ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.284947

J. Algebra Comb. Discrete Appl.
4(2) • 131–140

Received: 12 June 2015
Accepted: 17 February 2016

Journal of Algebra Combinatorics Discrete Structures and Applications

Codes over an infinite family of algebras

Research Article

Irwansyah∗, Intan Muchtadi-Alamsyah∗∗, Ahmad Muchlis, Aleams Barra∗∗, Djoko
Suprijanto

Abstract: In this paper, we will show some properties of codes over the ring Bk = Fp[v1, . . . , vk]/(v2i = vi, ∀i =
1, . . . , k). These rings, form a family of commutative algebras over finite field Fp. We first discuss
about the form of maximal ideals and characterization of automorphisms for the ring Bk. Then, we
define certain Gray map which can be used to give a connection between codes over Bk and codes over
Fp. Using the previous connection, we give a characterization for equivalence of codes over Bk and
Euclidean self-dual codes. Furthermore, we give generators for invariant ring of Euclidean self-dual
codes over Bk through MacWilliams relation of Hamming weight enumerator for such codes.

2010 MSC: 11T71

Keywords: Gray map, Equivalence of codes, Euclidean self-dual, Hamming weight enumerator, MacWilliams relation,
Invariant ring

1. Introduction

Codes over finite rings has been an interesting topic in algebraic coding theory since the discovery
of codes over Z4, see [4]. An example of finite rings which has interesting properties is the ring Ak =
F2[v1, . . . ,vk], where v2i = vi, for 1 ≤ i ≤ k, because it has two Gray maps which relate codes over such
ring and binary codes, see [2]. This ring also has non-trivial automorphisms which can be used to define
skew-cyclic codes, for example in [1], skew-cyclic codes over the ring A1 = F2 + vF2, where v2 = v,
which give some optimal Euclidean and Hermitian self-dual codes. Furthermore, Abualrub et al. show
that skew-cyclic codes over A1 have a connection to left submodules over a skew-polynomial ring and

∗ The author is supported by Beasiswa Unggulan BPKLN Direktorat Jenderal Pendidikan Tinggi.
∗∗ The authors are supported by Hibah Desentralisasi DIKTI 2016.
Irwansyah (Corresponding Author); Algebra Research Group, Institut Teknologi Bandung, Bandung, Indonesia,

and Department of Mathematics, Universitas Mataram, Mataram, Indonesia (email: irw@unram.ac.id).
Intan Muchtadi-Alamsyah, Ahmad Muchlis, Aleams Barra; Algebra Research Group, Institut Teknologi Bandung,

Bandung, Indonesia (email: ntan@math.itb.ac.id, muchlis@math.itb.ac.id, barra@math.itb.ac.id).
Djoko Suprijanto; Combinatorial Research Group, Institut Teknologi Bandung, Bandung, Indonesia (email:

djoko@math.itb.ac.id).

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Irwansyah et al. / J. Algebra Comb. Discrete Appl. 4(2) (2017) 131–140

give skew-polynomial generators for these codes. In [6], skew-cyclic codes over the ring A1 have been
characterized using a Gray map. This characterization gives a way to construct skew-cyclic codes over the
ring A1 from binary cyclic or quasi-cyclic codes, and also gives decoding algorithm for some codes over
such ring. Meanwhile, Gao [3] consider skew-cyclic codes over the ring B1 = Fp + vFp, where v2 = v, and
found that these codes are equivalent to either cyclic codes or quasi-cyclic codes. Using this connection,
Gao is able to give an enumeration for skew-cyclic codes which are constructed using an automorphism
with order relatively prime to the length of the codes.

In this paper, we consider codes over the ring Bk = Fp[v1, . . . ,vk], where v2i = vi for 1 ≤ i ≤ k,
which is a generalization of the ring Ak in [2] and B1 in [3]. We study its maximal ideals, automorphisms,
equivalence codes, and Euclidean self-dual codes over these rings, including the generators for its invariant
ring. This paper is organized as follows: Section 2 describes some properties of the ring Bk such as
maximal ideals and automorphisms. Meanwhile, in Section 3, we describe a Gray map for the ring Bk, and
we characterize linear codes and equivalent codes over the ring Bk. Finally, in Section 4, we characterize
Euclidean self-dual codes, give the shape of MacWilliams relation and generators of invariant rings for
Euclidean self-dual codes.

2. The ring Bk

As we readily see, the ring Bk forms a commutative algebra over prime field Fp. Let Ω = {1, 2, . . . ,k}
and 2Ω is the collection of all subsets of Ω. Also, let wi be an element in the set {vi, 1−vi}, for 1 ≤ i ≤ k.
Then, we will prove the following observation.

Lemma 2.1. ω ∈ Bk is a zero divisor if and only if ω ∈ 〈w1,w2, . . . ,wk〉.

Proof. (⇐=) It is clear that, vi(1−vi) = 0, for all i = 1, . . . ,k. Therefore, if ω ∈ 〈w1,w2, . . . ,wk〉, then
it is a zero divisor in Bk.

(=⇒) Consider the equation,

(α + βvk)(γ + �vk) = a + bvk

given α + βvk,a + bvk ∈ Bk, for some α,β,a,b ∈ Bk−1. We have γ = aα−1 and � = (b−βa)(α(β + α))−1.
Therefore, if a + bvk = 1, then γ = 1 and � = −β(α(β + α))−1. Which implies, α + βvk is a unit if and
only if α and α + β are also units. Considering this observation for elements in Bk−1,Bk−2, . . . ,B1, we
have α + βv ∈ B1 is a unit if and only if α,α + β ∈ Fp are non zero elements. Since, every element in
finite commutative ring is either a unit or a zero divisor, we can see that the only zero divisors in B1 are
the elements in the ideals generated by βv or α(1 − v). By generalizing this result recursively, we have
the intended conclusion.

Also, we can easily show that I = 〈w1,w2, . . . ,wk〉 is a maximal ideal in Bk.

Lemma 2.2. Let I = 〈w1,w2, . . . ,wk〉. Then I is a maximal ideal in Bk.

Proof. Consider quotient ring Bk/I. If vi ∈ I, then 1 − vi ≡ 1 mod I, and if 1 − vi ∈ I, then
vi = 1 − (1 − vi) ≡ 1 mod I. Consequently, Bk/I is a field. So, I is a maximal ideal. Moreover,
Bk/I ∼= Fp.

The following lemma is needed to prove Proposition 2.4.

Lemma 2.3. αp = α, for all α ∈ Bk.

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Irwansyah et al. / J. Algebra Comb. Discrete Appl. 4(2) (2017) 131–140

Proof. Let α =
∑
A⊆{1,...,k}αAvA, for some αA ∈ Fp, where vA =

∏
j∈A vj. Then, consider

αp =

p∑
i=0

(
p

i

)
αiA1vA1


 ∑
A6=A1

αAvA


p−i = αA1vA1 +


 ∑
A 6=A1

αAvA


p

since Fp has characteristic p and βp−1 = 1 for all β ∈ Fp. If we continue this procedure, then we have
αp = α.

The following result shows that the ring Bk is a principal ideal ring.

Proposition 2.4. Let I = 〈α1, . . . ,αm〉 be an ideal in Bk, for some α1, . . . ,αm ∈ Bk. Then,

I = 〈
∑

A⊆{1,...,m},A 6=∅

(−1)|A|+1(
∏
j∈A

αj)
p−1〉.

Proof. Consider αi
∑
A⊆{1,...,m},A 6=∅(−1)

|A|+1(
∏
j∈A αj)

p−1. For any A ⊆{1, . . . ,m}, if i ∈ A, then

αi(−1)|A|+1(
∏
j∈A

αj)
p−1 = (−1)|A|+1αi(

∏
j∈A−{i}

αj)
p−1

since αpi = αi by Lemma 2.3. Consequently, there is a unique A
′ = A−{i}⊆{1, . . . ,m}, such that

αi


(−1)|A|+1(∏

j∈A
αj)

p−1 + (−1)|A
′|+1(

∏
j∈A

αj)
p−1


 = 0.

Otherwise, if i 6∈ A, then there is a unique A′′ = A∪{i}⊆{1, . . . ,m} such that

αi


(−1)|A|+1(∏

j∈A
αj)

p−1 + (−1)|A
′′|+1(

∏
j∈A

αj)
p−1


 = 0.

So, every term will be vanish except αiα
p−1
i = αi. Therefore,

I ⊆〈
∑

A⊆{1,...,m},A6=∅

(−1)|A|+1(
∏
j∈A

αj)
p−1〉.

It is clear that

〈
∑

A⊆{1,...,m},A 6=∅

(−1)|A|+1(
∏
j∈A

αj)
p−1〉⊆ I.

Thus, I = 〈
∑
A⊆{1,...,m},A6=∅(−1)

|A|+1(
∏
j∈A αj)

p−1〉.

The following proposition shows that the ideal in Lemma 2.2 is the only maximal ideal in Bk.

Proposition 2.5. An ideal I in Bk is maximal if and only if I = 〈w1,w2, . . . ,wk〉.

Proof. (⇐=) It is clear by Lemma 2.2.

(=⇒) Let J be a maximal ideal in Bk. By Proposition 2.4, Bk is a principal ideal ring. Then, let J = 〈ω〉,
for some ω ∈ Bk. Note that, ω is not a unit in Bk, so it is a zero divisor. By Lemma 2.1, ω is an element
of some mi = 〈w1,w2, . . . ,wk〉, which means J ⊆ mi. Consequently, J = mi, because J is a maximal
ideal.

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Using the above result, we have the following lemmas.

Lemma 2.6. The ring Bk can be viewed as an Fp-vector space with dimension 2k whose basis consists
of elements of the form wS =

∏
i∈S wi, where S ∈ 2

Ω.

Proof. As we can see, every element a ∈ Bk can be written as a =
∑
S∈2Ω αSvS, for some αS ∈ Fp,

where vS =
∏
i∈S vi and v∅ = 1. So, Bk is a vector space over Fp whose basis consists of elements of the

form vS =
∏
i∈S vi, where v∅ = 1 and there are

∑k
j=0

(
k
j

)
= 2k elements of basis. Now, we will show that

the set {1,wS2, . . . ,wS2k} is also a basis. Consider,

α1 + α2wS2 + · · · + α2kwS2k = 0

for some αi ∈ Fp, for all i = 1, . . . , 2k, which gives,

−α1 = α2wS2 + · · · + α2kwS2k .

If α1 6= 0, then ξ1 =
(
α2wS2 + · · · + α2kwS2k

)
is a unit, a contradiction to the fact that ξ1 ∈ 〈w1, . . . ,wk〉.

So, α1 = 0, which means,

−
(
α2wS2 + · · · + αk+1wSk+1

)
= αk+2wSk+2 + · · · + α2kwS2k .

If
(
α2wS2 + · · · + αk+1wSk+1

)
6= 0, then it is a contradiction to the fact that |Sj| ≥ 2, for all j =

k + 2, . . . , 2k. Consequently,
(
α2wS2 + · · · + αk+1wSk+1

)
= 0. We have to note that, the set with elements

of the wS, where S ∈ 2Ω, is also linearly independent over Fp, because Sk is a vector space over Fp
with element of basis are of the form vS, where S ⊆ Ω. Therefore,

(
α2wS2 + · · · + αk+1wSk+1

)
= 0 gives

α2 = · · · = αk+1 = 0. By continuing this process, we have α1 = · · · = α2k = 0, which means they are
linearly independent over Fp.

Lemma 2.7. The ring Bk has characteristic p and cardinality p2
k

.

Proof. It is immediate since characteristic of Fp is p, and Bk can be viewed as a Fp-vector space with
dimension

∑k
i=0

(
k
i

)
= 2k. So, |Bk| = p2

k

.

The following theorem characterizes the shape of automorphisms in the ring Bk.

Theorem 2.8. Let θ be an endomorphism in Bk. Then, θ is an automorphism if and only if θ(vi) = wj,
for every i ∈ Ω, and θ, when restricted to Fp, is an identity map.

Proof. (=⇒) Let J = 〈v1, . . . ,vk〉 and Jθ = 〈θ(v1), . . . ,θ(vk)〉. Consider the map

λ :
Bk
J

→
Bk
Jθ

a + J 7→ θ(a) + Jθ

We can see that the map λ is a ring homomorphism. For any a,b ∈ Bk/J where λ(a) = λ(b), let
a = a1 + J and b = b1 + J for some a1,b1 ∈ Bk. As we can see, θ(a1 − b1) ∈ Jθ, so a1 − b1 ∈ J.
Consequently, a − b = 0 + J, which means a = b, in other words, λ is a monomorphism. Moreover,
for any a′ ∈ Bk/Jθ, let a′ = a2 + Jθ for some a2 ∈ Bk, then there exists a = θ−1(a2) + J such that
λ(a) = a′. Therefore, Fp ' Bk/J ' Bk/Jθ, which implies Jθ is also a maximal ideal. By Proposition 2.5,
Jθ = 〈w1, . . . ,wk〉, where wi ∈{vi, 1 −vi} for 1 ≤ i ≤ k. By Proposition 2.4,

Jθ = 〈
∑

A⊆Ω,A 6=∅

(−1)|A|+1(
∏
j∈A

wj)
p−1〉 = 〈

∑
A⊆Ω,A 6=∅

(−1)|A|+1(
∏
j∈A

θ(vj))
p−1〉

which means,
∑
A⊆Ω,A 6=∅(−1)

|A|+1(
∏
j∈A wj)

p−1 and
∑
A⊆Ω,A6=∅(−1)

|A|+1(
∏
j∈A θ(vj))

p−1 are associate.

Therefore, θ(vi) = βwj for some unit β which satisfies
(
β|A|

)p−1
= β, for all A 6= ∅. Consequently, we

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Irwansyah et al. / J. Algebra Comb. Discrete Appl. 4(2) (2017) 131–140

have βp−1 = β, but by Lemma 2.3, βp = β. Since β is a unit, we have that βp−1 = 1. Therefore, β must
be equal to 1. Moreover, since θ is an automorphism, θ(vi) 6= θ(vj) whenever i 6= j. Also, since the only
automorphism in Fp is identity map, we have the conclusion.

(⇐=) Suppose that θ(vi) = wj, and θ(vi) 6= θ(vj) whenever i 6= j. By Lemma 2.6, we can see that θ is
also an automorphism.

Now, we have to note that every element a in Bk can be written as

a =
∑
S∈2Ω

αSwS

for some αS ∈ Fp, where wS =
∏
i∈S wi. Define a map ϕ as follows.

ϕ : Bk → F2
k

p

a =
∑2k
i=1 αSiwSi 7→ (

∑
S⊆S1 αS,

∑
S⊆S2 αS, . . . ,

∑
S⊆S

2k
αS)

We can show that this map ϕ is a bijection map. Furthermore, this map can be extended n tuples of Bk
as follows.

ϕ : Bnk → F
n2k

p

(a1, . . . ,an) 7→ (ϕ(a1), . . . ,ϕ(an)).

Since ϕ is a bijection map, we also have ϕ is a bijection map. We have to note that, the map ϕ is a
permutation, based on the choice of subsets Si ∈ 2Ω, of Gray maps in [2].

3. Codes over the ring Bk

A subset C ⊆ Bnk is called code over Bk of length n. If C is a Bk-submodule of B
n
k , then C called

linear code. The following proposition gives a characterization of Bk-linear codes using the map ϕ.

Proposition 3.1. C is a linear code over Bk if and only if there exist linear codes C1, . . . ,C2k over Fp
such that C = ϕ−1(C1, . . . ,C2k ).

Proof. (=⇒) Since ϕ is a bijection, there exist C1, . . . ,C2k such that C = ϕ−1(C1, . . . ,C2k ). Now, we
only need to show that Ci is a linear code over Fp for all i = 1, . . . , 2k. For any Ci, let c1 and c2 be two
codewords in Ci. For l = 1, 2, let cl = (α

(l)
1 , . . . ,α

(l)
n ), for some α

(l)
j in Fp. Consider

c′l = ϕ
−1(0, . . . ,0,λlcl,0,0)

=
(
ϕ−1(0, . . . , 0,λlα

(l)
1 , 0, . . . , 0), . . . ,ϕ

−1(0, . . . , 0,λlα
(l)
n , 0, . . . , 0)

)
=
(
λlα

(l)
1

(
wSl −

∑
j∈{1,...,k}−Sl wSl∪{j}

)
, . . . ,λlα

(l)
n

(
wSl −

∑
j∈{1,...,k}−Sl wSl∪{j}

))
,

for any λl in F×p for all l = 1, 2. The last equality holds since

ϕ


α(l)t


wSl − ∑

j∈{1,...,k}−Sl

wSl∪{j}




 = (0, . . . , 0,α(l)t , 0, . . . , 0)

for all 1 ≤ t ≤ n. Since C = ϕ−1(C1, . . . ,C2k ), we have c′l is in C for all l = 1, 2, and c
′
1 + c

′
2 is also in C.

Then, consider

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Irwansyah et al. / J. Algebra Comb. Discrete Appl. 4(2) (2017) 131–140

ϕ(c′1 + c
′
2) =




0 · · · 0 · · · 0
... · · ·

... · · ·
...

0 · · · 0 · · · 0
λ1α

(1)
1 + λ2α

(2)
1 · · · λ1α

(1)
l + λ2α

(2)
l · · · λ1α

(1)
n + λ2α

(2)
n

0 · · · 0 · · · 0
... · · ·

... · · ·
...

0 · · · 0 · · · 0



.

Hence, λ1c1 + λ2c2 is also in Ci.

(⇐=) Take any two codewords c3 and c4 in C. Let

c3 =

( ∑
S∈2Ω

α
(1)
S wS, . . . ,

∑
S∈2Ω

α
(n)
S wS

)

and

c4 =

( ∑
S∈2Ω

β
(1)
S wS, . . . ,

∑
S∈2Ω

β
(n)
S wS

)
,

for some αi,βi in Fp, where i = 1, . . . , 2k. For any λ3 and λ4 in F×p we have

ϕ(λ3c3 + λ4c4) =




λ3α
(1)
S1

+ λ4β
(1)
S1

· · · λ3α
(n)
S1

+ λ4β
(n)
S1

λ3
∑
S⊆S2 α

(1)
S + λ4

∑
S⊆S2 β

(1)
S · · · λ3

∑
S⊆S2 α

(n)
S + λ4

∑
S⊆S2 β

(n)
S

...
...

...
λ3
∑
S⊆S

2k
α

(1)
S + λ4

∑
S⊆S

2k
β

(1)
S · · · λ3

∑
S⊆S

2k
α

(n)
S + λ4

∑
S⊆S

2k
β

(n)
S




is also in (C1, . . . ,C2k ) , since Ci is a linear code for every i = 1, . . . , 2k. Therefore, λ3c3 + λ4c4 is also in
C.

Now, following [5], we define permutation equivalence of codes as follows.

Definition 3.2. Two codes are permutation equivalent if one can be obtained from the other by permuting
the coordinates.

Using Definition 3.2, we can define the following notion of equivalence between two codes.

Definition 3.3. Two codes C and C′ over Bk are equivalent if either they are permutation-equivalent
or C is permutation equivalent to the code θ(C′) for some automorphism θ in Bk, i.e. the code θ(C′)
obtained from C′ by changing α with θ(α) in all coordinates.

Note that, the above definition is similar to the one in [5]. Now, let Πθ be a permutation on 2k
tuples of Fp induced by automorphism θ. Then we have

(Πθ ◦ϕ) (c) = ϕ(θ(c)) (1)

for any c ∈ Bnk . Then, we have the following characterization.

Theorem 3.4. Let C and C′ be two codes over Bk. Then, C and C′ are equivalent if and only if there
exists a permutation which sends (C1, . . . ,C2k ) to (C

′
1, . . . ,C

′
2k

) or to (Πθ(C′1), . . . , Πθ(C
′
2k

)).

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Irwansyah et al. / J. Algebra Comb. Discrete Appl. 4(2) (2017) 131–140

Proof. (=⇒) Let C = ϕ−1(C1, . . . ,C2k ) and C′ = ϕ−1(C′1, . . . ,C′2k ), where Ci and C
′
i are codes over

Fp, for all 1 ≤ i ≤ 2k. If there exists an automorphism θ such that C is permutation equivalent to θ(C′),
then by equation 1, we have C = ϕ−1(C1, . . . ,C2k ) is permutation equivalent to

(
Πθ(C

′
1), . . . , Πθ(C

′
2k

)
)
.

(⇐=) If there exists a permutation which sends (C1, . . . ,C2k ) to

(Πθ(C
′
1), . . . , Πθ(C

′
2k )) ,

for some bijective map Πθ, then we can have the automorphism θ using the equation 1.

4. Invariant ring

In this section, we describe some aspect of Euclidean self-dual codes as well as MacWiliams identity
and invariant ring.

Related to Euclidean self-dual codes over the ring Bk, we have the following result.

Proposition 4.1. Let C = ϕ−1(C1,C2, . . . ,C2k ), for some p-ary codes C1, . . . ,C2k. Then, C is Eulidean
self-dual codes over Bk if and only if Ci is also Euclidean self-dual codes, for 1 ≤ i ≤ 2k.

Proof. (=⇒) For any ci ∈ Ci, let ci = (α
(0)
Si
, . . . ,α

(n−1)
Si

), for some α(j)Si ∈ Fp, where 0 ≤ j ≤ n− 1. Let
c = ϕ−1(0, . . . , 0,ci, 0, . . . , 0) ∈ C, then we have 〈c,c′〉 = 0 for every c′ ∈ C. To make the representation
for any element in the ring Bk easier, we will use the basis whose elements are of the form vS, for all
S ⊆{1, 2, . . . ,k}. Now, let

c′ =


β(0)Si vSi + ∑

S∈2Ω,S 6=Si

β
(0)
S vS, . . . ,β

(n−1)
Si

vSi +
∑

S∈2Ω,S 6=Si

β
(n−1)
S vS


 .

Consider,

c = ϕ−1(0, . . . , 0,ci, 0, . . . , 0)

=
(
α

(0)
Si

(vSi −
∑
j∈{1,...,k}−Si vSi∪{j}), . . . ,α

(n−1)
Si

(vSi −
∑
j∈{1,...,k}−Si vSi∪{j})

)
.

Since 〈c,c′〉 = 0 for every c′ ∈ C and v2S = vS for every S ∈ 2
Ω, we have

n−1∑
j=0


α(j)Si β(j)Si vSi − ∑

j∈{1,...,k}−Si

α
(j)
Si
β

(j)
Si∪{j}

vSi∪{j} = 0


 .

Consequently,
∑n−1
j=0 α

(j)
Si
β

(j)
Si

= 0.

Take any c′i ∈ Ci. Let c
′
i = (γ

(0)
Si
, . . . ,γ

(n−1)
Si

), for some γ(j)Si ∈ Fp, where 0 ≤ j ≤ n − 1. Since
c′ = ϕ−1(0, . . . , 0,ci, 0, . . . , 0) ∈ C, we have 〈c,c′〉 = 0. So

〈ci,c′i〉 =
n−1∑
j=0

α
(j)
Si
γ

(j)
Si

= 0.

Therefore Ci ⊆ C⊥i .
For any c1 ∈ C⊥i , let c1 = (ζ0, . . . ,ζn−1) for some ζj ∈ Fp, where 0 ≤ j ≤ n− 1. Since 〈c1,ci〉 = 0,

we have
∑n−1
j=0 ζjα

(j)
Si

= 0. We can see that

c′1 = ϕ
−1(0, . . . , 0,c1, 0, . . . , 0)

=
(
ζ0(vSi −

∑
j∈{1,...,k}−Si vSi∪{j}), . . . ,ζn−1(vSi −

∑
j∈{1,...,k}−Si vSi∪{j})

)
.

137



Irwansyah et al. / J. Algebra Comb. Discrete Appl. 4(2) (2017) 131–140

Now, since
∑n−1
j=0 ζjα

(j)
Si

= 0, we also have 〈c′1,c2〉 = 0 for every c2 ∈ C. Remember that C = C⊥, which
gives c′1 ∈ C. So, c1 ∈ Ci, or in other words C⊥i ⊆ Ci. Thus, Ci is a Euclidean self-dual code, for all
i = 1, . . . , 2k.

(⇐=) Take any c1,c2 ∈ C. For every i = 1, 2, let

ci =


 ∑
S⊆{1,...,k}

c
(i,0)
S , . . . ,

∑
S⊆{1,...,k}

c
(i,n−1)
S


 ,

for some c(i,j)S ∈ Fp, where i = 1, 2, and j = 0, . . . ,n− 1. Consider,

ϕ(ci) =
(∑

S⊆S1 c
(i,0)
S , . . . ,

∑
S⊆S1 c

(i,n−1)
S , . . .

. . . ,
∑
S⊆Sl c

(i,0)
S , . . . ,

∑
S⊆Sl c

(i,n−1)
S , . . .

. . . ,
∑
S⊆S

2k
c
(i,0)
S , . . . ,

∑
S⊆S

2k
c
(i,n−1)
S

)
,

where i = 1, 2. Since Cl is a Euclidean self-dual code, for all l = 1, . . . , 2k, we have

〈c1,c2〉 =
∑n−1
j=0

∑
Sl∈2Ω

∑
S⊆Sl c

(1,j)
S c

(2,j)
S vS

= 0.

So, C ⊆ C⊥.
Now, take any c3 ∈ C⊥. Since 〈c3,c〉 = 0 for all c ∈ C, we have

n−1∑
j=0

∑
S⊆Sl

c
(1,j)
S c

(2,j)
S vS = 0,

for all S ∈ 2Ω. Remember that Cl is a Euclidean self-dual code, for all l = 1, 2, . . . , 2k, which give
n−1∑
j=0

∑
S⊆Sl

c
(1,j)
S c

(2,j)
S vS = 0,

for all S ∈ 2Ω, and moreover c3 ∈ C. So, C⊥ ⊆ C. Therefore, C is a Euclidean self-dual code.

The following lemma gives MacWilliams identity for codes over the ring Bk.

Lemma 4.2. The MacWilliams identity for Hamming weight enumerators for codes over Bk is :

WC⊥(X,Y ) =
1

|C|
WC(X + (p

2k − 1)Y,X −Y ) (2)

Proof. The identity follows from [7, Theorem 8.3] and Proposition 4.1.

As we can see from Lemma 4.2, MacWilliams identity gives a transformation between polynomial
representing a code and polynomial representing its corresponding dual code. We have to note that if C
is an Euclidean self-dual code, then the weight enumerator of C is invariant under this transformation.
The above transformation can be formulated as an action ’◦’ by a matrix group G generated by matrices

T =


 1p2k−1 p2k−1p2k−1

1

p2
k−1

−1
p2

k−1


 and D = ( −1 0

0 −1

)
. The action of any g =

(
a1 a2
a3 a4

)
∈ G to a polynomial

f(X,Y ) is written as

g ◦f(X,Y ) = f(a1X + a2Y,a3X + a4Y ).

138



Irwansyah et al. / J. Algebra Comb. Discrete Appl. 4(2) (2017) 131–140

Note that the matrix T is derived from the identity in Lemma 4.2 and the matrix D is derived from the
condition that n is always even. Also, it is easy to see that G = {I,D,T,−T}. Formally, we have the
following result.

Lemma 4.3. If WC(X,Y ) is a Hamming weight enumerator for an Euclidean self-dual code C over Bk,
then WC(X,Y ) is invariant under the action of G.

Let RG be a set of all polynomials in two variables which are invariant under the action ◦ of G.
We can easily prove that RG is a ring, and by the above Lemma we can see that every Hamming
weight enumerator of Euclidean self-dual codes must be inside RG. This ring RG called invariant ring
for Euclidean self-dual codes over Bk. The following theorem gives generators for RG.

Theorem 4.4. Invariant ring of G is generated by

WC0 (x,y) = x
2 + (p2

k

− 1)y2

and

f̃(x,y) =
1

4


2p2k−1 + 2

p2
k

x2 +
4
(
p2

k

− 1
)

p2
k−1 xy +

2(p2
k

− 1)2

p2
k−1 y

2


 .

Proof. Consider the Molien series,

Φ(λ) = 1|G|
∑
A∈G

1
det(I−λA)

= 1
4

(
1

(1+λ)2
+ 1

(1−λ)2 +
2

(1−λ2)

)
= 1

(1−λ2)2

= 1 + 2λ2 + 3λ4 + 4λ6 + 5λ8 + · · · + nλ2(n−1) + . . .

we can see that, the invariant ring generated by two invariants of degree 2. Consider the weight enumerator
for self-dual code

C0 = {cc|∀c ∈ Ak}

i.e. WC0 (x,y) = x
2 + (p2

k

− 1)y2. This weight enumerator is of degree 2 and invariant under the action
of G. So, this weight enumerator is one of the generator. We use averaging method to find the other one.
Let f(x) = x2, then by averaging method, we have

f̃(x,y) =
1

4


2p2k−1 + 2

p2
k

x2 +
4
(
p2

k

− 1
)

p2
k−1 xy +

2(p2
k

− 1)2

p2
k−1 y

2




f̃(x,y) are algebraically independent.

References

[1] T. Abualrub, N. Aydin, P. Seneviratne, On θ−cyclic codes over F2 + vF2, Australas. J. Combin. 54
(2012) 115–126.

[2] Y. Cengellenmis, A. Dertli, S. T. Dougherty, Codes over an infinite family of rings with a Gray map,
Des. Codes Cryptogr. 72(3) (2014) 559–580.

[3] J. Gao, Skew cyclic codes over Fp + vFp, J. Appl. Math. Inform. 31(3–4) (2013) 337–342.

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http://dx.doi.org/10.1109/18.312154
http://dx.doi.org/10.1109/18.312154
http://dx.doi.org/10.2991/iccst-15.2015.27
http://dx.doi.org/10.2991/iccst-15.2015.27
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http://dx.doi.org/10.1353/ajm.1999.0024
http://dx.doi.org/10.1353/ajm.1999.0024

	Introduction
	The ring Bk
	Codes over the ring Bk
	Invariant ring
	References