Content uploaded by Ismail Gokhan Kelebek
Author content
All content in this area was uploaded by Ismail Gokhan Kelebek on Sep 28, 2017
Content may be subject to copyright.
Bull. Korean Math. Soc. 54 (2017), No. 4, pp. 1111–1122
https://doi.org/10.4134/BKMS.b150764
pISSN: 1015-8634 / eISSN: 2234-3016
CYCLIC AND CONSTACYCLIC SELF-DUAL CODES
OVER Rk
Suat Karadeniz, Ismail Gokhan Kelebek, and Bahattin Yildiz
Abstract. In this work, we consider constacyclic and cyclic self-dual
codes over the rings Rk. We start with theoretical existence results for
constacyclic and cyclic self-dual codes of any length over Rkand then
construct cyclic self-dual codes over R1=F2+uF2of even lengths from
lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual
codes over R1of even lengths for which non-trivial such codes exist. In
particular we demonstrate that our constructions provide a counter ex-
ample to a claim made by Batoul et al. in [1] and we explain why their
claim fails.
1. Introduction
Both cyclic codes and self-dual codes over different alphabets have been the
focus of many works related to coding theory for a long time now. Cyclic codes
have a rich algebraic structure, making them relatively easier to study and to
encode and decode. Self-dual codes have a combinatorial aspect which makes
them quite popular as well as connections with several fields such as lattices,
invariant theory, cryptography and designs.
Starting with [16], a combination of these two structures has become some-
thing of interest to study by researchers. In [3], a complete classification of
cyclic self-dual codes of odd lengths over the ring F2+uF2was given. Re-
searchers have considered the same problem over different fields and rings in
[1], [2], [6], [15].
In this work, we consider constacyclic and cyclic self-dual codes over Rk,
the family of rings introduced in [9], that extends such rings as R1=F2+uF2
and R2=F2+uF2+vF2+uvF2. Being a family of Frobenius rings ([17]),
they have been studied quite extensively from many different angles; see [8],
[9], [10], [13], [18] for more.
Received September 17, 2015; Accepted January 14, 2016.
2010 Mathematics Subject Classification. Primary 94B05, 94B99; Secondary 11T71,
13M99.
Key words and phrases. cyclic codes, constacyclic codes, cyclic self-dual codes, lift,
projection.
c
2017 Korean Mathematical Society
1111
1112 S. KARADENIZ, I. G. KELEBEK, AND B. YILDIZ
The main motivation for the current work is twofold: First we consider
a theorem in [2] in which it was stated that a necessary condition for a λ-
constacyclic self-dual code over Fqto exist is that λ2= 1. Since in Rk, all units
satisfy this property, our first goal was to determine whether λ-constacyclic self-
dual codes over Rkof all lengths exist for all units λ∈Rk. The second is that
since cyclic self-dual codes over R1of odd lengths were classified completely,
and binary cyclic self-dual codes up to certain lengths were characterized in
[11], we considered cyclic self-dual codes over R1of even lengths. In doing so
we introduced the notion of independence in Rkand laid down the theoretical
background on lifting binary cyclic self-dual codes to cyclic self-dual codes over
R1(and Rkin general). Using lifts of binary cyclic self-dual codes were able
to classify all free cyclic self-dual codes over R1of even lengths for which non-
trivial such codes exist. In due process we obtained many good cyclic self-dual
codes over R1from such lifts, providing a counter example to a claim by Batoul
et al. in [1] (Theorem 4.14) in which they claimed that a self-dual code of even
length over a finite chain ring with residue field F2cannot be the lift of a binary
self-dual cyclic code.
The rest of the work is organized as follows. Section 2 contains the prelimi-
naries about the rings Rk, self-dual, cyclic and constacyclic codes. In Section
3 we settle the question of existence of cyclic and constacyclic codes over Rkof
any length. In Section 4 we consider the projections, lifts and independence in
Rk, laying the ground for computational results. Section 5 contains the com-
putational results and examples; in particular it contains tables of good cyclic
self-dual codes of certain even lengths over the ring R1, obtained from lifts of
binary cyclic self-dual codes. We finish with a remark on the claim in [1] we
have disproved and explain why the claim fails.
2. Preliminaries
The family of rings denoted by Rkhave been introduced in [9]. Leaving the
details of these rings to the aforementioned work, we recall some of the basic
properties, the proofs of which can be found in [9]. For k≥1, let
(2.1) Rk=F2[u1, u2,...,uk]/hu2
i= 0, uiuj=ujuii.
We actually take R0=F2, the binary field. The basis elements of Rkcan be
viewed, using subsets A⊆ {1,2,...,k}by letting
(2.2) uA:= Y
i∈A
ui
with the convention that u∅= 1.Then any element of Rkcan be represented
as
(2.3) X
A⊆{1,...,k}
cAuA, cA∈F2.
CONSTACYCLIC SELF-DUAL CODES 1113
The ring Rkis a local ring with maximal ideal hu1, u2,...,ukiand |Rk|=
2(2k). It is neither a principal ideal ring nor a chain ring when k≥2, but is a
Frobenius ring for all k≥0.
An element of Rkis a unit if and only if the coefficient of u∅is 1 and each
unit is also its own inverse. We also have the following:
(2.4) ∀a∈Rka2=1 if ais a unit
0 otherwise.
A linear code of length nover Rkis defined to be an Rk-submodule of Rn
k.
We attach the usual inner product on this ambient space Rn
k, that is ha, bik=
Paibi. The dual C⊥is defined as C⊥={y∈Rn
k| hy, xik= 0 for all x∈C}.
We say that a code is self-orthogonal if C⊆C⊥and self-dual if C=C⊥.
We define the Gray map inductively, extending it naturally from the Gray
map on R1from [7] as follows.
For c∈Rn
k,write c=c1+ukc2with c1, c2∈Rn
k−1, then we can define
φk(c) = (φk−1(c2), φk−1(c1) + φk−1(c2)) ,
with φ0being the identity map on F2. For example on R1we have φ1(c1+uc2) =
(c2, c1+c2) and on R2,
φ2(a+ub +vc +uvd) = (d, c +d, b +d, a +b+c+d).
The Lee weight wLof a codeword is the Hamming weight of the image of
the codeword under φk. Then the Gray map is a linear weight preserving map
from Rn
kto Fn2k
2.
If all the codewords of a self-dual code have doubly-even Lee weight, then
the code is said to be Type II, otherwise it is said to be Type I. The following
lemma from [10] shows that the Gray map is duality-preserving:
Lemma 2.1. If Cis a self-dual code over Rkof length n, then φk(C)is a
binary self-dual code of length 2kn. The Lee weight distribution of Cand the
Hamming weight distribution of φk(C)are the same. In particular, if Cis Type
I, then so is φk(C)and the same is true for Type II codes as well.
A cyclic shift on Rn
kis a permutation τsuch that
(2.5) τ(c0, c1,...,cn−1) = (cn−1, c0, c1,...,cn−2).
A linear code Cover Rkof length nis said to be a cyclic code if it is invariant
under the cyclic shift, i.e., τ(C) = C.
For a unit λ∈Rk, the λ-constacyclic shift on Rn
kis the map
(2.6) τλ(c0, c1,...,cn−1) = (λcn−1, c0, c1,...,cn−2).
A linear code Cover Rkof length nis said to be a λ-constacyclic code if it
is invariant under the constacyclic shift, i.e., τλ(C) = C.
Using the natural polynomial representation of codewords over Rkin Rk[x],
that is representing the vector (a0, a1,...,an−1) by the polynomial Paixi, we
see that for a codeword c∈Rn
k,τ(c) corresponds to xc(x) in Rk[x]/(xn−1),
1114 S. KARADENIZ, I. G. KELEBEK, AND B. YILDIZ
whereas τλ(c) corresponds to xc(x) in Rk[x]/(xn−λ). The following then is
clearly a characterization of constacyclic and cyclic codes over Rk:
Proposition 2.2. A subset Cof Rn
kis a linear cyclic code of length nover
Rkif and only if its polynomial representation is an ideal of the ring Rk,n :=
Rk[x]/(xn−1).Cis λ-constacyclic if and only if it corresponds to an ideal in
Rk,n,λ := Rk[x]/(xn−λ).
The binary image of a cyclic code over Rkis equivalent to a quasi-cyclic
code by the following theorem from [8]:
Theorem 2.3. Let Cbe a cyclic code of length nover Rk. Then φk(C)is
equivalent to a binary quasi-cyclic code of length 2knand index 2k.
3. The existence of cyclic and constcyclic self-dual codes over Rk
We begin with the following theorem:
Theorem 3.1. Let Rbe a finite commutative Frobenius ring. Assume that self-
dual codes of length 1exist over R. Then cyclic self-dual codes of all lengths
exist over R.
Proof. Assume that r∈Rgenerates a self-dual code of length 1. Then the
direct sum of the length 1 code, i.e., the code generated by the matrix r·Inis
a cyclic self-dual code over Rof length n.
Now, since in Rk,uigenerates a self-dual code of length 1 for all i=
1,2,...,k we have the following corollary:
Corollary 3.2. Cyclic self-dual codes of all lengths exist over Rk, for all k≥1.
The following theorem from [2] describes the duals of constacyclic codes over
rings:
Theorem 3.3 ([2, Theorem 1]).Let Rbe a ring and λa unit in R. Then the
dual of a λ-constacyclic code over Ris λ−1-constacyclic.
The theoretical consequence of this theorem is that in Rk, where each unit λ
satisfies λ2= 1, self-dual constacyclic codes might exist. But in fact it is easy to
see that since huiiand hλuiiare the same self-dual code of length 1, Corollary
3.2 can easily be extended to λ-constacyclic codes for any unit λ∈Rk:
Corollary 3.4. Let λbe any unit in Rk. Then λ-constacylic self-dual codes
over Rkof any length nexist.
In [13], it was shown that when nis odd, the map that takes xto (1 + v)x
establishes a ring isomorphism between the quotient rings R2[x]/(xn−1) and
R2[x]/(xn−(1 + v)) resulting in the statement that cyclic codes of odd length
are isomorphic to (1 + v)-constacyclic codes of the same length. The idea used
in the proof can easily be generalized without much change to include any unit
λsuch that λ2= 1 and any commutative Frobenius ring of characteristic 2. In
CONSTACYCLIC SELF-DUAL CODES 1115
particular, since every unit λin Rksatisfies λ2= 1, we can extend the result
to all λ-constacyclic codes over Rk. What remains is to mention orthogonality.
But since λ2= 1, we have (λr1)(λr2) = r1r2for all r1, r2∈Rkand any unit
λ∈Rk. Thus we have proven the following theorem:
Theorem 3.5. Suppose nis odd and λis any unit in Rk. Then the map
x7→ λx establishes an isomorphism between cyclic self-dual codes of length n
over Rkand λ-constacyclic self-dual codes over Rkof the same length.
We will actually say something more about constacyclic codes for particular
units. The subgroup Ub=h1 + u1,1 + u2,...,1 + ukiof the unit group of Rkis
called the subgroup of basic units. In [9], it is shown that multiplying by basic
units corresponds to a permutation of coordinates in the Gray image, thus we
have the following lemma:
Lemma 3.6 (Lemma 2.1 in [12]).(1) The Lee weight of each basic unit is 1.
In fact, any element in Rkof Lee weight 1must be a basic unit.
(2) If for α, β ∈Rk, we have α=r·βfor some r∈ Ub, then wL(α) = wL(β).
Thus in the case of basic units we can say something more:
Theorem 3.7. Suppose nis odd and λis a basic unit in Rk. Assume that C
is a cyclic self-dual code of length nover Rk. Then applying the map x7→ λx
to elements of C, we obtain a λ-constacyclic self-dual code ˜
Cof length nover
Rk. Moreover, the Lee weight distribution of Cand ˜
Care the same.
We finish this section with the construction for a family of cyclic and con-
stacyclic self-dual codes over Rkof minimum Lee weight 4, for k≥2.
Theorem 3.8. For n≥2, the polynomial whose coefficient vector is the length
nvector (u, v, v, . . . , v)generates a cyclic self-dual code over R2of minimum
Lee weight 4. In fact the polynomial with coefficients (ui, uj, uj,...,uj),i6=j
also generates a cyclic self-dual code of minimum Lee weight 4over Rkfor all
k≥2.
Proof. The proof will be done for the case when k= 2. The same ideas can
be used to extend the proof for the general case as well. Now, let Cbe the
code generated over R2by the vector c= (u,v,v,...,v) and all its cyclic shifts.
Note that hτi(c), τ j(c)i= 2uv + (n−2)v2= 0 for all i, j with i6=jand it is
equal to u2+ (n−1)v2= 0 if i=j. So, we easily see that Cis self-orthogonal.
Then, since R2is Frobenius, we must have |C|·|C⊥|= 16n. Self-orthogonality
of Cimplies that C⊆C⊥, which implies |C| ≤ |C⊥|. Putting this into the
equation above, we get
(3.1) |C| ≤ 4n.
Now consider, the set
S={c, τ (c),...,τn−1(c), v ·c, v ·τ(c),...,v·τn−1(c)}.
1116 S. KARADENIZ, I. G. KELEBEK, AND B. YILDIZ
It is clearly seen that Sis an independent set over F2. Since Cis linear over
R2, all F2-linear combinations of vectors in Sbelong to C. This means we have
(3.2) |C| ≥ 22n= 4n.
Now, combining (3.1) and (3.2) we get |C|= 4n=|C⊥|. Thus Cmust be a
cyclic self-dual code. Note that v·chas Lee weight 4 and it is easy to see that
there is no codeword of weight 2. Thus the minimum weight is 4. In fact more
can be said when nis even. Note that in that case the generators in Shave
weights divisible by 4, which means that the cyclic self-dual code of even length
obtained in this way is a Type II code.
Remark 3.9.For the R2-case, the previous result can easily be extended to
(1 + uv)-constacyclic self-dual codes as well. In general, it can be extended to
(1 + uiuj)-constacyclic self-dual codes.
Corollary 3.10. Putting n= 2,3,4,5in the theorem, thus taking (u, v),
(u, v, v),(u, v , v, v)and (u, v, v, v, v)as the base vectors, we obtain Type II
extremal binary self-dual codes of lengths 8and 16 and Type I extremal binary
self-dual codes of length 12 and 20 from both cyclic and constacyclic self-dual
codes over R2.
4. Projections, lifts and independence
Let µk:Rk→F2be defined as the projection modulo hu1, u2,...,uki. Then
µkpreserves orthogonality and cyclicity. We will need notions of independence
over the ring Rkand free codes. By a free code over Rkwe mean a code that
is generated as a free Rk-module with a basis. The row operations and the
properties of the ring Rkimplies that any free code over Rkcan be brought
to an equivalent form where it is generated by the rows of the matrix [Im|A],
with Imdenoting the m×midentity matrix. Let’s start with the following
definition:
Definition 4.1. Let S={c1, c2,...,cm}be a set of vectors in Rn
k. We say
that Sis linearly independent over Rkif α1c1+···+αmcm= 0 with αi∈Rk
implies αi= 0 for all i. Throughout the paper, when we say independent we
mean linearly independent.
Note that the rows of a standard generating matrix of a free code will be
independent. One should also observe that if Sis linearly independent over
Rk, then we must have ci6∈ Iu1,u2,...,uk. Thus each vector in Smust contain
some units. In other words 06∈ µk(S).
When µk(c) = x∈Fn
2, we will say cis an Rk-lift of the binary vector x.
When taking lifts we just replace 0 by any non-unit in Rkand 1 by any unit
in Rk.
The following theorem starts us on independence and pro jections:
CONSTACYCLIC SELF-DUAL CODES 1117
Theorem 4.2. Let S={c1, c2, . . . , cm}be a subset of Rn
1such that the pro-
jection µ1(S)is an independent subset of Fn
2. Then Sis independent over
R1.
Proof. Since µ1(S) is independent, 0 6∈ µ1(S). So all vectors in Sare free
vectors, meaning that they each have units in them. Now suppose that
α1c1+···+αmcm= 0
for αi∈R1. Apply the pro jection µ1to this equation and we get
µ1(α1)µ1(c1) + ···+µ1(αm)µ1(cm) = 0
in F2. Since µ1(S) is independent we must have µ1(αi) = 0 for all i=
1,2,...,m. But this means αi= 0 or ufor each i= 1,2,...,m. Now, if
αi= 0 for all i= 1,2,...,m, then we are done. So assume without loss
of generality that α1=α2=··· =αℓ=ufor some ℓ > 0. Then go-
ing back to the original equation we get u(c1+··· +cℓ) = 0 in R1, which
implies that c1+· · · +cℓ∈ {0, u}n. But then taking the pro jections yields
µ1(c1) + ···+µ1(cℓ) = 0 in Fn
2, contradicting the independence of µ1(S).
Considering the inductive construction of Rkfrom Rk−1and the correspond-
ing projections, we can extend the previous theorem to any Rk, using an in-
ductive argument:
Theorem 4.3. Let S={c1, c2, . . . , cm}be a subset of Rn
ksuch that the pro-
jection µk(S)is an independent subset of Fn
2. Then Sis independent over
Rk.
Note that the converse of the previous theorem is also true if the vectors
in Sare free vectors. Assume for example that S={c1, c2, . . . , cm}is an
independent subset of Rn
kwith µk(ci)6= 0, and let
α1µk(c1) + ···+αmµk(cm) = 0
be given with αi∈F2.Now if αi= 0 for all i= 1,2,...,m, then we are done.
So assume without loss of generality that α1=α2=··· =αℓ= 1 for some
ℓ > 0. Then the equation would reduce to
µk(c1) + ···+µk(cℓ) = 0,
which implies that
c1+c2+···+cℓ∈In
u1,u2,...,uk.
But this would imply that
u1u2...uk(c1+c2+···+cℓ) = 0
in Rn
k, contradicting the independence of S. Thus we have the following useful
corollary:
Corollary 4.4. Suppose that Cis a free cyclic or a λ-constacyclic self-dual
code over Rkof length n. Then nmust be even and µk(C)must be a cyclic
self-dual binary code.
1118 S. KARADENIZ, I. G. KELEBEK, AND B. YILDIZ
Because of the properties of µk, we know that if Cis a cyclic or λ-constacyclic
code over Rk, then µk(C) is a binary cyclic code. Since µkpreserves duality
as well, we can easily say that if Cis a cyclic or λ-constacyclic self-dual code
over Rk, then µk(C) is a self-orthogonal binary cyclic code. Of course in many
cases µk(C) might turn out to be zero.
Now assume that C=hf(x)iis a cyclic or constacyclic self-dual code over
Rkof length n. First of all, by [9], we know that µk(f(x)) cannot be relatively
prime to xn−1. But in fact we can say more.
Theorem 4.5. Suppose that C=hf(x)iis a cyclic or a λ-constacyclic self-dual
code over Rkof length n, for some unit λin Rk. Then
Rank (hµk(f(x))i)≤n
2.
Proof. If C=hf(x)iis a cyclic or a λ-constacyclic self-dual code over Rk
of length n, then µk(C) = hµk(f(x))iis a self-orthogonal binary code. This
means µk(C)⊆µk(C)⊥. But, since |µk(C)| · |µk(C)⊥|= 2n, we must have
|µk(C)|2≤2n, leading to the required assertion.
Thus we have the following quite useful corollary:
Corollary 4.6. Suppose that C=hf(x)iis a cyclic or a λ-constacyclic self-
dual code over Rkof length n. If GCD(µk(f(x)), xn−1) = d(x), then we must
have deg(d(x)) ≥n/2.
5. Computational results
We begin with a few examples of general constructions and then proceed to a
systematic construction of cyclic self-dual codes over R1of non-trivial lengths.
The constructions have all been carried out using Magma Computational Al-
gebra ([5]).
Example 5.1. Let f(x) = 1 + ux +ux2+ux3+x4+vx5+uvx6+vx7. The
cyclic code C=hf(x)iof length 8 generated over R2is a cyclic self-dual code
of length 8 whose binary image is the extremal Type II code of length 32 with
weight enumerator
WC(z) = 1 + 620z8+ 13888z12 + 36518z16 + 13888z20 + 620z24 +z32
with an automorphism group of order 215 ∗32∗5∗7.
Example 5.2. Let f(x) = x2+x3+x4+ux5+ (1 + u)x6be a polynomial
in R1[x]. The (1 + u)-constacyclic code generated by f(x) is a constacyclic
self-dual code of length 7 whose binary image is an extremal Type I code of
length 14 and weight enumerator
1 + 14z4+ 49z6+ 49z8+ 14z10 +z14 .
CONSTACYCLIC SELF-DUAL CODES 1119
Cyclic self-dual codes over R1=F2+uF2of odd lengths have been charac-
terized and studied in [3]. Since in [4] they also gave a decoding algorithm for
cyclic codes of over R1of odd lengths, they have been of interest to study. We
will now focus on free cyclic self-dual codes over R1of even length. In doing so
we will make use of Theorem 4.2 together with the characterization of binary
cyclic self-dual codes that can be found in [11]. Thus, we know that if Cis a
free cyclic self-dual code of length nover R1, then nmust be even and µ1(C)
must be a binary cyclic self-dual code. Taking binary cyclic self-dual codes of
even length nwith high minimum weight, we can obtain a cyclic self-dual code
over R1of high minimum distance from the lifts. The following theorem, which
can be found in [14] narrows the search field considerably:
Theorem 5.3. Suppose Cis a linear code over F2+uF2and that C′=µ1(C)
is its projection to F2with µ1(C)6= 0. If dand d′represent the minimum Lee
and Hamming distances of Cand C′respectively, then we have d≤2d′.
More generally, if dis the minimum Lee weight of a linear code Cover Rk
and d′is the minimum hamming weight of µk(C)6= 0, then d≤2kd′.
With this in mind, looking at the binary cyclic self-dual codes from [11], we
see that the lengths we should study are 14, 28, 30, 42 and 46.
5.1. Cyclic self-dual codes over R1of length 14
Taking f(x) = u+ux +ux2+ux3+ux4+ux5+x6+x7+ux8+ux9+
x10 +x11 +x12 +x13, the cyclic code generated by f(x) over R1is a cyclic
self-dual code of length 14 and minimum Lee weight 4. The Gray image is then
a 2-quasi-cylic self-dual code of length 28 and minimum distance 4.
5.2. Cyclic self-dual codes over R1of length 28
In [11], it is shown that there are two non-trivial binary cyclic self-dual codes
of length 28, which we lift to obtain cyclic self-dual codes of length 28 over R1.
The lifts of the first one all have minimum Lee distance 4, which we discard
as the binary images then would be binary codes of length 56 and minimum
distance 4. However the search through lifts of the second generator turns
out to be more fruitful. We obtain codes with 6 different weight enumerators
which we describe below. To save space, 1 + uis replaced by 3 in writing the
generators.
Table 1. Cyclic self-dual codes over R1of length 28
Generator Type Partial Weight Enumerator
133u2103120212010u02Type II 1 + 147z8+ 9072z12 + 610113z16 +···
1321u23u131021201110 Type I 1 + 56z6+ 315z8+ 1512z10 + 7840z12 +···
1321u2103120212012uType II 1 + 315z8+ 10080z12 + 596169z16 +···
1330210130212010u0uType I 1 + 147z8+ 224z10 + 5040z12 +···
12310u3u312021209u202Type I 1 + 427z8+ 10752z12 + 16384z14 +···
133023u3210212010u20 Type II 1 + 427z8+ 10752z12 + 586873z16 +···
1120 S. KARADENIZ, I. G. KELEBEK, AND B. YILDIZ
5.3. Cyclic self-dual codes over R1of length 30
Our search through lifts of the only non-trivial binary cyclic self-dual code of
length 30 have resulted in the following Type I codes all of which have minimum
Lee distance 8:
Table 2. Cyclic self-dual codes over R1of length 30
Generator Partial Weight Enumerator
1201231u0u1u013013u1 + 60z8+ 180z10 + 2975z12 + 33720z14 +···
13u3212u023u013011u021 + 150z8+ 216z10 + 4245z12 + 32400z14 +···
13012320u23u013010u031 + 60z8+ 396z10 + 2975z12 + 31560z14 +···
130133u2030213010u02u1 + 690z8+ 11865z12 + 28800z14 +···
5.4. Cyclic self-dual codes over R1of length 42
In [11] 9 different binary cyclic self-dual codes of length 42 were found.
One of these is trivial which we have discarded. The remaining 8 are grouped
into reciprocal pairs. Thus we considered the four possible generators to lift
to R1. Only in two of the generators the search resulted in codes with high
minimum distances. The generators to lift are gen1 = 140210130210130212020
and gen2 = 1201013010414013020 . From the lifts of gen1 we found 20 cyclic
self-dual codes over R1of length 42 and minimum Lee distance 8 and from lifts
of gen2 we managed to find 5 cyclic self-dual codes over R1of length 42 and
minimum Lee distance 8. We tabulate these results together with the partial
weight distributions.
Table 3. Cyclic self-dual codes over R1of length 42 from the
lifts of gen1
Generator Partial Weight Enumerator
1321u230130u10130212019u1 + 21z8+ 637z12 + 1224z14 +···
1321u03u13u030130212018u0 1 + 21z8+ 217z12 + 2688z14 +···
140u1u13u230130212018u21 + 21z8+ 252z10 + 301z12 + 1008z14 +···
133u230130u10130212017u021 + 21z8+ 217z12 + 3408z14 +···
1330u1u13u230130212017u20 1 + 21z8+ 217z12 + 1512z14 +···
133u03u13u030130212017u31 + 21z8+ 217z12 + 3360z14 +···
1232021u131u010130212016u0u0 1 + 21z8+ 217z12 + 2232z14 +···
12310u101310u30130212016u2021 + 21z8+ 217z12 + 3024z14 +···
1312u23u131u210130212016u30 1 + 21z8+ 553z12 + 2904z14 +···
1231021u131u010130212016u41 + 21z8+ 84z10 + 637z12 + 4752z14 +· · ·
133021u13u230130212015u02u21 + 567z8+ 21679z12 + 384z14 +···
1321u030130u10130212015u0u021 + 21z8+ 84z10 + 637z12 + 5040z14 +· · ·
13120u1u130210130212013u05u1 + 21z8+ 217z12 + 2400z14 +···
1312021013u230130212013u04u0 1 + 21z8+ 217z12 + 840z14 +···
1231u23013u030130212013u04u21 + 21z8+ 637z12 + 2520z14 +···
133021u1310u30130212013u02u031 + 21z8+ 217z12 + 2352z14 +···
133u23u1310230130212013u02u02u1 + 21z8+ 553z12 + 3192z14 +···
14u23u1310230130212013u02u2021 + 21z8+ 637z12 + 1176z14 +···
13210210131u010130212013u0u20u21 + 21z8+ 637z12 + 2184z14 +···
13210u3u130210130212013u20u02u1 + 21z8+ 301z12 + 5760z14 +···
CONSTACYCLIC SELF-DUAL CODES 1121
Table 4. Cyclic self-dual codes over R1of length 42 from the
lifts of gen2
Generator Partial Weight Enumerator
12030130303u1312013019u1 + 84z8+ 1540z12 + 3024z14 +···
13u3u3210303u3131013018u0 1 + 84z8+ 1540z12 + 2352z14 +···
13u1u3210104131013018uu 1 + 84z8+ 252z10 + 1540z12 + 11472z14 +···
13u10312u102u23212013017 u20 1 + 210z8+ 5026z12 + 7776z14 +···
1301u33u1u0u21312013015u02u21 + 336z8+ 21112z12 + 424284z16 + 225792z18 +···
Remark 5.4.The last entry in Table 4 is an interesting example of a code as
the first non-zero weight not divisible by 4 is 18. It seems to have few non-zero
weights, which would make it of interest for combinatorial reasons.
5.5. Cyclic self-dual codes over R1of length 46
Our search through lifts of the non-trivial binary cyclic self-dual code of
length 46 have all resulted in Type I codes of minimum Lee distance 8, meaning
that the Gray images are 2-quasi-cyclic self-dual codes of parameters [92,46,8].
We just give one such example together with its weight enumerator. Let f(x)
have coefficients as 14061602120212022. Then the cyclic code generated by f(x)
is a cyclic self-dual code of length 46 over R1whose partial weight enumerator
is given by 1 + 2024z8+ 5152z12 + 128018z14 +···.
6. A remark about Theorem 4.14 in [1]
Batoul et al. claimed in [1] that a self-dual code of even length over a chain
ring with residue field F2cannot be the lift of a binary cyclic self-dual code.
What we have done in Section 4 and Section 5 contradict this claim. Because
all the self-dual codes we obtained in Section 5 are of even length and they are
all lifts of binary cyclic self-dual codes.
The reason why this claim is false can be seen in the proof of Theorem 4.14
in [1]. In the proof they use the statement that “the residue code of a self-dual
code over such a ring must be a doubly even binary code”. While this statement
is certainly true for the ring Z4(since the characteristic is 4, self-dual Z4-code
means the number of units in each row is divisible by 4, thus the residue code
is indeed doubly even), it is not true for the case F2+uF2, which is also a chain
ring with residue field F2. Since the characteristic is 2, the residue code of a
self-dual code over F2+uF2does not have to be doubly even as can be seen in
many of the examples above.
References
[1] A. Batoul, K. Guenda, and T. A. Gulliver, On self-dual cyclic codes over finite chain
rings, Des. Codes Cryptogr. 70 (2014), no. 3, 347–358.
[2] T. Blackford, Isodual constacyclic codes, Finite Fields Appl. 24 (2013), 29–44.
[3] A. Bonnecaze and P. Udaya, Cyclic codes and Self-dual codes over F2+uF2, IEEE
Trans. Inform. Theory 45 (1999), no. 4, 1250–1255.
1122 S. KARADENIZ, I. G. KELEBEK, AND B. YILDIZ
[4] , Decoding of cyclic codes over F2+uF2, IEEE Trans. Inform. Theory 45 (1999),
no. 6, 2148–2157.
[5] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user lan-
guage, J. Symbolic Comput. 24 (1997), 235–265.
[6] B. Chan and H. Q. Dinh, A note on isodual constacyclic codes, Finite Fields Appl. 29
(2014), 243–246.
[7] S. T. Dougherty, P. Gaborit, M. Harada, and P. Sol´e, Type II codes over F2+uF2, IEEE
Trans. Inform. Theory 45 (1999), no. 1, 32–45.
[8] S. T. Dougherty, S. Karadeniz, and B. Yildiz, Cyclic codes over Rk, Des. Codes Cryp-
togr. 63 (2012), no. 1, 113–126.
[9] S. T. Dougherty, B. Yildiz, and S. Karadeniz, Codes over Rk, Gray maps and their
binary images, Finite Fields Appl. 17 (2011), no. 3, 205–219.
[10] ,Self-dual codes over Rkand binary self-dual codes, Eur. J. Pure Appl. Math.
6(2013), no. 1, 89–106.
[11] Y. Jia, S. Ling, and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans.
Inform. Theory 57 (2011), no. 4, 2243–2251.
[12] S. Karadeniz, S. T. Dougherty, and B. Yildiz, Constructing formally self-dual codes over
Rk, Discrete Appl. Math. 167 (2014), 188–196.
[13] S. Karadeniz and B. Yıldız, (1 + v)-constacylic codes over F2+uF2+vF2+uvF2, J.
Franklin Inst. 348 (2011), no. 9, 2625–2632.
[14] S. Karadeniz, B. Yıldız, and N. Aydın, Extremal binary self-dual codes of lengths 64
and 66 from four-circulant constructions over codes F2+uF2, FILOMAT to appear.
[15] V. Pless, P. Sole, and Z. Qian, Cyclic self-dual Z4-codes, Finite Fields Appl. 3(1997),
no. 1, 48–69.
[16] N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes, IIEEE Trans. Inform.
Theory 29 (1983), no. 3, 364–366.
[17] J. Wood, Duality for modules over finite rings and applications to coding theory, Amer.
J. Math. 121 (1999), no. 3, 555–575.
[18] B. Yıldız and S. Karadeniz, Cyclic codes over F2+uF2+vF2+uvF2, Des. Codes
Cryptogr. 58 (2011), no. 3, 221–234.
Suat Karadeniz
Department of Mathematics
Fatih University
34500, Istanbul, Turkey
E-mail address:skaradeniz@fatih.edu.tr
Ismail Gokhan Kelebek
Department of Mathematics
Fatih University
34500, Istanbul, Turkey
E-mail address:gkelebek@fatih.edu.tr
Bahattin Yildiz
Department of Mathematics
Fatih University
34500, Istanbul, Turkey
E-mail address:byildiz@fatih.edu.tr
