Construction of Z4-linear Reed-Muller codes

0.26 MB·

by  J. Pujol

Construction of Z -linear Reed-Muller codes∗
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J. Pujol, J. Rif`a, F. I. Solov’eva †
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0
0
2 Abstract

n
NewquaternaryPlotkinconstructionsaregivenandareusedtoobtain
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new families of quaternary codes. Theparameters of the obtained codes,
J
suchasthelength,thedimensionandtheminimumdistancearestudied.
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Using these constructions new families of quaternary Reed-Muller codes
1
arebuiltwiththepeculiaritythatafterusingtheGraymap theobtained

] Z4-linearcodes havethesame parameters and fundamentalproperties as
T the codes in the usual binary linear Reed-Muller family. To make more
I evident the duality relationships in the constructed families the concept
.
s of Kroneckerinnerproduct is introduced.
c
[

1 Introduction
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v
4 In[13]NechaevintroducedtheconceptofZ -linearityofbinarycodesandlater,
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2 in [7], Hammons, Kumar, Calderbank, Sloane and Sol´e showed that several
0 families of binary codes are Z -linear. In [7] it is proved that the binary linear
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Reed-Muller code RM(r,m) is Z -linear for r = 0,1,2,m− 1,m and is not
. 4
1 Z -linear for r = m−2 (m ≥ 5). In a subsequent work, Hou, Lahtonen and
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0 Koponen [8], proved that RM(r,m) is not Z -linear for 3≤r ≤m−2.
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In [7] it is introduced a construction of codes, called QRM(r,m), based on
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: Z4-linear codes, such that after doing modulo two we obtain the usual binary
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linear Reed-Muller (RM) codes. In [2, 3] such family of codes is studied and
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X their parametersare computed as well as the dimension ofthe kerneland rank.
r In [16] a kind of Plotkin construction was used to build a family of additive
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Reed-Muller codes and also in [19] it was used a Plotkin constructionto obtain
a sequence of quaternary linear Reed-Muller like codes. In both last quoted
constructions,the imagesofthe obtainedcodesunder the Graymaparebinary
codes with the same parameters as the binary linear RM codes. Moreover, on
theotherhand,in[5,9,10,14]wereclassifiedallthenon-equivalentZ Z -linear
2 4
extended 1-perfect codes and their duals, the Z Z -linear Hadamard codes. It
2 4
∗ThisworkhasbeenpartiallysupportedbytheSpanishMECandtheEuropeanFEDER
Grant MTM2006-03250. Part of the material in Section III of this paper was presented at
the 17th Symposium on Appliedalgebra, Algebraicalgorithms, and ErrorCorrecting Codes
(AAECC),Bangalore,India,December2007.
†J.PujolandJ.Rifa`arewiththeDepartmentofInformationandCommunicationsEngi-
neering,UniversitatAuto`nomadeBarcelona,08193-Bellaterra,Spain. F.I.Solov’evaiswith
theSobolevInstituteofMathematics andNovosibirskState University,Novosibirsk,Russia.
1
is a natural question to ask for the existence of families of quaternary linear
codessuchthat,aftertheGraymap,thecorrespondingZ -linearcodeshavethe
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same parameters as the well known family of binary linear RM codes. In these
new families, like in the usual RM(r,m) family, the code with (r,m) = (1,m)
should be a Hadamard code and the code with (r,m) = (m−2,m) should be
an extended 1-perfect code.
It is well known that an easy way to built the binary RM family of codes
is by using the Plotkin construction [12]. So, it seems a good matter of study
to try to generalizethe Plotkinconstructionto the quaternarylinearcodes and
try to obtain new families of codes which contain the above mentioned Z Z -
2 4
linear Hadamardcodes andZ Z -linear extended 1-perfectcodes andfulfill the
2 4
same properties from a parameters point of view (length, dimension, minimum
distance, inclusion and duality relationship) than the binary RM family.
InthispaperwebeginbystudyingtheZ -linearcaseandweorganizeitinthe
4
followingway. InSection2weintroducetheconceptofquaternarycodeandgive
some constructions that could be seen as quaternary generalizationsof the well
knownbinaryPlotkinconstruction. InSection3,weconstructseveralfamiliesof
Z -linearReed-Mullercodesandprovethattheyhavesimilarparametersasthe
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classicalbinary RM codes but they are not linear. In Section 4, we discuss the
concept of duality for the constructed Z -linear Reed-Muller codes and, finally,
4
in Section 5 we give some conclusions and further researchin the topic.
2 Constructions of quaternary codes
2.1 Quaternary codes
LetZ andZ betheringofintegersmodulotwoandmodulofour,respectively.
2 4
Let Zn be the set of all binary vectors of length n and ZN be the set of all
2 4
quaternary vectors of length N. Any non-empty subset C of Zn is a binary
2
code and a subgroup of Zn is called a binary linear code. Equivalently, any
2
non-empty subset C of ZN is a quaternary code and a subgroupof ZN is called
4 4
a quaternary linear code. In general, any non-empty subgroup C of Zα×Zβ is
2 4
an additive code.
The Hamming weight w (u) of a vector in Zn is the number of its nonzero
H 2
coordinates. The Hamming distance d(u,v) between two vectors u,v ∈ Zn is
2
d(u,v) = w (u−v). For quaternary codes it is more appropriate to use the
H
Lee metric [11]. In Z the Lee weight coincides with the Hamming weight, but
2
in Z the Lee weight of their elements is w (0) = 0,w (1) = w (3) = 1, and
4 L L L
w (2)= 2. The Lee weight w (u) of a vector in ZN is the addition of the Lee
L L 4
weight of all the coordinates. The Lee distance d (u,v) between two vectors
L
u,v∈ZN is d (u,v)=w (u−v).
4 L L
LetC beanadditive code,soasubgroupofZα×Zβ andletC =Φ(C),where
2 4
Φ : Zα ×Zβ −→ Zn, n = α+2β, is given by Φ(u,v) = (u,φ(v)) for any u
2 4 2
from Zα and any v from Zβ, where φ : Zβ −→ Z2β is the usual Gray map,
2 4 4 2
so φ(v ,...,v ) = (ϕ(v ),...,ϕ(v )), and ϕ(0) = (0,0),ϕ(1) = (0,1),ϕ(2) =
1 β 1 β
2
(1,1), ϕ(3)= (1,0). We will use the symbols 0, 1 and 2 for the all zeroes, the
all ones and the all twos vectors, respectively (by the context it will be always
clear we speak about the binary vectors0, 1 or quaternary,it will also be clear
the length of the vectors).
Hamming and Lee weights, as well as Hamming and Lee distances, can be
generalized,inanaturalway,tovectorsinZα×Zβ byaddingthecorresponding
2 4
weights (or distances) of the Zα part and the Zβ part.
2 4
SinceC isasubgroupofZα×Zβ,itisalsoisomorphictoanabelianstructure
2 4
like Zγ ×Zδ. Therefore, we have that |C|= 2γ4δ and the number of order two
2 4
codewordsin C is 2γ+δ. We call suchcode C an additive code of type (α,β;γ,δ)
and the binary image C = Φ(C) a Z Z -linear code of type (α,β;γ,δ). In the
2 4
specific case α = 0 we see that C is a quaternary linear code and its binary
imageis calleda Z -linear code. Note thatthe binarylengthofthe binarycode
4
C =Φ(C) is n=α+2β.
The minimumHammingdistancedofaZ Z -linearcode C is theminimum
2 4
value of d(u,v), where u,v∈C andu6=v. Notice that the Hamming distance
of a Z Z -linear code C coincides with the Lee distance defined in the additive
2 4
code C = Φ−1(C). From now on, when we work with distances it must be
understoodthatwedealwithHamming distancesinthe caseofbinarycodesor
Lee distances in the case of additive codes.
Although C could not have a basis, it is appropriate to define a generator
matrix for C as
B Q
G = 2 2 ,
(cid:18) B1 Q1 (cid:19)
where B is a γ×α matrix; Q is a γ×β matrix; B is a δ×α matrix and Q
2 2 1 1
is a δ×β matrix. Matrices B ,B are binary and Q ,Q are quaternary, but
1 2 1 2
the entries in Q are only zeroes or twos.
2
TwoadditivecodesC andC bothofthesamelengtharesaidtobemonomi-
1 2
ally equivalent, if one can be obtained from the other by permuting the coordi-
natesandmultiplyingby−1ofcertaincoordinates. Additivecodeswhichdiffer
only by a permutation of coordinates are said to be permutation equivalent.
ForZ Z -linearcodesisusualtousethefollowingdefinitionofinnerproduct
2 4
in Zα×Zβ that we will call the standard inner product [18, 4]:
2 4
α α+β
hu,vi=2( u v )+ u v ∈Z , (1)
i i j j 4
Xi=1 j=Xα+1
where u,v∈Zα×Zβ. We can also write the standard inner product as
2 4
hu,vi=u·J ·vt,
N
2I 0
where J = α , N =α+β, is a diagonalmatrix overZ . Note that
N (cid:18) 0 Iβ (cid:19) 4
when α = 0 the inner product is the usual one for vectors over Z and when
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β =0 it is twice the usual one for vectors over Z .
2
3
For α = 0 and N = β = 2i, i = 1,2,3,..., we can define the inner product
1 0
in an alternative way. Let K = be a matrix over Z and define
2 (cid:18) 0 3 (cid:19) 4
K = log2(N)K where denotes the Kronecker product of matrices. We
N j=1 2
call theNKronecker inner prNoduct the following:
hu,vi =u·K ·vt. (2)
⊗N N
The additive dual code of C, denoted by C⊥, is defined in the standard way
as
C⊥ ={u∈Zα×Zβ |hu,vi=0 for all v∈C}
2 4
or, using the Kronecker inner product
C⊥ ={u∈Zα×Zβ |hu,vi =0 for all v∈C}.
2 4 ⊗N
The definition and notations will be the same for the Z -duality obtained
4
by using the standard inner product or the Kronecker inner product and the
difference will be clear from the context.
Note that hu,vi = u·K ·vt = hu,v·K i. Hence, both additive dual
⊗N N N
codes by using the standard inner product or the Kronecker inner product,
respectively, are monomially equivalent and so they have the same weight dis-
tribution. Forbothinnerproducts,theadditivedualcodeC⊥ isalsoanadditive
code,thatisasubgroupofZα×Zβ. Itsweightenumeratorpolynomialisrelated
2 4
tothe weightenumeratorpolynomialofC bytheMacWilliamsidentity[6]. The
corresponding binary code Φ(C⊥) is denoted by C and called the Z Z -dual
⊥ 2 4
code of C. In the case α = 0, the code C⊥ is also called the quaternary dual
code of C and C the Z -dual code of C. Notice that C and C are not dual
⊥ 4 ⊥
in the binary linear sense but the weight enumerator polynomial of C is the
⊥
McWilliams transform of the weight enumerator polynomial of C. Given an
additivecodeC oftype(α,β,γ,δ)itisknownthetypeoftheadditivedualcode
([4] for additive codes with α6=0 and [7] for additive codes with α=0).
In the present paper, as we will see later, the duality concept using the
Kronecker inner product will make more visible the property that if a code C
belongstoafamilyofReed-Mullercodesthenitsdualcodebelongstothesame
family.
From now on, we focus our attention specifically to additive codes with
α = 0, so quaternary linear codes such that after the Gray map they give rise
to Z -linear codes. Given a quaternary linear code of type (0,β;γ,δ), we will
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write (N;γ,δ) to say that α=0 and β =N.
2.2 The Plotkin construction
In this section, we show that the well-known binary Plotkin construction can
be generalized to quaternary linear codes.
Let A and B be two quaternarylinear codes of types (N;γ ,δ ) and
A A
(N;γ ,δ ) and minimum distances d , d , respectively. Given u ∈ ZN define
B B A B 4
supp(u)⊂{1,...,N} as the set of nonzero coordinates of vector u.
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Definition 1 (Plotkin Construction) Given two quaternary linear codes A
and B, we define a quaternary linear code as
PC(A,B)={(u |u +u ):u ∈A,u ∈B}.
1 1 2 1 2
ItiseasytoseethatifG andG aregeneratormatricesofAandB,respectively,
A B
then the matrix
G G
G = A A
PC (cid:18) 0 GB (cid:19)
is a generator matrix of the code PC(A,B).
Proposition 2 The quaternary linear code PC(A,B) defined using the Plotkin
construction is of type (2N;γ,δ), where γ = γ +γ and δ = δ +δ ; the
A B A B
binary length is n = 4N; the size is 2γ+2δ and the minimum distance is d =
min{2d ,d }.
A B
Proof: The type, the binary length and the size of PC(A,B) can be easily
computedfromthedefinitionofthecode. The minimumdistancecanbe estab-
lishedasinthe binarycase[12]but, bycompleteness,weinclude the proof. Let
us consideranyvectoru∈PC(A,B)suchthat u=(u |u +u ), where u ∈A
1 1 2 1
and u ∈ B. Since PC(A,B) is a quaternary linear code, it is enough to prove
2
that the weight w (u) is not less than d.
L
If u =0, then w (u)=2w (u )≥2d .
2 L L 1 A
If u 6=0, by using the triangle inequality we immediately obtain
2
w (u)=w (u )+w (u +u )≥w (u )≥d .
L L 1 L 1 2 L 2 B
Hence d ≥ min{2d ,d }. The equality holds because taking the specific
A B
vectors u ∈A with minimum weight d and u ∈B with minimum weight d
1 A 2 B
we obtain w (u |u )=2d and w (0|v )=d . (cid:3)
L 1 1 A L 2 B
2.3 The quaternary Plotkin construction
A useful generalization of the above construction to obtain quaternary linear
codes is the following construction, called the quaternary Plotkin construction.
Such construction was used, for example, in [10] for the classification of all
Z -linear Hadamard codes.
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Definition 3 (Quaternary Plotkin Construction) Giventwoquaternarylin-
ear codes A and B, we define the quaternary linear code
QP(A,B)={(u |u +u |u +2u |u +3u ):u ∈A,u ∈B}.
1 1 2 1 2 1 2 1 2
It is easy to see that if G and G are generator matrices of A and B, then
A B
the matrix
G G G G
G = A A A A
QP (cid:18) 0 GB 2GB 3GB (cid:19)
is a generator matrix of the code QP(A,B).
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Proposition 4 The quaternary linear code QP(A,B) given in Definition 3 is
of type (4N;γ,δ), where γ = γ +γ and δ = δ +δ ; the binary length is
A B A B
n=8N; the size is 2γ+2δ and the minimum distance is d≥min{4d ,2d }.
A B
Proof: The type, the binary length and the size of QP(A,B) can be easily
computed from the definition of the code. To check the minimum distance of
QP(A,B)letusconsideranyvectoru∈QP(A,B). Vectorucanberepresented
by u = (u |u |u |u ) + (0|u |2u |3u ), where u ∈ A and u ∈ B. Since
1 1 1 1 2 2 2 1 2
QP(A,B) is a quaternary linear code it is enough to show that the weight of u
is at least d.
Ifu =0,thenw (u)=4w (u )≥4d . Theequalityholdstakingavector
2 L L 1 A
u ∈A of minimum weight.
1
For u 6=0 we have
2
w (u)=w (u |u +u |u +2u |u +3u )
L L 1 1 2 1 2 1 2
=(w (u )+w (u +u ))+(w (u +2u )+w (u +2u +u ))
L 1 L 1 2 L 1 2 L 1 2 2
≥w (u )+w (u )(by using the triangle inequality)
L 2 L 2
≥2d .
B
(cid:3)
The Plotkinandthe quaternaryPlotkinconstructionscanbecombinedina
double Plotkin construction. Let A, B, C and D be four quaternary linear
codes of types (N;γ ,δ ), (N;γ ,δ ), (N;γ ,δ ), and (N;γ ,δ ) and
A A B B C C D D
minimum distances d , d , d , d , respectively.
A B C D
Definition 5 (Double Plotkin Construction) Given A, B, C and D four
quaternary linear codes, we define the quaternary linear code
DP(A,B,C,D)={(u |u +u |u +2u +u |u +3u +u +u ):u ∈A,u ∈B,u ∈C,u ∈D}.
1 1 2 1 2 3 1 2 3 4 1 2 3 4
It is easy to see that if G , G , G and G are generator matrices of A, B,
A B C D
C and D, then the matrix
G G G G
A A A A
G = 0 GB 2GB 3GB 
DP 0 0 G G
 C C 
 0 0 0 GD 
 
is a generator matrix of the code DP(A,B,C,D).
Proposition 6 The quaternary linear code DP(A,B,C,D) given in Defini-
tion5isoftype(4N;γ,δ),whereγ =γ +γ +γ +γ andδ =δ +δ +δ +δ ;
A B C D A B C D
the binary length is n = 8N; the size is 2γ+2δ and the minimum distance is
d≥min{4d ,2d ,2d ,d }.
A B C D
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Proof: The type, the binary length and the size of the code DP(A,B,C,D)
can be easily computed from the definition.
To check the minimum distance of the code DP(A,B,C,D) let us consider
any vector u from this code. It can be represented as u = (u |u |u |u )+
1 1 1 1
(0|u |2u |3u )+(0|0|u |u )+(0|0|0|u ), where u ∈ A, u ∈ B, u ∈ C and
2 2 2 3 3 4 1 2 3
u ∈D. Since DP(A,B,C,D) is a quaternary linear code it is enough to show
4
that the weight of u is, at least, d.
If u = 0 then we can write u = (u |u |u |u )+(0|0|u |u +u ) so that
2 1 1 1 1 3 3 4
u∈PC((A|A),PC(C,D)),where(A|A)isthecodegeneratedby(G |G ). Using
A A
Proposition 2 we obtain
w (u)=min{2d ,d }=min{4d ,min{2d ,d }}=min{4d ,2d ,d }.
L (A|A) P(C,D) A C D A C D
Ifu 6=0thenwedistinguishtwocases. Ifu =0thenw (u)=w (u |u +
2 4 L L 1 1
u )+w (u +2u +u |u +3u +u )≥w (u )+w (u )≥2d using twice
2 L 1 2 3 1 2 3 L 2 L 2 B
the triangle inequality.
Ifu 6=0thenw (u)=w (u |u +u )+w (u +2u +u |u +3u +u +
4 L L 1 1 2 L 1 2 3 1 2 3
u )≥w (u )+w (u +u )≥w (u )≥d . (cid:3)
4 L 2 L 2 4 L 4 D
NotethatincaseB =Ctheboundistightbecaused =d andtheminimum
B C
distanced=min{4d ,2d ,d }canbe obtainedtakingspecific vectorsfromA,
A C D
C or D.
2.4 The BQ-Plotkin construction
We slightly change the construction given in Definition 5 in order to obtain
a tight bound for the minimum distance. We call this new construction the
BQ-Plotkin construction.
Let A, B and C be three quaternary linear codes of types (N;γ ,δ ),
A A
(N;γ ,δ ), (N;γ ,δ ), with minimum distances d , d and d , respectively.
B B C C A B C
Definition 7 (BQ-Plotkin Construction) Let G , G and G be generator
A B C
matrices of the quaternary linear codes A, B and C, respectively. We define a
new code BQ(A,B,C) as the quaternary linear code generated by
G G G G
A A A A
 0 G′ 2G′ 3G′ 
G = B B B ,
BQ 0 0 Gˆ Gˆ
 B B 
 0 0 0 G 
 C 
where G′ is the matrix obtained from G after switching twos by ones in their
B B
γ rows ofordertwoandGˆ is thematrixobtained fromG afterremovingtheir
B B B
γ rows of order two.
B
Proposition 8 The quaternary linear code BQ(A,B,C) is of type (4N;γ,δ),
where γ = γ +γ and δ = δ +γ +2δ +δ ; the binary length is n = 8N;
A C A B B C
the size is 2γ+2δ and the minimum distance d=min{4d ,2d ,d }.
A B C
7
Proof: The type, the length and the size of BQ(A,B,C) can be easily com-
puted from the definition of the code.
TochecktheminimumdistanceofBQ(A,B,C)letusconsideranyvectoru=
(u |u |u |u )+(0|u |2u |3u )+(0|0|u |u )+(0|0|0|u )∈BQ(A,B,C),where
1 1 1 1 2 2 2 3 3 4
u ∈A; u ∈B′; u ∈Bˆand u ∈C. Codes B′ and Bˆare the quaternarylinear
1 2 3 4
codes generated by G′ and Gˆ , respectively. Since BQ(A,B,C) is a quaternary
B B
linear code it is enough to show that the weight of u is at least d.
If u = 0 then by using the same arguments as in Proposition 6 we have
2
w (u)≥min{4d ,2d ,d } because d ≥d .
L A B C Bˆ B
If u 6=0 then we distinguish two cases. If u =0 then
2 4
w (u)=w (u |u +u |u +2u +u |u +3u +u )
L L 1 1 2 1 2 3 1 2 3
=(w (u )+w (u +u ))+(w (u +2u +u )+w (u +2u +u +u ))
L 1 L 1 2 L 1 2 3 L 1 2 3 2
=(w (u )+w (u +2u +u ))+(w (u +u ))+w (u +2u +u +u ))
L 1 L 1 2 3 L 1 2 L 1 2 3 2
≥w (2u +u )+w (2u +u )(by the triangle inequality)
L 2 3 L 2 3
≥2w (2u +u ).
L 2 3
Note that 2u ∈ B and u ∈ Bˆ⊂ B. If u 6= 2u then w (2u +u ) ≥ d
2 3 3 2 L 2 3 B
and w (u) ≥ 2d . If 2u = u then u ∈ Bˆ and w (u ) ≥ d . So, w (u) =
L B 2 3 2 L 2 B L
w (u |u +u |u |u +u )≥2w (u )≥2d .
L 1 1 2 1 1 2 L 2 B
Using twice the triangle inequality, the case u 6=0 easily gives
4
w (u)=w (u |u +u )+w (u +2u +u |u +3u +u +u )
L L 1 1 2 L 1 2 3 1 2 3 4
≥w (u )+w (u +u )≥w (u ).
L 2 L 2 4 L 4
Hence, d ≥ min{4d ,2d ,d }. But the equality holds after the following
A B C
considerations.
Taking the specific vector u ∈ A with minimum weight d we obtain
1 A
w (u |u |u |u )=4d .
L 1 1 1 1 A
Taking the specific vector u ∈ C with minimum weight d we obtain
4 C
w (0|0|0|u )=d .
L 4 C
Taking the specific vector u ∈ B with minimum weight d we obtain the
2 B
following. NotethatB ⊂B′ andsowecanwrite the vectoru asu =vˆ+2w′,
2 2
wherevˆ∈Bˆandw′ ∈B′\Bˆ. Takethe vectoru =vˆ+2w′ ∈B′ and,moreover,
2
the vector uˆ = 2vˆ ∈ Bˆ and compose the vector (0|u |2u + uˆ|3u + uˆ) =
2 2 2
(0|u |0|u )whichbelongstoBQ(A,B,C). ThisvectorhasminimumLeeweight
2 2
2d . (cid:3)
B
3 Quaternary Reed-Muller codes
The usual binary linear RM family of codes is one of the oldest and inter-
esting family of codes. The codes in this family are easy to decode and their
combinatorialproperties are of great interest to produce new optimal codes.
8
For any integer m≥1 the family of binary linear RM codes is given by the
sequence RM(r,m), where 0 ≤ r ≤ m. The code RM(r,m) is called the r-th
order binary linear Reed-Muller code of length n=2m and it is true that
RM(0,m)⊂RM(1,m)⊂...⊂RM(r−2,m)⊂RM(r−1,m)⊂RM(r,m).
Let 0≤r ≤m, m≥1. Following [12] the RM(r,m) code of order r can be
constructed by using the Plotkin construction in the following way:
RM(0,m) = {0,1}, RM(m,m) = Z2m,
2
RM(r,m) = {(u |u +u ):u ∈RM(r,m−1), u ∈RM(r−1,m−1)(}3.)
1 1 2 1 2
Itisimportanttonotethatifwefixm,onceweknowthesequenceRM(r,m)
for all 0≤r ≤m, then it is easy to obtain the new sequence RM(r,m+1) by
using the Plotkin construction (3).
Moreover,thecodesintheRM familyfulfillthebasicpropertiessummarized
in the following theorem:
Theorem 9 ([12]) ThebinarylinearReed-Mullerfamilyofcodes{RM(r,m)},
0≤r ≤m, has the following properties:
1. the length n=2m, m≥1;
2. the minimum distance d=2m−r;
r
m
3. the dimension k = ;
(cid:18)i(cid:19)
Xi=0
4. the code RM(r − 1,m) is a subcode of RM(r,m), r > 0. The code
RM(0,m) is the repetition code with only one nonzero codeword (the all
ones vector). The code RM(m,m) is the whole space Z2m and RM(m−
2
1,m) is the even code (that is, the code with all the vectors of even weight
from Z2m);
2
5. thecode RM(1,m)is thebinary linearHadamard code andRM(m−2,m)
is the extended binary Hamming code of parameters (2m,2m−m−1,4);
6. the code RM(r,m) is the dual code of RM(m−1−r,m) for 0≤r <m.
Intherecentliterature[7,20,2,3]severalfamiliesofquaternarylinearcodes
have been proposed and studied trying to generalize the RM codes. However,
when we take the corresponding Z -linear codes, they do not satisfy all the
4
properties in Theorem 9. This last requirement is the main goal of the present
work, to construct new families of quaternary linear codes such that, after the
Graymap,weobtainZ -linearcodeswiththeparametersandpropertiesquoted
4
in Theorem 9. The result of the present paper generalizes the results in [19].
Further we will refer to these quaternary linear Reed-Muller codes as RM
to distinguish them from the binary linear Reed-Muller codes RM. Contrary
to the binary linearcase, wherethere is only oneRM family, in the quaternary
case we have ⌊m+1⌋ families for each value of m. We will distinguish these
2
families by using subindexes s from the set {0,...,⌊m−1⌋}.
2
9
Table 1: RM(r,m) codes for m=1
(r,m)
(0,1) (1,1)
N (γ,δ)
1 (1,0) (0,1) RM (r,1)
0
3.1 The family of RM(r,1) codes
We begin by considering the trivial case of m = 1, that is, the case of codes
of binary length n = 21. The quaternary linear Reed-Muller code RM(0,1)
is the repetition code with only one nonzero codeword (the vector with only
one quaternary coordinate of value 2). This quaternary linear code is of type
(1;1,0). The code RM(1,1)is the whole space Z1, so a quaternarylinear code
4
of type (1;0,1).
These two codes, RM(0,1) and RM(1,1), after the Gray map, give binary
codeswiththesameparametersofthecorrespondingbinarycodesRM(r,1)and
withthesamepropertiesdescribedinTheorem9. Inthiscase,whenm=1,not
onlythese codeshavethe sameparameters,but they havethe same codewords.
We willrefertothesecodesasRM (0,1)andRM (1,1),respectively,asit
0 0
is shown in Table 1. In each entry of this table there are the parameters (γ,δ)
of the corresponding code of type (N;γ,δ).
Since we will need an specific representation for these codes in Table 1, we
will agree in using further the following matrices as the generator matrices for
each one of them. The generator matrix of RM (0,1) is G (0,1) = 2 and
0 0
the generator matrix of RM (1,1) is G (1,1)= 1 . (cid:0) (cid:1)
0 0
(cid:0) (cid:1)
3.2 Plotkin and BQ-Plotkin constructions
The first important point is to apply the Plotkin construction to quaternary
linear Reed-Muller codes.
LetRM (r,m−1)andRM (r−1,m−1),0≤s≤⌊m−2⌋,beanytwoRM
s s 2
codes of type (N;γs ,δs ) and (N;γs ,δs ); binary length
r,m−1 r,m−1 r−1,m−1 r−1,m−1
n=2m−1; number of codewords 2kr and 2kr−1; minimum distance 2m−r−1 and
2m−r respectively, where
r r−1
m−1 m−1
k = , k = .
r (cid:18) i (cid:19) r−1 (cid:18) i (cid:19)
Xi=0 Xi=0
Theorem 10 For any r and m≥2, 0<r <m, the code obtained by using the
Plotkin construction
RM (r,m)={(u |u +u ):u ∈RM (r,m−1), u ∈RM (r−1,m−1)},
s 1 1 2 1 s 2 s
10