u ∈F .
 0 m−1 i0...im−1 0 m−1 i0...im−1 p
 iX0=0 imX−1=0 
Consider a set of weights {w ,...,w } and let ω be an integer such that
0 m−1
0≤ω ≤(p−1)(w +...+w ).
0 m−1
WhenconsideringP(m,p)andAasvectorspacesoverF ,wehavethefollowing
p
isomorphism:
ψ : P(m,p)−→A
p−1 p−1
(5)
P(Y ,...,Y )7−→ ··· P(i ,...,i )xi0...xim−1
0 m−1 0 m−1 0 m−1
iX0=0 imX−1=0
The set
B := (x −1)i0...(x −1)im−1 |0≤i ,...,i ≤p−1 (6)
0 m−1 0 m−1
(cid:8) (cid:9)
4
is called the Jennings basis of A.
Set [0,pm−1]={0,1,2,...,pm−1}.
Let i∈[0,pm−1]. Consider its p-adic expansion
m−1
i= i pk
k
k=0
X
with 0≤i ≤p−1 for all k =0,...,m−1.
k
We need the following notations and definitions:
i:=(i ,...,i ),
0 m−1
the p-weight of i is defined by wt (i):= m−1i ,
p k=0 k
andthep-weightofiwithrespecttothesetofweights{w ,...,w }isdefined
0 m−1
P
by
m−1
Wwt (i):= i w . (7)
p k k
k=0
X
j ≤iifj ≤i foralll =0,1,...,m−1wherej :=(j ,...,j )∈([0,p−1])m,
l l 0 m−1
x:=(x ,...,x ),
0 m−1
xi :=xi0...xim−1.
0 m−1
Consider the polynomial
B (x):=(x −1)i0...(x −1)im−1 ∈A. (8)
i 0 m−1
The following proposition is from [1].
3.1 Proposition. We have H (Y)=ψ−1(B (x)), where ψ is the isomorphism
i i
defined in (5), i.e.
B (x)= H (j)xj
i i
j≤i
X
where
m−1
H (Y):= H (Y )
i il l
l=0
Y
and
p−1−i
H (Y)=α (Y +j),
i i
j=1
Y
with α =−i! mod p.
i
3.2 Corollary. We have
m−1
deg (H (Y))=(p−1) w −Wwt (i).
̟ i l p
l=0
X
WenowpresentadescriptionoftheweightedReed-MullercodeWRMC (m,p)
ω
in the algebra A.
5
3.3Theorem. Considerasetofweights{w ,...,w }andletω beaninteger
0 m−1
such that 0≤ω ≤(p−1) m−1w . Then, the set
l=0 l
P m−1 m−1
B :={(x −1)i0...(x −1)im−1 |0≤i ≤p−1, w i ≥(p−1) w −ω}
ω 0 m−1 k k k k
k=0 k=0
X X
forms a linear basis of the weighted Reed-Muller code WRMC (m,p) over F
ω p
in A.
Proof. It is clear that B is a set of linearly independant elements because
ω
B ⊆B.
ω
Let Bi(x) := (x0 −1)i0...(xm−1 −1)im−1 ∈ Bω, i.e. 0 ≤ ik ≤ p−1, for all
k =0,...,m−1, and m−1w i ≥(p−1) m−1w −ω.
k=0 k k k=0 k
By the Proposition 3.1 and the Corollary 3.2, we have B (x) = H (j)xj
P P i j≤i i
with Hi(Y)= ml=−01Hil(Yl), Hi(Y)=αi pj=−11−i(Y +j), and αiP=−i! mod p.
We have deg (H (Y))=(p−1) m−1w −Wwt (i)≤ω.
̟Qi l=0 lQ p
Thus H (Y)∈V (ω).
i ̟
P
Therefore, B (x)∈WRMC (m,p).
i ω
Itis clearthatdimFp(WRMCω(m,p))=card({i∈[0,pm−1]|Wwtp(i)≤ω}).
On the other hand, we have card(B ) = card({i ∈ [0,pm −1] | Wwt (i) ≥
ω p
(p−1) m−1w −ω}).
k=0 k
Consider the bijection
P
θ :[0,pm−1]−→[0,pm−1]
m−1 m−1
i= i pk 7−→θ(i)= (p−1−i )pk.
k k
k=0 k=0
X X
We have Wwt (θ(i)) = m−1w (p−1−i ) = (p−1) m−1w −Wwt (i),
p k=0 k k k=0 k p
i.e. Wwt (i)=(p−1) m−1w −Wwt (θ(i)).
p Pk=0 k p P
Thus, we have Wwt (i)≤ω ⇐⇒Wwt (θ(i))≥(p−1) m−1w −ω.
p P p k=0 k
Hence,card({i∈[0,pm−1]|Wwt (i)≤ω})=card({i∈[0,pm−1]|Wwt (i)≥
p p
(p−1) m−1w −ω}). P
k=0 k
ThePfollowing Corollaryis the famous resultof Berman-Charpin([1],[4],[5]).
3.4 Corollary. Consider the weights w = ... = w = 1 and an integer ω
0 m−1
such that 0≤ω ≤m(p−1). Then, the set
m−1
B :={(x −1)i0...(x −1)im−1 |0≤i ≤p−1, i ≥m(p−1)−ω}
ω 0 m−1 k k
k=0
X
forms a linear basis of the GRM code C (m,p)=Pm(p−1)−ω over F , where P
ω p
is the radical power of A.
6
4 The homogeneous Reed-Muller codes
Inthissection,we recallthe definitionandsomepropertiesofthe homogeneous
Reed-Mullercodes[3],[10]. F denotethefieldofq =pr elementswithpaprime
q
number and r ≥1 an integer. For n=qm−1, let {0,P ,...,P } be the set of
1 n
points in (F )m ordered in a fixed order.
q
Let F [Y ,...,Y ]0 be the vector space of homogeneous polynomials in m
q 0 m−1 d
variables over F of degree d.
q
Nowddenoteanintegersuchthat0≤d≤m(q−1). Thedthorderhomogeneous
Reed-Muller (HRM) codes of length qm over F is defined as
q
HRMC (m,q):={(F(0),F(P ),...,F(P ))|F ∈F [Y ,...,Y ]0}. (9)
d 1 n q 0 m−1 d
Thus HRMC (m,q) is a proper subcode of the GRM code C (m,q).
d d
The following theorem can be found in [3].
4.1 Theorem. Let d such that 1 ≤ d ≤ (m − 1)(q − 1). The HRM code
HRMC (m,q) is an [n+1,k,δ] linear code with n+1=qm,
d
m
m t−jq+m−1
k = (−1)j ,
j t−jq
t≡dmod(Xq−1),0<t≤dXj=0 (cid:18) (cid:19)(cid:18) (cid:19)
and
δ =(q−1)(q−s)qm−r−2,
where d−1=r(q−1)+s and 0≤s<q−1.
5 Description of the binary HRM codes in A
First, we recall some results in the Proposition 3.1 for the special case p = 2.
In this section, we consider the ambiant space
A=F [X ,...,X ]/(X2−1,...,X2 −1).
2 0 m−1 0 m−1
We have
B (x)=(x −1)i0...(x −1)im−1 = H (j)xj
i 0 m−1 i
j≤i
X
where 0≤i ≤1, for all k,
k
m−1
H (Y):= H (Y )
i il l
l=0
Y
and
1−i
H (Y)=α (Y +j),
i i
j=1
Y
7
with α =−i! mod 2.
i
Note that B (x)=(x −1)1...(x −1)1 =ˆ1 is the "all one" word.
(1,1,...,1) 0 m−1
Let d be an integer such that 0 ≤ d ≤ m. The dth order homogeneous Reed-
Muller (HRM) codes of length 2m over F is defined as
2
HRMC (m,2):={(F(0),F(P ),...,F(P ))|F ∈F [Y ,...,Y ]0},
d 1 n 2 0 m−1 d
where n = 2m − 1. We now give the description of the binary HRM code
HRMC (m,2) in A.
d
5.1 Theorem. Let d be an integer such that 1≤d≤m. The set
m−1
{(x −1)i0...(x −1)im−1 +ˆ1|0≤i ≤1, m> i ≥m−d}
0 m−1 k k
k=0
X
forms a linear basis for the binary HRM code HRMC (m,2).
d
Proof. Let d such that 1≤d≤m.
Consider the element
B (x)+ˆ1=(x −1)i0...(x −1)im−1 +ˆ1,
i 0 m−1
where 0≤i ≤1 for all k and m> m−1i ≥m−d.
k k=0 k
Set D(i):={j ∈({0,1})m |j ≤i} and C(i):=({0,1})m−D(i).
P
We have H (j)=0 for j ∈C(i).
i
Thus
B (x)= H (j)xj = H (j)xj .
i i i
Xj≤i j∈({X0,1})m
We have
m−1
H (Y):= H (Y ),
i il l
l=0
Y
where
H (Y)=1,H (Y)=Y +1. (10)
1 0
Since m−1i ≥ m −d, then B (x) ∈ Pm−d where P is the radical of the
k=0 k i
modular algebra A.
And siPnce Pm−d =C (m,2), then H (Y)∈P (m,2) where P (m,2) is a linear
d i d d
space generated by the set
m−1
{Yi0...Yim−1 |0≤i ≤1,0≤ i ≤d}. (11)
0 m−1 k k
k=0
X
We have
B (x)+ˆ1= (H (j)+1)xj .
i i
j∈({X0,1})m
8
By (10) and (11), we have H (Y)+1∈F [Y ,...,Y ]0.
i 2 0 m−1 d
Note that F [Y ,...,Y ]0 is a linear space generated by the set
2 0 m−1 d
m−1
S :={Yi0...Yim−1 |0≤i ≤1,0< i ≤d}.
0 m−1 k k
k=0
X
Thus B (x)+ˆ1∈HRMC (m,2).
i d
Note also that m−1i =m if and only if i =1 for all k =0,...,m−1.
k=0 k k
SetR:={(x0−P1)i0...(xm−1−1)im−1+ˆ1|0≤ik ≤1,m> mk=−01ik ≥m−d}.
WWee hwailvleshdoimwFt2h(HatRdMimCF2d((HmR,2M))C=d(dmim,2F2))(F=2[cYa0r,d.(.R.,)Y.m−1]0d)P=card(S).
Consider the bijection
β : ({0,1})m −→({0,1})m
(i ,...,i )7−→(1−i ,...,1−i )
0 m−1 0 m−1
Set R′ :={i=(i ,...,i )∈({0,1})m | m−1i ≥m−d}
0 m−1 k=0 k
and S′ :={i=(i ,...,i )∈({0,1})m | m−1i ≤d}.
0 m−1 Pk=0 k
It is clear that S′ =β(R′). Thus card(R′)=card(S′).
Since card(R) = card(R′) − 1 and card(SP) = card(S′) − 1, then card(R) =
card(S).
6 Decoding procedure for the binary HRM codes
In this section, we will follow Landrock-Manz as in [9].
Let d be an integer such that 1≤ d≤ m. The HRM code HRMC (m,2) is of
d
type 2m, d m ,2m−d over F .
t=1 t 2
Set b(h{i1,.P..,it}(cid:0)):(cid:1)=(xi1 −i1)...(xit −1),
where {i ,...,i }⊆{0,1,...,m−1}.
1 t
B := {b(η)+ˆ1 | η ⊆ {0,1,...,m−1},m > card(η) ≥ m−d} is a linear
m−d
basis of HRMC (m,2).
d
General results of the following Proposition can be found in [2].
6.1 Proposition. We have
1. b({})=1.
2.
0 if η∩κ6={},
b(η).b(κ)=
(b(η∪κ) otherwise.
3. The weight of the codeword w(b({i ,...,i }))=2t.
1 t
4. b({0,1,...,m−1})=ˆ1 the "all one" word.
5. ˆ1.b({η})=0 if η 6={}.
9
Set ηc :={0,1,...,m−1}−η.
Let c ∈ HRMC (m,2) be a transmitted codeword and v ∈ A the received
d
vector, where
A=F [X ,...,X ]/(X2−1,...,X2 −1)
2 0 m−1 0 m−1
Since HRMC (m,2) is (2m−d−1−1)-error correcting, we write v =c+f with
d
w(f)≤2m−d−1−1.
We have
c= τ(η)(b(η)+ˆ1)
m−d≤card(η)<mX,η⊆{0,1,...,m−1}
with τ(η)∈F .
2
We now present the decoding procedure to determine the coefficients τ(η).
Step1:
Let κ be a subset of {0,1,...,m−1} such that card(κ)=m−d. We have
v.b(κc)=(c+f).b(κc)
=( τ(η)(b(η)+ˆ1)+f).b(κc)
m−d≤card(η)<mX,η⊆{0,1,...,m−1}
= τ(η)(b(η).b(κc)+ˆ1.b(κc))+f.b(κc)
m−d≤card(η)<mX,η⊆{0,1,...,m−1}
=τ(κ).ˆ1+f.b(κc)
We havew(f.b(κc))≤w(f).w(b(κc))≤(2m−d−1−1).2d =2m−1−2d <2m−1 =
12m.
2
Then it is easy to see that
τ(κ)=0 if and only if w(v.b(κc))<2m−1.
We next subtract τ(κ).(b(κ)+ˆ1) from v and obtain v′ =v+τ(κ).(b(κ)+ˆ1)=
c′+f where c′ =c+τ(κ).(b(κ)+ˆ1)∈HRMC (m,2).
d
Consider another set η of cardinalitycard(η)=m−d, multiply v′ by b(ηc) and
find so τ(η). Having eventually run through all sets of cardinality m−d, we
end up with v′′ =c′′+f, where c′′ ∈HRMC (m,2).
d−1
Step2:
We now fix a set κ of cardinalitym−d+1 and by using the same technique as
inthe firststep,wecanfindthe coefficientτ(κ). We repeatthe sametreatment
for all set η of cardinality m−d+1. We eventually determine
c= τ(η)(b(η)+ˆ1).
m−d≤card(η)<mX,η⊆{0,1,...,m−1}
If another step is needed, we must pick a set κ of cardinality m−d+2 and
determine τ(κ), and we continue in this way.
10