ADDITIVE CYCLIC CODES
OVER FINITE COMMUTATIVE CHAIN RINGS
arXiv:1701.06672v1 [cs.IT] 23 Jan 2017
EDGAR MARTINEZ-MORO, KAMİL OTAL AND FERRUH ÖZBUDAK
Abstract. Additive cyclic codes over Galois rings were investigated in [3]. In this paper, we
investigate the same problem but over a more general ring family, finite commutative chain
rings. When we focus on non-Galois finite commutative chain rings, we observe two different
kinds of additivity. One of them is a natural generalization of the study in [3], whereas the
other one has some unusual properties especially while constructing dual codes. We interpret
the reasons of such properties and illustrate our results giving concrete examples.
Keywords: Cyclic codes, additive codes, codes over rings, finite commutative chain
rings, Galois rings.
AMS classification: 11T71, 94B99, 81P70, 13M10
1. Introduction
Additive codes are a direct and useful generalization of linear codes, and they have applications in quantum error correcting codes. There are several studies using different approaches
on them and their applications (see, for example, [1, 4, 12, 20]).
Cyclic codes are one of the most attractive code families thank to their rich algebraic
structure and easy implementation properties. There are many generalizations of cyclic codes
in different directions, e.g., [7, 13, 14, 15].
Codes over rings have been of interest in the last quarter after the discovery that some
linear codes over Z4 are related to non-linear codes over finite fields (see, for example, [6,
5, 11, 17, 18]). The first family of the rings used in this perspective was Zpn , where p is a
prime and n is a positive integer. The most important property of such rings is the linearity
of their ideals under inclusion. Therefore, the generalizations on Galois rings, or moreover
finite chain rings are immediate. Some recent works on codes over such rings are [3, 7, 9, 22].
Finite chain rings, besides their practical importance, are quite rich mathematical objects
and so they have also theoretical attraction. They have connections in both geometry (Pappian Hjelmslev planes) and algebraic number theory (quotient rings of algebraic integers).
These connections have also been interpreted in applications (as an example of application
in coding theory, see [9]). Some main sources about finite commutative chain rings in the
literature are [2, 16, 19, 23].
Edgar Martı́nez-Moro is with the Mathematics Research Institute, University of Valladolid, Castilla, Spain.
Partially supported by MINECO MTM2015-65764-C3-1-P project and MTM2015-69138-REDT Network.
Kamil Otal and Ferruh Özbudak are with Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Dumlupınar Bulvarı No. 1, 06800, Ankara, Turkey; e-mail:
{kotal,ozbudak}@metu.edu.tr.
1
2
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
1.1. Related work and our contribution. Additive cyclic codes over Galois rings were
investigated in [3]. In this paper, we investigate the same problem but over a more general
ring family, finite commutative chain rings. When we focus on non-Galois finite commutative
chain rings, we observe two different kinds of additivity.
The first one, so-called Galois-additivity, is a natural generalization of the study in [3],
anyway our way of construction in this generalization is slightly different from the one in that
paper. The authors in [3] were using some linear codes over the base ring and their generator
matrices, but we just make use of ideals and do not get involved generator matrices. Our
main result with this approach is Theorem 4.9. Also we have some further results related
to the code size relations (Corollary 4.11) and self-duality (Corollary 4.12) as well as we
illustrate these results in a concrete example (Example 4.13).
The second one, so-called Eisenstein-additivity, is set up in Theorem 5.2. Eisensteinadditivity has some unusual properties especially while constructing dual codes. It is because,
some chain rings are not free over their coefficient rings and hence we can not define a trace
function for some elements over the base ring (Lemma 5.1). Hence one can not make use of
the equivalence of Euclidean orthogonality and duality via the trace map (see [21, Lemma
6]). Thus we use the general idea of duality, constructed via annihilators of characters
(see [24]). We have adapted this character theoretic duality notion to Eisenstein-additive
codes (see Section 5.1) and hence we observe one-to-oneness between a code and its dual
(a MacWilliams identity) as expected. We again provide a concrete example for our result
regarding the character theoretic duality (see Example 5.5).
Notice that the idea in [3] has been generalized recently also in [22]. However, the generalization in [22] is from Galois rings to free R-algebras, where R is a finite commutative chain
ring. Recall that we consider also non-free algebras (a finite commutative chain ring does not
have to be a free module over a subring of it). On the other hand, our paper does not cover
[22] since every free R-algebra does not have to be a chain ring.
1.2. Organization of the paper. In Section 2, we introduce finite commutative chain
rings constructing them step by step, as from Zpn to Galois rings and then to arbitrary finite
commutative chain rings. Section 3 provides basic definitions, notations and facts of a code
concept over rings, by focusing mainly on cyclic codes and additive codes.
In Section 4, firstly we give some lemmas and then construct the main theorem (Theorem
4.9) of Galois-additivity using them. We also give some corollaries regarding the code size
relations between dual codes (Corollary 4.11), and characterization of self-duality (Corollary
4.12). We have also provided an illustration (Example 4.13) about our results.
In Section 5, we directly give the main result (Theorem 5.2) about Eisenstein-additive
codes since it comes from the lemmas in Section 4. A direct character theoretic duality is
constructed in Section 5.1 separately. Also an example (Example 5.5) is available to illustrate
our results.
ADDITIVE CYCLIC CODES OVER FINITE COMMUTATIVE CHAIN RINGS
3
2. Finite Commutative Chain Rings
A ring R is called local if it has only one maximal ideal. A local ring R is called a chain
ring if its ideals form a chain under inclusion. Saying “finite” we will refer to having finitely
many elements (not being finitely generated). A finite chain ring is a principal ideal ring,
and its maximal ideal is its nil-radical (i.e. the set of nil-potent elements). Hence, the chain
of ideals of a finite chain ring R is of the form
R ) N ilRad(R) =< x >) ... )< xm−1 >)< xm >=< 0 >
for some idempotent x ∈ R and a positive integer m (see, for example [19, Theorem 3.2]).
The simplest example of finite chain rings is Zpn with the maximal ideal < p >, where p is a
prime and n is a positive integer. We may construct all finite chain rings using Zpn .
2.1. Galois Rings. Consider the ring homomorphism Zpn → Fp given by a 7→ a, where a is
the remainder of a modulo p and Fq denotes the finite field of q elements. We may extend
this homomorphism in a natural way from the polynomial ring Zpn [X] to the polynomial ring
P
P
Fp [X] by ( i ai X i ) = i ai X i . A polynomial over Zpn is called basic irreducible (primitive)
if its image under this homomorphism is irreducible (primitive) over Fp .
Let f (X) ∈ Zpn [X] be a basic irreducible polynomial of degree r. The quotient ring
Zpn [X]/ < f (X) > is a finite chain ring with the maximal ideal < p >. This kind of rings
are known as Galois rings and denoted by GR(pn , r). In a Galois ring GR(pn , r), pn is
called characteristic and r is called rank. Galois rings are unique up to isomorphism for a
given characteristic and rank. We will use the notation Zpn [ω] to denote Galois rings, taking
ω = X+ < f (X) >. Note that f (X) is the unique monic polynomial of degree less than or
equal to r such that f (ω) = 0. We may extend the “overline homomorphism” defined above
as from Zpn [ω] to Fpr such that ω satisfies f (ω) = 0 and so Fpr = Fp [ω]. Therefore, in a
Galois ring GR(pn , r), there exists an element of multiplicative order pr − 1, which is a root
r
of a basic primitive polynomial of degree r over Zpn and dividing X p −1 − 1 in Zpn [X]. If ω
is a this kind of “basic primitive element”, then the set
T = {0, 1, ω, ω 2 , ..., ω p
r −2
}
is called a Teichmuller set of the Galois ring GR(pn , r). Any element a ∈ GR(pn , r) can be
written uniquely as
a = a0 + a1 p + ... + an−1 pn−1 ,
(2.1)
where a0 , ..., an−1 ∈ T . Here, a is unit if and only if a0 6= 0. Conversely, if a is not unit, then
it is either zero or a zero divisor.
When we compare two Galois rings having the same characteristic, we observe that,
GR(pn , r ′ ) is an extension of GR(pn , r) if and only if r divides r ′ . Let r ′ = rs where s is a positive integer, the extension GR(pn , rs) of GR(pn , r) can be constructed by a basic primitive
polynomial g(X) ∈ GR(pn , r)[X] = Zpn [ω][X] of degree s with g(X) ∈ Fpr [X] = Fp (ω)[X].
Taking ζ = X+ < g(X) > in GR(pn , r)[X]/ < g(X) >, we can see that ζ has multiplicative
prs −1
order prs − 1, ω = ζ pr −1 and Fprs = Fp (ζ). For the proofs of these facts and more detailed
information, see, for example [23].
4
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
2.2. Finite Commutative Chain Rings. We have constructed all Galois rings from Zpn ,
and now we construct all finite commutative chain rings from Galois rings. The following
theorem gives a complete characterization of finite commutative chain rings. For the proof
and more detail, see, for example [16, Theorem XVII.5].
Theorem 2.1. Consider the ring
(2.2)
R=
GR(pn , r)[x] = Zpn [ω][x]
,
< g(x) = xk + p(ak−1 xk−1 + ... + a0 ), pn−1 xt >
where g(x) is an Eisenstein polynomial (i.e., a0 is a unit element), 1 ≤ t ≤ k when n ≥ 2,
and t = k when n = 1. R is a finite commutative chain ring, and conversely, any finite
commutative chain ring is of the form (2.2).
Notation 2.2. Since now, we fix p, n, r, k, t, g and x, ω as the corresponding parameters and
elements respectively of a finite commutative chain ring given in Theorem 2.1.
The following proposition gives some elementary facts about finite commutative chain
rings. For the proofs and more detail, see, for example [16, 19].
Proposition 2.3. Let R denote the finite commutative chain ring with the parameters in
Notation 2.2.
• The chain of ideals of R is of the form
R )< x >) ... )< xk(n−1)+t−1 >)< xk(n−1)+t >= 0.
Moreover, < xk >=< p >.
• |R| = pr(k(n−1)+t) and | < x > | = pr(k(n−1)+t−1) . Also, R/ < x >∼
= Fpr .
n
• The largest Galois ring in R is GR(p , r) and it is called the coefficient ring of R.
• The Teichmuller set T of the coefficient ring S of a finite commutative chain ring R
is also considered as the Teichmuller set of R. Using T and the generator x of the
maximal ideal, each element of R can be written uniquely as
a0 + a1 x + ... + ak(n−1)+t−1 xk(n−1)+t−1 ,
for some a0 , ..., ar−1 ∈ T . Let T = {0, 1, ω, ..., ω p
of elements in R using ω can be given as
r −2
}, another unique representation
a0 + a1 ω + ... + ar−1 ω r−1 ,
for some a0 , ..., ar−1 ∈ Zpn [x]/ < g(x), pn−1 xt >. Accordingly, we can give also the
unique representation
r−1 X
k−1
X
ai,j ω i xj ,
i=0 j=0
where ai,j ∈ Zpn for 0 ≤ i ≤ r − 1 and 0 ≤ j ≤ k − 1, but 0 ≤ ai,j < pn−1 when
t ≤ j ≤ k − 1.
Now, we give an important idea we use in this paper. For the proof, see, for example [2,
Theorem 4.3.1] or [7, Lemma 3.1].
ADDITIVE CYCLIC CODES OVER FINITE COMMUTATIVE CHAIN RINGS
5
Theorem 2.4. Let S = GR(pn , r) and Se = GR(pn , rs) = S[ζ] for some basic primitive
element ζ of degree s > 1 over S. Let R = S[x]/ < g(x), pn−1 xt > be a finite commutative
chain ring constructed over S. Then Re = Se [ζ] is a finite commutative chain ring satisfying
Re = Se [x]/ < g(x), pn−1 xt >.
In other words, we can extend a finite chain ring using the extension of its coefficient ring.
We can draw the tree of extensions as in Figure 1.
Figure 1. Tree of extensions of a finite commutative chain ring
Re =
Se [x]
<g(x),pn−1 xt >
= R[ζ]
k − k−t
n
(by g(x) and t)
s (by ζ)
Se = S[ζ]
R=
S[x]
<g(x),pn−1 xt >
s (by ζ)
k−
k−t
n
(by g(x) and t)
S = GR(pn , r)
Remark 2.5. Let S be a finite commutative chain ring and R be an extension of it. It makes
sense to determine a kind of “degree of extension” considering Log|S| (|R|). Remark that this
degree of extension may not be an integer. See Figure 1 and as an example see Figure 2.
Remark 2.6. Notice that a finite commutative chain ring is a Galois ring if and only if
k = 1. In case n = 1, a finite commutative chain ring is of the form Fpr [x]/ < xk > and
it is called quasi-Galois ring. Similar to Galois rings, quasi-Galois rings are unique up to
isomorphism for a given characteristic and cardinality. However, a proper (neither Galois nor
quasi-Galois) finite commutative chain ring may not be unique up to isomorphism for given
characteristic, rank and cardinality. Example 2.7(ii) below illustrates this fact.
Example 2.7.
i) Consider
Z4 [x]
R=
2
< x + 2, 2x >
Z4 [x]
=
2
< x + 2x + 2, 2x >
.
R has 8 elements and is additively equivalent to Z2 ⊕ Z4 . It has 4 units and its
multiplicative group is isomorphic to Z4 . Also R/ < x >∼
= F2 .
ii) Consider
Z4 [x]
Z4 [x]
R1 =
=
.
< x2 + 2, 2x2 >
< x2 + 2 >
6
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
R1 has 16 elements, and additively equivalent to Z4 ⊕ Z4 . Similarly,
Z4 [x]
Z4 [x]
R2 =
=
< x2 + 2x + 2, 2x2 >
< x2 + 2x + 2 >
has 16 elements and additively equivalent to Z4 ⊕ Z4 . Moreover, R1∗ ∼
= Z2 ⊕ Z4 ,
= R2∗ ∼
∗
∼
where Ri denotes the set of units for 1 ≤ i ≤ 2. Also R1 / < x >= R2 / < x >∼
= F2 .
However, R1 and R2 are not isomorphic as rings.
iii) Consider
Z4 [ζ][x]
,
Re =
< x2 + 2, 2x >
where ζ is a root of the basic primitive polynomial f (X) = X 2 + X + 1 ∈ Z4 [X]. Re
has 64 elements, and is additively equivalent to Z2 ⊕ Z2 ⊕ Z4 ⊕ Z4 . Moreover, Re is
an extension of R given in (i) with Re = R[ζ], and also Re / < x >∼
= F4 = F2 (ζ).
Similarly, Se = Z4 [ζ] is a Galois ring as an extension of S = Z4 . See Figure 2 for the
tree of extension (remark that the extension degree of Re over Se is not an integer).
Figure 2. Tree of extensions: a proper example
Re =
Z4 [ζ][x]
<x2 +2,2x>
3
2
2
Se = Z4 [ζ]
R=
Z4 [x]
<x2 +2,2x>
2
3
2
S = Z4
Polynomials over finite commutative finite chain rings have a kind of “unique factorization”
under some assumptions. The following theorem, which is also known as Hensel’s lemma,
demonstrates this property. It is the adapted version of Hensel’s result in [8] to our context.
Theorem 2.8. Let R be a finite commutative chain ring and I be its maximal ideal. Also
let h(X), h1 (X), ..., hl (X) ∈ R[X] be monic polynomials such that
• h(X) = h1 (X) · · · hl (X),
• hi (X) ∈ R[X]/I are pairwise coprime for all 1 ≤ i ≤ l, and
• hi (0) 6= 0 for all 1 ≤ i ≤ l.
Then, the factorization h(X) = h1 (X) · · · hl (X) is unique up to reordering.
ADDITIVE CYCLIC CODES OVER FINITE COMMUTATIVE CHAIN RINGS
7
The polynomials hi in Theorem 2.8 are known as the Hensel’s lifting of the polynomials
hi , for all 1 ≤ i ≤ l.
Now we give automorphisms of finite chain rings. Let
R=
GR(pn , rs)[x]
GR(pn , r)[x]
and
R
=
e
< g(x), pn−1 xt >
< g(x), pn−1 xt >
be two finite commutative chain rings. Hence, Re is an extension of R. Let Re = R[ζ] for
some basic primitive element ζ ∈ GR(pn , rs) over GR(pn , r) of degree s, the automorphisms
of Re keeping the elements in R fixed are of the form
(2.3)
i
a0 + a1 ζ + ... + as−1 ζ s−1 7→ a0 + a1 ζ q + ... + as−1 ζ (s−1)q
i
for some 0 ≤ i ≤ s − 1, where q = pr and aj ∈ R for all 0 ≤ j ≤ s − 1 (for the proofs and
more detail, see, for example [2, Corollary 5.1.5]). We will denote the automorphism group
(i.e., the set of such maps with composition) by AutR (Re ). Remark that the roots of a basic
irreducible polynomial over Re are conjugate each other under this automorphism.
3. Codes over finite commutative chain rings
Let R be a ring and N be a positive integer, a non-empty subset C of RN = {(a1 , ..., aN ) :
ai ∈ R for all i = 1, .., N } is called a code over R of length N . An element of a code
C is called codeword. The function d on RN × RN given by d(a, b) = |{i : ai 6= bi }| for
all a = (a1 , ..., aN ), b = (b1 , ..., bN ) ∈ RN is a metric and it is called Hamming distance.
Minimum distance d(C) of a code C is defined by d(C) = min{d(a, b) : a, b ∈ C and a 6= b}.
The weight of a codeword a ∈ C is defined by w(a) = d(a, 0). Similarly, the minimum weight
of a code C is defined as the minimum non-zero weight in C and denoted by w(C).
Let S be a subring of R, then we say that a code C is S-linear if it is a module over S. Such
codes are also known as additive codes, and in particular, R-linear codes are known as linear
codes. Note that, if C is an additive code, then d(C) = w(C). This property and some other
properties make additive codes quite interesting and useful in both theory and application.
P
The Euclidean inner product of RN is defined by (a, b)E = N
i=1 ai bi for all a = (a1 , ..., aN )
N
N
and b = (b1 , ..., bN ) in R . When a code C ⊆ R is given, the Euclidean orthogonal code
C ⊥E of C is given by C ⊥E = {a ∈ RN : (a, b)E = 0 for all b ∈ C}. It can be shown directly
that the Euclidean orthogonal code of a code is a linear code and that in the case of linear
codes over Galois rings it coincides with the dual code, see [21].
A code C is cyclic if (aN , a1 , ..., aN −1 ) ∈ C for all (a1 , ..., aN ) ∈ C. Cyclic codes correspond to
a subset of R[X]/ < X N −1 >, which is closed under multiplication by X. In particular, there
is a one to one correspondence between a linear cyclic code and an ideal of R[X]/ < X N −1 >.
In the literature, the term “cyclic code” is generally understood as “linear cyclic code”.
However, there are non-linear cyclic code constructions, too. A recent example is [3], in
which the authors investigated additive cyclic codes over Galois rings. In detail, the authors
constructed Zpn -linear cyclic codes over Zpn [ω], where ω is a basic primitive element of degree
prime r. They also investigated the duality of such codes. In the following two sections, we
generalize their results from Galois rings to finite commutative chain rings.
8
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
In Section 4, we construct S = Zpn [x]/ < g(x), pn−1 xt >-linear cyclic codes over R =
Zpn [ω][x]/ < g(x), pn−1 xt > and investigate the duality of such codes, where g(x) ∈ Zpn [x]
is a monic Eisenstein polynomial of degree k and ω is a basic primitive element over Zpn of
degree prime r. Our results in this set up occur as a natural generalization of the results in
[3]. However, the method we use in our generalization is slightly different from the one in [3].
We just make use of ideals and do not get involved generator matrices, whereas the authors
in [3] were using some linear codes over the base ring S and their generator matrices. We
call such codes Galois-additive codes because R is a Galois extension of S.
In Section 5, we construct S = Zpn -linear cyclic codes over R = Zpn [x]/ < g(x), pn−1 xt >
and investigate the duality of such codes, where g(x) ∈ Zpn [x] is a monic Eisenstein polynomial of degree k. Our results in this set up are not very similar to the ones in Section 4,
especially considering duality. The main reason here is the fact that R is not a free module
over S. Therefore, we can not ensure that we can generate all the additive characters by
means of trace functions. Hence, in the case we use Euclidean inner product (Euclidean
orthogonality) we observe that some properties are not satisfied as expected. Thus we use
directly the duality notion in Section 5.1 as stated in [24] using character theory. Since R
is obtained by an Eisenstein polynomial from S, we call such additive codes as Eisensteinadditive codes.
4. Galois-additive cyclic codes
Z
n [ω][x]
Z
n [x]
p
p
Let R = <g(x),p
n−1 xt > and S = <g(x),pn−1 xt > , where p is a prime, n is a positive integer, ω
is a basic irreducible element over Zpn of degree r (which is a prime) and g(x) ∈ Zpn [x] is an
Eisenstein polynomial of degree k. Also we denote the nilpotency index by m = k(n − 1) + t.
Clearly S is a subring of R. Let R = R[X]/ < X N − 1 > and S = S[X]/ < X N − 1 >,
where N is a positive integer satisfying gcd(N, p) = 1. Similarly S is a subring of R. On
the other hand, R is a module over R (and also over S). In addition, there is a one to one
correspondence between R and RN given by a0 + a1 X + ... + aN −1 X N −1 ↔ (a0 , a1 , ..., aN −1 ).
On the grounds of this correspondence, we will say “code” for both a non-empty subset of R
and a non-empty subset of RN . In this section, we will construct S-linear cyclic codes C ⊆ R
and investigate their duality.
(b)
(p)
(pr )
Let Ci = {ibj mod N : j ∈ Z}, where b ∈ {p, pr }. Also let κi = |Ci | and κi,j = |Cipj |,
for 0 ≤ j ≤ r − 1. Then we have the following.
(p)
(pr )
(1) If gcd(κi , r) = 1, then Ci = Ci .
Sr−1 (pr )
(pr )
(p)
(2) If r divides κi , then κi = r|Ci | and Ci = j=0
Cipj .
b = R[ζ], where ζ is a basic primitive
Let s = min{j ∈ Z+ : (pr )j ≡ 1 mod N } and R
prs −1
element over R of degree s. Notice that the multiplicative order of ζ is prs −1 and ω = ζ pr −1 .
rs
b For each element a ∈ R,
b consider
The set Tb = {0, 1, ζ, ..., ζ p −2 } is a Teichmuller set of R.
m−1
b
the unique representation a = a0 + a1 x + ... + am−1 x
of a, where ai ∈ T for 0 ≤ i ≤ m − 1.
p
p
p
b
b = b if and
Using this representation, let φ : a 7→ a0 + a1 x + ... + am−1 xm−1 . Notice that φ(b)
P
P
b i )X i .
b
b
only if b ∈ R. φb can be extended as R[X]
→ R[X]
by i ai X i 7→ i φ(a
ADDITIVE CYCLIC CODES OVER FINITE COMMUTATIVE CHAIN RINGS
Similarly, the set T = {0, 1, ω, ..., ω p
b R.
φ = φ|
r −2
9
} is a Teichmuller set of R. Also we may define
prs −1
(p)
(p)
(p)
Let η = ζ N , clearly it is a primitive N.th root of unity. Also let Cj0 , Cj1 , ..., Cjv be all
distinct p−cyclotomic cosets modulo N , where j0 = 0 and 1 ≤ j1 < ... < jv ≤ N − 1 such that
(p)
(pr )
gcd(κji , r) = 1 for 0 ≤ i ≤ u, and κji is a multiple of r for u + 1 ≤ i ≤ v. Hence, Cji = Cji
(pr )
(p)
(pr )
(p)
(pr )
(pr )
for 0 ≤ i ≤ u, Cji 6= Cji and Cji = Cji ∪ Cji p ∪ ... ∪ Cji pr−1 for u + 1 ≤ i ≤ v.
Q
Let mi (X) = j∈C (p) X − η j and Ri = S[X]/ < mi (X) > for 0 ≤ i ≤ v. mi (X) is a basic
ji
irreducible polynomial over S. Now, let
ǫi (X) =
N −1
1 X X −jl j
b
η X ∈ R[X],
N
(p)
j=0 l∈C
j
i
b i (X)) = ǫi (X) and so ǫi (X) ∈ S[X], for all 0 ≤ i ≤ v. Let
for 0 ≤ i ≤ v. Here, φ(ǫ
also Ki = ǫi (X)S, for 0 ≤ i ≤ v. The following lemma can be proved by straightforward
computations.
Lemma 4.1. The following hold for all 0 ≤ i, j ≤ v.
(1) ǫi (X)2 = ǫi (X) and ǫi (X)ǫj (X) = 0 in S when i 6= j. Also ǫ0 (X) + ǫ1 (X) + ... +
ǫv (X) = 1 in S.
(2) S = K0 + K1 + ... + Kv = K0 ⊕ K1 ⊕ ... ⊕ Kv , and ǫi (X) is the multiplicative identity
of Ki .
(3) The map ψi : Ri → Ki given by f (X)+ < mi (X) >7→ ǫi (X)f (X)+ < X N − 1 > is a
ring isomorphism. Hence, Ki has a basis {ǫi (X)X j : j = 0, 1, ..., κji − 1} over S.
Q
Now let mi,h = j∈C (pr ) X − η j for 0 ≤ h ≤ r − 1 and u + 1 ≤ i ≤ v. Similarly, mi,h (X)
ji ph
is a basic irreducible polynomial over R. Also let
N −1
1 X X −jl j
b
ǫi,h (X) =
η X ∈ R[X],
N
r
(p )
j=0 l∈C
ji ph
for u + 1 ≤ i ≤ v and 0 ≤ h ≤ r − 1. Here, φbr (ǫi,h (X)) = ǫi,h (X) and so ǫi,h (X) ∈ R[X], for
all u + 1 ≤ i ≤ v and 0 ≤ h ≤ r − 1. Let also Li = ǫi (X)R and Li,h = ǫi,h (X)R, the following
lemma, similar to Lemma 4.1, can be proved directly by computations.
Lemma 4.2. The following hold for all u + 1 ≤ i ≤ v and 0 ≤ h, j ≤ r − 1.
(1) ǫi,h (X)2 = ǫi,h (X) and ǫi,j (X)ǫi,h (X) = 0 in R, when j 6= h. Also ǫi,0 (X) + ǫi,1 (X) +
... + ǫi,r−1 (X) = ǫi (X) in R.
(2) R = L0 + L1 + ... + Lv = L0 ⊕ L1 ⊕ ... ⊕ Lv , and ǫi (X) is the multiplicative identity
of Li .
(3) Li = Li,0 +Li,1 +...+Li,r−1 = Li,0 ⊕Li,1 ⊕...⊕Li,r−1 , and ǫi,h (X) is the multiplicative
identity of Li,h .
(4) The map ψi : R[X]/ < mi (X) >→ Li given by f (X)+ < mi (X) >7→ ǫi (X)f (X)+ <
X N −1 > is a ring isomorphism and hence Li has a basis {ǫi (X)X j : j = 0, 1, ..., κji −
1}. Similarly, the map ψi,h : R[X]/ < mi,h (X) >→ Li,h given by f (X)+ < mi,h (X) >7→
10
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
ǫi,h (X)f (X)+ < X N −1 > is a ring isomorphism and hence Li,h has a basis {ǫi,h (X)X j :
j = 0, 1, ..., κji ,h − 1}.
Corollary 4.3. In R, ǫi (X)ǫj,h (X) = 0 when i 6= j, for all 0 ≤ i ≤ v, u + 1 ≤ j ≤ v and
0 ≤ h ≤ r − 1.
Proof. For any arbitrary 0 ≤ i ≤ v, u + 1 ≤ j ≤ v and 0 ≤ h ≤ r − 1, Lemma 4.1(1) and
Lemma 4.2(1) implies
X
ǫi (X)ǫj,h (X) = −
ǫi (X)ǫl,h .
l6=j
Multiplying both sides by ǫj,h (X), we obtain the statement of the corollary, by Lemma
4.2(1).
Clearly, Ki = Li ∩ S for all 0 ≤ i ≤ v. However, when we define Ki,j = ǫi,j (X)S, we obtain
Ki,j 6= Li,j ∩ S, for all u + 1 ≤ i ≤ v and 0 ≤ j ≤ r − 1. The following lemma expresses the
idea behind it in a precise way.
Lemma 4.4. Ki,j = Li,j for all u + 1 ≤ i ≤ v and 0 ≤ j ≤ r − 1.
Proof. Firstly notice that
(4.1)
Li,h =
=
=
=
ǫi,h (X)R
Pr−1 j
ǫi,h (X) j=0
ω S
Pr−1 j
ω ǫi,h (X)S
Pj=0
r−1 j
j=0 ω Ki,h ,
for all u + 1 ≤ i ≤ v and 0 ≤ j ≤ r − 1. Equation (4.1) says that, Ki,j = Li,j if and only
if Ki,h = ωKi,h . Hence, we will equivalently prove Ki,h = ωKi,h , for all u + 1 ≤ i ≤ v and
0 ≤ h ≤ r − 1.
Now, assume the contrary, i.e., assume Ki0 ,h0 6= ωKi0 ,h0 for some u + 1 ≤ i0 ≤ v and
0 ≤ h0 ≤ r − 1. Then, since ǫi,j (X) 6= ω l ǫi,h (X) when j 6= h and 1 ≤ l ≤ r − 1, the equation
(4.1) says that Li0 = ǫi0 (X)R has a basis over Ki0 = ǫi0 (X)S including more than r elements,
this is a contradiction. Then, Ki,h = ωKi,h , for all u + 1 ≤ i ≤ v and 0 ≤ j ≤ r − 1.
It can be shown directly that φ satisfies the following, for all 0 ≤ i ≤ v and 0 ≤ h ≤ r − 1.
(1) φ(ǫi (X)) = ǫi (X) and φ(ǫi,h (X)) = ǫi,h+1
(2) φ(Li ) = Li and φ(Li,h ) = Li,h+1 mod r .
mod r (X).
The following lemma is immediate.
Lemma 4.5. Let C be a non-empty subset of R. C is an S-linear code over R of length N
L
if and only if there is a unique Ki -submodule Ci of Li such that C = vi=0 Ci . Hence we have
Q
also |C| = vi=0 |Ci |.
L
Let C be a code and Ci be given as in Lemma 4.5. The unique decomposition C = vi=0 Ci
is called canonical decomposition of C.
Euclidean inner product of a(X) = a0 + a1 X + ... + aN −1 X N −1 and b(X) = b0 + b1 X + ... +
PN −1
bN −1 X N −1 in R is naturally given by (a(X), b(X))E = i=0
ai bi . Accordingly, Euclidean
ADDITIVE CYCLIC CODES OVER FINITE COMMUTATIVE CHAIN RINGS
11
orthogonal code C ⊥E of a code C is given by C ⊥E = {a ∈ R : (a(X), b(X))E = 0 for all b(X) ∈
C}.
Pr−1 j
Let T r : R → S be given by c 7→ j=0
φ (c). T r is called the generalized trace of R relative
PN −1
to S. Trace inner product of R over S is defined by (a(X), b(X))T r = j=0
T r(ai bi ) for
N
−1
N
−1
a(X) = a0 +a1 X +...+aN −1 X
, b(X) = b0 +b1 X +...+bN −1 X
∈ R. Accordingly, Trace
⊥
⊥
orthogonal code C of a code C is given by C = {a ∈ R : (a(X), b(X))T r = 0 for all b(X) ∈
C}.
Let µ : R → R be given by a(X) 7→ X N a(X −1 ) for all a(X) ∈ R. Clearly, µ is an
automorphism on R of order 2. Also µ can be used as the permutation on {0, 1, ..., v} given
by i 7→ i′ such that Cji′ = C−ji . The following lemma is straightforward.
Lemma 4.6. The map µ defined in the previous paragraph has the following properties.
(1) µ(ǫi (X)) = ǫµ(i) (X) in R for all 0 ≤ i ≤ v.
(2) µ(0) = 0, 1 ≤ µ(i) ≤ u for 1 ≤ i ≤ u, and u + 1 ≤ µ(i) ≤ v for u + 1 ≤ i ≤ v.
In addition, µ also determines a ring isomorphism Ri → Rµ(i) given by f (X) 7→ µ(f (X))
mod mµ(i) (X). Hence, µ(Ki ) = Kµ(i) and µ(Li ) = Lµ(i) for all 0 ≤ i ≤ v.
The following lemma is one of the main arguments we use to prove our main theorem in
this section.
Lemma 4.7. In R, a(X)µ(b(X)) = 0 if and only if (a(X), X h b(X))E = 0 for all 0 ≤ h ≤
N − 1.
Proof. Let a(X) =
PN −1
PN −1
ai X i and b(X) = i=0
bi X i . Then
PN −1 PN −1
i−j mod N
a(X)µ(b(X)) =
j=0 ai bj X
Pi=0
P
N −1
N −1
h
=
i=0
h=0 ai bi−h mod N X
PN −1 PN −1
h
=
i=0 ai bi−h mod N X
h=0
PN −1
h
h
=
h=0 a(X), X b(X) E X .
i=0
Recall that {1, ω, ..., ω r−1 } is an S-basis of R. Let {θ0 , θ1 , ..., θr−1 } be another S-basis of
R. These two basis are called trace dual of each other if T (ω j θj ) = 1 and T (ω j θh ) = 0 for all
0 ≤ j 6= h ≤ r − 1. When a basis is given, its trace dual can be constructed by the following
lemma.
Pr−1
γ(X)
Lemma 4.8. Let γ ′ (X) = X−ω
= j=0
γj X j ∈ R[X], where γ(X) is the primitive irreducible
γj
polynomial of ω over S. Let also θj = γ ′ (ω) ∈ R for 0 ≤ j ≤ r − 1. Then {θ0 , ..., θr−1 } is the
trace dual of {1, ω, ..., ω r−1 }.
Proof. It can be proved directly by the interpolation idea of polynomials.
Now we give one of the main results in this paper. Note that, as we are in a Galois
extension, trace orthogonality can be translated one to one to duality [21, Lemma 6] thus we
will use the term of trace duality.
12
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
Theorem 4.9. Consider the definitions and notations given above. Any S-linear cyclic code
C ⊆ R over R of length N is of the form
(4.2)
C=
r−1
u X
X
xei,j ω j Ki +
r−1
v
X
X
xei,j Ki,j ,
i=u+1 j=0
i=0 j=0
for some 0 ≤ ei,j ≤ m − 1 given for 0 ≤ i ≤ v and 0 ≤ j ≤ r − 1.
On the other hand, the trace dual code of C given in (4.2) is of the form
C⊥ =
(4.3)
r−1
u X
X
xm−ei,j θj Ki +
r−1
v
X
X
xm−ei,j Ki,j ,
i=u+1 j=0
i=0 j=0
where T r(ω i θi ) = 1 and T r(ω i θj ) = 0 for all 0 ≤ i 6= j ≤ r − 1.
Proof. By Lemmas 4.1, 4.2, 4.3 and 4.4, the canonical decomposition of R is given by
R=
u X
r−1
X
ω j Ki +
v
r−1
X
X
i=u+1 j=0
i=0 j=0
Ki,j =
u M
r−1
M
ω j Ki ⊕
i=0 j=0
v
r−1
M
M
Ki,j .
i=u+1 j=0
Notice that any S-linear subset of ω j Ki (or Ki,j ) is an ideal xei,j ω j Ki (or xei,j Ki,j ) of itself,
for some 0 ≤ ei,j ≤ m − 1 given for 0 ≤ i ≤ v and 0 ≤ j ≤ r − 1. Therefore, the construction
of C is clear by Lemma 4.5.
Now, let
r−1
v
r−1
u X
X
X
X
m−ei,j
xm−ei,j Ki,j ,
x
θj Ki +
D=
i=u+1 j=0
i=0 j=0
T r(ω i θ
T r(ω i θ
where
i ) = 1 and
j ) = 0 for all 0 ≤ i 6= j ≤ r − 1. Also let a(X) ∈ C and
b(X) ∈ D be arbitrary. From the properties Lemma 4.1(1), Lemma 4.2(1), Corollary 4.3 and
the trace duality between {1, ω, ..., ω r−1 } and {θ0 , θ1 , ..., θr−1 }, we deduce
r−1
X
φl (a(X)b(X)) = 0,
l=0
which implies (a(X), b(X))T r = 0 by Lemma 4.7. Therefore, D ⊆ C ⊥ . Now, let b′ (X) ∈ R
but b′ (X) ∈
/ D. Then similar arguments above can be used to show (a(X), b′ (X))T r 6= 0 for
some a(X) ∈ C. Therefore, C ⊥ ⊆ D. Conclusively, C ⊥ = D.
Remark 4.10. The special case of Theorem 4.9 for t = k = 1 corresponds to the construction
in [3]. However, we use a different language without mentioning generator matrices of some
subcodes over S.
As expected for a dual code we can also deduce the following results from Theorem 4.9
related to cardinality and self-duality.
Corollary 4.11. Let C and C ⊥ be given as in Theorem 4.9. Then we obtain the equation
logp |C| + logp |C ⊥ | = logp |R|.
Proof. Follows from the equations (4.2), (4.3) and Lemma 4.5.
ADDITIVE CYCLIC CODES OVER FINITE COMMUTATIVE CHAIN RINGS
13
Corollary 4.12. Let C be a code given as in (4.2). Then, C is self-dual if and only if m is
even and ei,j = m
2 for all 0 ≤ i ≤ v and 0 ≤ j ≤ r − 1.
Proof. The evenness of m and the property ei,j = m
2 for all 0 ≤ i ≤ v and 0 ≤ j ≤ r − 1 can
be derived from (4.2) and (4.3). Remark that the basis {1, ω, ..., ω r−1 } is also the basis of
Zpn [ω] over Zpn , hence the dual basis {θ0 , ..., θr−1 } of it exists, and this existence is enough
to complete the proof (since all ei,j ’s are the same).
Now, we illustrate Theorem 4.9 and its corollaries above in the following example.
Example 4.13. Let S = Z4 [x]/ < x2 + 2, 2x > and R = S[ω], where ω is a root of the
polynomial X 2 + X + 1 ∈ S[X]. That is, p = 2, n = 2, r = 2, g(x) = x2 + 2, k = 2, t = 1 and
b = R and ζ = ω = η.
so m = 3. Let also N = 3 (which is relatively prime to p). Hence R
Then we have
(p)
(pr )
C0 = C0
= {0},
(p)
C1 = {1, 2},
(pr )
(pr )
C1,0 = {1} and C1,1 = {2}.
That is, u = 0 and v = 1. Accordingly,
m0 (X) = X + 3,
m1,0 (X) = X + 3ω,
2
m1 (X) = X + X + 1, m1,1 (X) = X + (ω + 1),
and
ǫ0 (X) = 3X 2 + 3X + 3, ǫ1,0 (X) = 3ωX 2 + (ω + 1)X + 3,
ǫ1 (X) = X 2 + X + 2,
ǫ1,1 (X) = (ω + 1)X 2 + 3ωX + 3.
Direct computations give that
K0 = {aX 2 + aX + a : a ∈ S},
K1 = {aX 2 + bX + (−a − b) : a, b ∈ S},
K1,0 = {(a + ωb)X 2 + (−b + ω(a − b))X + (b − a + ω(−a)) : a, b ∈ S},
K1,1 = {(a + ωb)X 2 + (b − a + ω(−a))X + (−b + ω(a − b)) : a, b ∈ S}.
Notice also that θ0 = ω + 3 and θ1 = 2ω + 1. Let
C = K0 + 2ωK0 + xK1,0 ,
then
C ⊥ = xθ1 K0 + 2K1,0 + K1,1 .
Observe that |C| = 28 , |C ⊥ | = 210 and |R| = (26 )3 . Also remark that any self-dual codes do
not exist in this ambient space R, since m is not even.
5. Eisenstein-additive cyclic codes
Z
n [x]
p
Consider the same definitions and notations in Section 4 but inserting R = <g(x),p
n−1 xt >
and S = Zpn , where p is a prime, n is a positive integer and g(x) ∈ Zpn [x] is an Eisenstein
polynomial of degree k. Notice that S is a Galois ring which is the coefficient ring of R. In
this case, we obtain u = v.
Now, before to give our main theorem, we give a lemma to clarify some further points.
Lemma 5.1. R is not a free module over S.
14
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
Proof. Clearly, the set B = {1, x, ..., xk−1 } is a minimal set spanning R over S. However, B
is not linearly independent, since pn−1 is non-zero but pn−1 xt = 0. Therefore, no bases of R
exist over S, i.e., R is not a free module over S.
The second main result in this paper is the following theorem.
Theorem 5.2. Any S-linear cyclic code C ⊆ R over R of length N is of the form
(5.1)
C=
v m−1
X
X
ai,j xj Ki ,
i=0 j=0
for some ai,j ∈ {0, 1} ⊆ R.
Proof. The proof can be done similar to the proof of Theorem 4.9. Remark that Lemma 4.7
works efficiently also here, since u = v.
Remark 5.3. Notice that, if ai,j0 = 1 for some 0 ≤ j0 ≤ m − 1, then ai,j0 +kl can be taken
both zero and one for l ≥ 1 (since < xk >=< p >⊆ S). However, any such situation does
not disturb the set up of Theorem 5.2.
Remark 5.4. We have not mention duality in Theorem 5.2, whereas we have done in Theorem 4.9. The reason is related to the profile of the extension of R over S. Since R is not
free over S, we can not determine any trace function for x in R over S, and hence we can not
define any trace inner products. In the following subsection, we examine the duality notion
for Eisenstein additivity separately.
5.1. Character Theoretic Duality for Eisenstein-Additive Codes. Eisenstein extension is not a free extension when t 6= k, hence the problem of a suitable inner product for
Eisenstein-additive codes occurs. The character theoretic approach in [24] provides a convenient inner product and a duality notion when we consider the one-to-oneness between a code
and its dual (a MacWilliams identity). Remark that the character theoretic duality notion
in [24] was given for Frobenius rings, so we may apply this notion to finite commutative
chain rings. In this subsection, we adjust the notion in [24] to our context assuming that the
reader has some basic knowledge about characters (otherwise, we suggest [24, Section 3] for
the sufficient information about characters we use in this paper).
Consider commutative chain ring R = Zpn [x]/ < g(x), pn−1 xt >, where g(x) is an Eisenstein polynomial of degree k. Clearly the additive structure of R is isomorphic to the finite
abelian group G of the form
k−t
t
M
M
Zpn−1 .
Zpn ⊕
G=
i=1
i=1
Consider the unique representation a = a0 + a1 x + ... + ak−1 xk−1 of elements a ∈ R, where
ai ∈ Zpn for 0 ≤ i ≤ t − 1 and ai ∈ {0, 1, ..., pn−1 − 1} ⊆ Zpn for t ≤ i ≤ k − 1. Corresponding
to each element a ∈ R, we define a map χa from R to C (the set of complex numbers) given
by
a z0 +...+at−1 zt−1 at zt +...+ak−1 zk−1
,
ηpn−1
χa : z = z0 + z1 x + ... + zk−1 xk−1 7→ ηpn0
ADDITIVE CYCLIC CODES OVER FINITE COMMUTATIVE CHAIN RINGS
15
where ηpn and ηpn−1 are the pn .th and the pn−1 .th root of unities respectively. This map is
clearly an additive character of R (that is, χa is a group homomorphism from the additive
structure of R to the group of non-zero complex numbers with multiplication). Let χ = {χa :
a ∈ R}, then χ with the point-wise multiplication is a group and isomorphic to the additive
structure of R (i.e. χ is a (additive) character group of R).
In addition, χa corresponds to the vector (χa (z))z∈R ∈ C|R| for each a ∈ R. Remark that
the addition a + b in R corresponds to the component-wise multiplication of (χa (z))z∈R and
(χb (z))z∈R , when we fix the order of elements z in R. We use the notation χa for both the
homomorphism χa and the corresponding vector (χa (z))z∈R when the meaning is clear. Now
we define an inner product between vectors χa and χb by
(5.2)
(χa , χb ) =
1 X
χa (z)χb (z),
|R|
z∈R
where χb (z) denotes the complex conjugate of χb (z). This inner product is indeed a positive
definite Hermitian product.
We consider ring S = Zpn as an additive subgroup of R, and hence define the annihilator
(χ : S) = {χa ∈ χ : χa (z) = 0 for all z ∈ S}. Then (χ : S) is isomorphic to the character
group of the quotient group R/S, and hence |(χ : S)| = |R|/|S|.
Let G1 and G2 be two finite abelian groups and χ1 and χ2 be their character groups
respectively. Then the character group of the group G1 × G2 with the component-wise
operations is indeed χ1 × χ2 .
All the set up about characters mentioned above allow us to define dual codes for Eisensteinadditive codes. An Eisenstein-additive code C was an additive subgroup of RN (or equivalently R), thus the dual of C is defined as the additive subgroup D of RN given by
D = {(a1 , ..., aN ) ∈ RN : (χa1 , ..., χaN ) ∈ (χN : C)}. In that way, the one-to-oneness between an Eisenstein-additive code and its dual is satisfied according to [24, Theorem 4.2.1].
Example 5.5. Let S = Z4 and R = S[x]/ < x2 + 2, 2x >. That is, p = 2, n = 2, r =
1, g(x) = x2 + 2, k = 2, t = 1 and so m = 3. Let also N = 3 (which is relatively prime to p).
b = R and ζ = η. Then we have
Hence R
(p)
(p)
C0 = {0},
C1 = {1, 2};
m0 (X) = X + 3,
m1 (X) = X 2 + X + 1;
ǫ0 (X) = 3X 2 + 3X + 3, ǫ1 (X) = X 2 + X + 2.
and hence
K0 = {aX 2 + aX + a : a ∈ S}, K1 = {aX 2 + bX + (−a − b) : a, b ∈ S}.
If we write the elements a ∈ R as a = a0 + a1 x where a0 ∈ Z4 and a1 ∈ {0, 1}, then we may
define the corresponding characters χa as χa = ia0 (−1)a1 where i is the primitive 4.th root of
unity (in the set of complex numbers). Considering the ordering
(0, 1, 2, 3, x, x + 1, x + 2, x + 3)
16
EDGAR MARTÍNEZ-MORO, KAMIL OTAL AND FERRUH ÖZBUDAK
of elements in R, we may write the elements of the character group χ of R in vector form as
follows:
χ0
= (1, 1, 1, 1, 1, 1, 1, 1),
χ1
= (1, i, −1, −i, 1, i, −1, −i),
χ2
= (1, −1, 1, −1, 1, −1, 1, −1),
χ3
= (1, −i, −1, i, 1, −i, −1, i),
χx
= (1, 1, 1, 1, −1, −1, −1, −1),
χx+1 = (1, i, −1, −i, −1, −i, 1, i),
χx+2 = (1, −1, 1, −1, −1, 1, −1, 1),
χx+3 = (1, −i, −1, i, −1, i, 1, −i).
Then the annihilator of S is (χ : S) = {χ0 , χx } and hence the dual of S is xS = {0, x}.
Duality of other subgroups of R is as follows.
R ↔ {0},
S ↔ xS,
xR ↔ 2R.
Now, let us define a code. Let
C = K0 + xK0 + xK1 .
Here, we can also write C = K0 + xK0 + x2 K0 + xK1 (recall Remark 5.3). In addition, we
can write C = L0 + xK1 = RK0 + xK1 . Then we have
C ⊥ = K1 .
Observe that |C| = 25 and |C ⊥ | = 24 , i.e., |C| · |C ⊥ | = |R| = 29 . Also remark that any self-dual
codes do not exist in this ambient space, since no subgroups of R are self-dual.
Acknowledgement
The skeleton of this study has been constructed during the second author’s visit to University of Valladolid between March 1-31 in 2016 by the support of COST Action IC 1104
Random Network Coding and Designs over GF(q).
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