An efficient method to construct self-dual cyclic codes of length ps
over Fpm+uFpm
Yuan Caoa, b, c, Yonglin Caoa, ∗, Hai Q. Dinhd, e, Somphong Jitmanf
aSchool of Mathematics and Statistics,
Shandong University of Technology, Zibo, Shandong 255091, China
bHubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China
cHunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, Hunan 410114, China
dDivision of Computational Mathematics and Engineering, Institute for Computational
Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
eFaculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City,
Vietnam
fDepartment of Mathematics, Faculty of Science, Silpakorn University, Nakhon
Pathom 73000, Thailand
Abstract
Let p be an odd prime number, Fpm be a finite field of cardinality pm and s a positive integer.
Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over Fp with a specific type.
On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length ps over the finite chain ring Fpm+uFpm (u2=0). Moreover,
We provide
an efficient method to construct every self-dual cyclic code of length ps over Fpm+uFpm
precisely.
keywords:
Cyclic code; Self-dual code; Linear code; Kronecker product of matrices; Finite chain ring
Mathematics Subject Classification (2000) 94B15, 94B05, 11T71
††journal: XXX
1 Introduction
The class of self-dual codes is an interesting topic in coding theory due to
their connections to other fields of mathematics such as Lattices, Cryptography, Invariant Theory, Block designs, etc.
An effective way for the construction of self-dual codes is the use of some specific algebraic structures.
Let Fpm be a finite field of pm elements, where p is a prime number, and denote
R=⟨u2⟩Fpm[u]=Fpm+uFpm (u2=0).
Then R is a finite chain ring and every invertible element in R is of the form: a+bu, a,b∈Fpm and a=0.
Let N be a fixed positive integer and
RN={(a0,a1,…,aN−1)∣a0,a1,…,aN−1∈R}
Then RN is an R-free module with the usual componentwise addition and scalar multiplication by elements of R.
Let C be an
R-submodule of RN and λ be an invertible element in R. Then C is called a linear code over
R of length N. Moreover, C is called a λ-constacyclic code if
[TABLE]
In particular, a λ-constacyclic code C is called a negacyclic code when λ=−1, and
C is called a cyclic code when λ=1.
Let ⟨xN−λ⟩R[x]={∑i=0N−1aixi∣a0,a1,…,aN−1∈R}
in which the arithmetic is done modulo xN−λ.
In this paper, λ-constacyclic codes over
R of length N are identified with ideals of the ring ⟨xN−λ⟩R[x], under the
identification map θ:RN→⟨xN−λ⟩R[x] defined by
θ:(a0,a1,…,aN−1)↦a0+a1x+…+aN−1xN−1 for all ai∈R and i=0,1,…,N−1.
The Euclidean inner
product on RN is defined by
[α,β]=∑i=0N−1aibi∈R
for
all α=(a0,a1,…,aN−1),β=(b0,b1,…,bN−1)∈RN. Then
the (Euclidean) dual code of a linear code C over R of length N is defined by
[TABLE]
which is also a linear code over R of length N. In particular, C is said to be
(Euclidean) self-dual if C⊥=C.
There were a lot of literature on linear codes, cyclic codes and
constacyclic codes of length N over rings Fpm+uFpm (u2=0) for various prime p, positive integer m and some positive integer N (see [1], [2], [4], [5] and [7]–[20], for examples).
Specifically,
all constacyclic codes of length 2s over the Galois extension
rings of F2+uF2 were classified and their detailed structures was also established in [11]. Dinh [12]
classified all constacyclic codes of length ps over Fpm+uFpm. Then
negacyclic codes of length 2ps, constacyclic codes of length 2ps and
constacyclic codes of length 4ps (pm≡1 (mod 4)) over Fpm+uFpm
were investigated by Dinh et al. [13], Chen et al. [9] and Dinh et al. [14], respectively.
We note that the representation and enumeration for self-dual cyclic codes and self-dual negacyclic codes were not studied in these papers.
Dinh et al. [15] determined the algebraic structures of all cyclic and negacyclic codes
of length 4ps over Fpm+uFpm, established the duals of all such codes and gave some special subclass of self-dual negacyclic codes of length 4ps over Fpm+uFpm by Theorems 4.2, 4.4
and 4.9 of [15]. But the representation and enumeration for all self-dual negacyclic codes and all self-dual cyclic codes
were not given.
Choosuwan et al. [6] done the following:
♢ In pages 9 and 10, they proved that every
(Euclidean) self-dual cyclic code over Fpm+uFpm
of length ps is given by
[TABLE]
where ps=i0+i1 and h=(h0,h1,…,hi1−1)tr∈Fpmi1
satisfying
[TABLE]
in which M(ps,i1) is an i1×i1 matrix over Fpm defined by
[TABLE]
♢ Using Theorem 3.3 of [21] for the nullity of
M(ps,i1), they obtained a formula to
count the number of self-dual cyclic codes over Fpm+uFpm
of length ps (cf. Corollary 22 of [6]), where p is an arbitrary prime number.
Also Dinh et al. determined
the number of self-dual cyclic codes of length ps over Fpm+uFpm
(u2=0) by Section 4 of [10].
But they didn’t give a method how to solve the equation
M(ps,i1)h=0 and didn’t obtain an representation for
solutions of this equation in [6] and the equation (2.1) in [10]. So they didn’t provide an explicit
representation for all distinct self-dual cyclic codes over Fpm+uFpm
of length ps.
In [7], we provided a new way different
from the methods used in [9]–[16] to determine the algebraic structures,
generators and enumeration of λ-constacyclic codes over Fpm+uFpm of length
nps, where n is an arbitrary positive integer satisfying gcd(p,n)=1 and λ∈Fpm×.
Then we gave an explicit representation for the dual code of every cyclic
code and every negacyclic code. Moreover, we provided a discriminant condition for the self-duality
of each cyclic code and negacyclic code over Fpm+uFpm of length
nps. On the basis of [7], we can consider to give an explicit representation for self-dual cyclic
codes and self-dual negacyclic codes over Fpm+uFpm.
Recently,
by a new way different from that of [6],
we [8] gave an efficient method for the construction of all distinct self-dual cyclic codes with length 2s over F2m+uF2m. In particular, we provide an exact formula to count the number of all these self-dual cyclic codes
and corrected a mistake in Corollary 22(ii) of [6]. However, the methods and results of [8] depend heavily on that
the characteristic of the field F2m is 2.
They can’t be used directly to the case for self-dual cyclic codes with length ps over Fpm+uFpm
where p is odd. Hence we need to develop a new approach to the latter situation.
The present paper is organized as follows. In Section 2, we review the
known results for self-dual cyclic codes of length ps over Fpm+uFpm and prove that these self-dual cyclic codes are determined by a special kind of subsets Ωl in the
residue class ring ⟨(x−1)l⟩Fpm[x] for certain integers l, 1≤l≤ps−1. In Section 3,
we give an explicit representation of the set Ωl by studying properties for
Kronecker product of matrices over Fp with a specific type. In Section 4, we provide an efficient method to construct
and represent all distinct self-dual cyclic codes of length ps over Fpm+uFpm
precisely. As an application, we list all distinct self-dual cyclic codes over F3m+uF3m
of length 3s for s=1,2,3 in Section 5.
Section 6 concludes the paper.
2 Preliminaries
In this section, we list some known results for cyclic codes of length ps over the ring Fpm+uFpm (u2=0)
needed in the following sections.
By Corollary 7.1 in [7], every cyclic code C over Fpm+uFpm of length
ps and its dual code C⊥ are
given by the following five cases.
Case I.
(pm)ps−⌈2ps⌉=p2ps−1m codes:
C=⟨(x−1)b(x)+u⟩ with ∣C∣=ppsm and
C⊥=⟨(x−1)⋅x−1b(x−1)+u⟩,
where b(x)∈(x−1)2ps−1⋅⟨(x−1)ps−1⟩Fpm[x].
Case II.
∑k=1ps−1p(ps−k−⌈21(ps−k)⌉)m codes:
C=⟨(x−1)k+1b(x)+u(x−1)k⟩ with ∣C∣=p(ps−k)m
and
[TABLE]
where b(x)∈(x−1)⌈2ps−k⌉−1⋅⟨(x−1)ps−k−1⟩Fpm[x]
and 1≤k≤ps−1.
Case III.
ps+1 codes:
C=⟨(x−1)k⟩ with ∣C∣=p2(ps−k)m and C⊥=⟨(x−1)ps−k⟩,
where 0≤k≤ps.
Case IV.
∑t=1ps−1p(t−⌈2t⌉)m codes:
C=⟨(x−1)b(x)+u,(x−1)t⟩ with ∣C∣=p(2⋅ps−t)m
and C⊥=⟨(x−1)ps−t+1⋅x−1b(x−1)+u(x−1)ps−t⟩,
where b(x)∈(x−1)⌈2t⌉−1⋅⟨(x−1)t−1⟩Fpm[x]
and 1≤t≤ps−1.
Case V.
∑k=1ps−2∑t=1ps−k−1p(t−⌈2t⌉)m codes:
C=⟨(x−1)k+1b(x)+u(x−1)k,(x−1)k+t⟩ with ∣C∣=p(2⋅ps−2k−t)m
and C⊥=⟨(x−1)ps−k−t+1⋅x−1b(x−1)+u(x−1)ps−k−t,(x−1)ps−k⟩,
where b(x)∈(x−1)⌈2t⌉−1⋅⟨(x−1)t−1⟩Fpm[x],
1≤t≤ps−k−1 and 1≤k≤ps−2.
As ∣(Fpm+uFpm)ps∣=(ppsm)2, every self-dual cyclic
code C over Fpm+uFpm of length ps must contain
∣C∣=ppsm codewords. From this, we deduce that there is no self-dual codes in
Cases II, III and IV.
Let C=⟨(x−1)b(x)+u⟩ be a code in Case I. Then C=C⊥ if and only if
b(x)∈(x−1)2ps−1⋅⟨(x−1)ps−1⟩Fpm[x] satisfying b(x)=x−1b(x−1), i.e.,
[TABLE]
Let C=⟨(x−1)k+1b(x)+u(x−1)k,(x−1)k+t⟩ be a code in Case V.
Then C=C⊥ if and only if 2⋅ps−2k−t=ps and b(x)∈(x−1)⌈2t⌉−1⋅⟨(x−1)t−1⟩F2m[x] satisfying b(x)=x−1b(x−1). The latter is equivalent to
[TABLE]
The former is equivalent to that
[TABLE]
In the light of the above discussion, we have the following conclusion.
Proposition 1 For any integer l, 1≤l≤ps−1, we denote
[TABLE]
Then all distinct self-dual cyclic codes over Fpm+uFpm of length
ps are given by the following two cases:
(i)
⟨(x−1)b(x)+u⟩, where b(x)=∑i=2ps−1ps−2bi(x−1)i∈Ωps−1.
(ii)
⟨(x−1)k+1b(x)+u(x−1)k,(x−1)ps−k⟩*,
where
1≤k≤2ps−1 and b(x)=∑i=2ps−1−kps−2k−2bi(x−1)i∈Ωps−1−2k*.
In order to present all self-dual cyclic codes over Fpm+uFpm of length
ps explicitly, by Proposition 1 we need to determine the following subsets of
⟨(x−1)l⟩Fpm[x]:
[TABLE]
Let A=(aij) and B be matrices over Fpm of sizes k×t and l×v respectively.
The Kronecker product of A and B is
defined by A⊗B=(aijB) which is a a matrix over Fpm of size kl×tv.
For any positive integer λ≤s, we define a pλ×pλ lower triangular matrix
Gpλ over Fp as follows
[TABLE]
where
[TABLE]
In fact, we have Gpλ=(−1)ps−pλM(ps,pλ) (cf. [6]). Precisely, we have
[TABLE]
(mod p), where \left(\begin{array}[]{c}0\cr 0\end{array}\right)=1. Moreover, we have the following property for the matrix Gpλ.
Proposition 2 Let λ be any positive integer and set Gp0=1. Then
[TABLE]
where
g_{i,i}^{(p)}=(-1)^{i-1}\left(\begin{array}[]{c}p-i\cr i-i\end{array}\right)=(-1)^{i-1},\ i=1,2,\ldots,p.
Proof. Let 1≤j≤i≤p and 1≤l≤k≤pλ−1, where λ≥2.
As pλ−1 is odd, we have (−1)(j−1)pλ−1+(l−1)=(−1)(j−1)+(l−1). Then by Lucas’s Theorem for
a combinatorial identity in number theory
(see [3], for examples),
we have
[TABLE]
From these, by Equation (2),
[TABLE]
and ((i−1)pλ−1+k)−((j−1)pλ−1+l)=(i−j)pλ−1+(k−l), we deduce
[TABLE]
This implies that Gpλ=Gp⊗Gpλ−1.
□
For examples, we have G_{3}=\left(\begin{array}[]{ccc}1&0&0\cr-1&-1&0\cr 1&-1&1\end{array}\right)
where −1=2, and
[TABLE]
3 Calculation and representation of the set Ωl
In this section, we consider how to calculate and represent the subset Ωl of
⟨(x−1)l⟩Fpm[x] defined by Equation (1), where 1≤l≤ps.
For any matrix A over Fpm and positive integer l, let Atr be the transpose of A
and Il be the identity matrix of order l. In the rest of this paper, we adopt
the following notation
[TABLE]
Then we have
[TABLE]
In order to present the
subset Ωl of polynomials in ⟨(x−1)l⟩Fpm[x], it is equivalent to determine the set Sl.
Theorem 1
Let 1≤l≤ps−1 and assume λ be the least positive integer
such that 1≤l≤pλ.
Let Gl be the
submatrix in the upper left corner of
Gpλ defined by
[TABLE]
where Gl is a lower triangular matrix over Fp of size l×l. Then we have the following
conclusions.
(i)
Gpλ2=Ipλ.
(ii)
Gl2=Il, rank(Gl−Il)=⌊2l⌋
and rank(Gl+Il)=⌈2l⌉.
(iii)
Denote x−1=xpλ−1 (mod (x−1)l). For any Bl=(b0,b1,…,bl−1)tr∈Fpml, we denote b(x)=∑i=0l−1bi(x−1)i. Then
[TABLE]
(iv)
Let Υ1,Υ2,…,Υl be the column vectors of Gl+Il, i.e.,
Υi∈Fpl for all i and Gl+Il=(Υ1,Υ2,…,Υl). Then
{Υ2t−1∣t=1,2,…,⌈2l⌉} is an
Fpm-basis of Sl. Precisely, we have
dimFpm(Sl)=⌈2l⌉ and
[TABLE]
Proof. (i) First, we prove that Gp2=Ip over Fp.
Since Gp is a lower triangular matrix of order p,
Gp2 is a lower triangular matrix and its
(i,i)-entry is
[TABLE]
by Equation (2) in Section 2. Now, let 1≤j≤p−1 and i=j+k where 1≤k≤p−j. By the
definition of Gp and the following combinatorial identities
[TABLE]
and \sum_{h=0}^{k}(-1)^{h}\left(\begin{array}[]{c}k\cr h\end{array}\right)=0, the (i,j)-entry of Gp2 is equal to
[TABLE]
As stated above, we conclude that Gp2=Ip. Now, let λ≥2 and assume that Gpλ−12=Ipλ−1
(mod p). Then by Propostion 2, it follows that
[TABLE]
(ii) By (i) and Equation (7) for the definition of Gl, it follows that
[TABLE]
This implies Gl2=Il, and hence (Gl−Il)(Gl+Il)=0. From this and by linear algebra theory,
we deduce that
[TABLE]
Since Gpλ is a lower triangular matrix of order pλ with diagonal entries:
[TABLE]
by Equation (7) we see that
Gl−Il is a lower triangular matrix with ⌊2l⌋ nonzero
diagonal entries:
[TABLE]
and
Gl+Il is a lower triangular matrix with ⌈2l⌉ nonzero
diagonal entries:
[TABLE]
These imply rank(Gl−Il)≥⌊2l⌋
and rank(Gl+Il)≥⌈2l⌉.
From these and by ⌊2l⌋+⌈2l⌉=l, we deduce
that rank(Gl−Il)=⌊2l⌋
and rank(Gl+Il)=⌈2l⌉.
(iii) In the following, we denote
[TABLE]
for any integer l: 1≤l≤pλ. Then b(x)=XlBl. By xpλ≡1 (mod (x−1)pλ)
in Fpm[x], it follows that
[TABLE]
From this and by Equation (2), we deduce that
[TABLE]
for all j=1,2,…,pλ. These imply
[TABLE]
by the definition of the matrix Gpλ in Section 2.
As l≤pl, by Equations (15) and (7) it follows that
[TABLE]
(iv) Let Hl be the solution space of following linear equations over Fpm:
[TABLE]
By (ii) we have dimFpm(Hl)=l−rank(Gl−Il)=l−⌊2l⌋=⌈2l⌉.
By (ii) we know that rank(Gl+Il)=⌈2l⌉. This implies
that ⌈2l⌉ is the rank of the vectors Υ1,Υ2,…,Υl.
From this, by
[TABLE]
and G_{p^{\lambda}}+I_{p^{\lambda}}=\left(\begin{array}[]{cccccc}2&&&&&\cr\ast&0&&&&\cr\ast&\ast&2&&&\cr\vdots&\vdots&\vdots&\ddots&&\cr\ast&\ast&\ast&\ldots&0&\cr\ast&\ast&\ast&\ldots&\ast&2\end{array}\right) we
deduce that the set of column vectors
{Υ2t−1∣t=1,2,…,⌈2l⌉}
is a
maximal independent system of Υ1,Υ2,…,Υl.
On the other hand, by (Gl−Il)(Gl+Il)=Gl2−Il=0 and
Gl+Il=(Υ1,Υ2,…,Υl), it follows that
[TABLE]
This implies Υi∈Hl for all i=1,2,…,l.
Then by dimFpm(Hl)=⌈2l⌉, we conclude that
{Υ2t−1∣t=1,2,…,⌈2l⌉} is an Fpm-basis of
Hl.
Now, let’s prove that Sl=Hl.
Let b(x)∈⟨(x−1)l⟩Fpm[x] where 1≤l≤pλ. Then b(x) has a unique (x−1)-expansion:
[TABLE]
From this and by (iii), we deduce that
[TABLE]
Therefore, by Equations (1) and (6) we see that
[TABLE]
As stated above. we conclude that Sl=Hl.
□
4 Construction and enumeration for self-dual cyclic codes
In this section, we determine all self-dual cyclic codes of length ps over Fpm+uFpm (u2=0).
Theorem 2
For any integers l and δ, 0≤δ<l≤ps−1, we denote
[TABLE]
and write
Gl+Il=(Υ1,Υ2,…,Υl),
where Υj∈Fpml for all j=1,2,…,l. Moreover, for any integer j,
⌈2δ⌉+1≤j≤⌈2l⌉, we set
[TABLE]
Then
[TABLE]
Hence ∣Sl[δ]∣=p(⌈2l⌉−⌈2δ⌉)m.
Proof. By Equation (7) and the proof of Theorem 1(iv), we have
[TABLE]
where 0(2j−2)×1 is the zero matrix of type (2j−2)×1 and gi,j∈Fp.
Let Bl=(b0,b1,…,bl−1)tr∈Fpml. Since {Υ2t−1∣t=1,2,…,⌈2l⌉} is an
Fpm-basis of Sl by Theorem 1(iv), we see that
Bl∈Sl if and only if there is a unique
row vector (a0,a2,…,a2⌈2l⌉−2) over Fpm such that
[TABLE]
From this we deduce that
[TABLE]
This implies
[TABLE]
Finally, by Equation (21) and
[TABLE]
we conclude that {Υ2j−1[δ;l)∣⌈2δ⌉+1≤j≤⌈2l⌉}
is a linear independent set of vectors in Fpml−δ.
Therefore, dimFpm(Sl[δ])=⌈2l⌉−⌈2δ⌉ and hence
∣Sl[δ]∣=p(⌈2l⌉−⌈2δ⌉)m.
□
Now, we list explicitly all self-dual cyclic codes of length ps over Fpm+uFpm (u2=0)
by the following theorem.
Theorem 3
Using the notation in Theorem 2, let p be any odd prime number and s be any positive integer. Then
we have the following conclusions.
(†)
If ps≡3 (mod 4), all
distinct self-dual cyclic codes of length ps over Fpm+uFpm are given by the following two cases.
∙ ∑ν=04ps+1−1(pm)4ps+1−1−ν* codes:*
[TABLE]
where b(x)=∑i=2ps−1−2νps−1−4ν−1bi(x−1)i with
[TABLE]
for a2j−2∈Fpm arbitrary and 0≤ν≤4ps+1−1.
∙ ∑ν=04ps+1−1(pm)4ps+1−1−ν* codes:*
[TABLE]
where b(x)=∑i=2ps−1−2ν−1ps−1−4ν−3bi(x−1)i with
[TABLE]
for a2j−2∈Fpm arbitrary and 0≤ν≤4ps+1−1.
In this case, the number of all self-dual cyclic codes of length ps over Fpm+uFpm
is equal to
[TABLE]
(‡)
If ps≡1 (mod 4), all
distinct self-dual cyclic codes of length ps over Fpm+uFpm are given by the following three cases:
∙ (pm)4ps−1* codes*:
[TABLE]
where
b(x)=∑i=2ps−1ps−2bi(x−1)i with
[TABLE]
for a2j−2∈Fpm arbitrary.
∙ ∑ν=14ps−1(pm)4ps−1−ν* codes:*
[TABLE]
where b(x)=∑i=2ps−1−2νps−1−4ν−1bi(x−1)i with
[TABLE]
for a2j−2∈Fpm arbitrary and 1≤ν≤4ps−1.
∙ ∑ν=14ps−1(pm)4ps−1−ν* codes:*
[TABLE]
where b(x)=∑i=2ps−1−2ν+1ps−4νbi(x−1)i with
[TABLE]
for a2j−2∈Fpm arbitrary and 1≤ν≤4ps−1.
In this case, the number of all self-dual cyclic codes of length ps over Fpm+uFpm
is equal to
[TABLE]
Proof.
By Proposition 1, all distinct
self-dual codes of length ps over Fpm+uFpm are given as follows:
[TABLE]
where
0≤k≤2ps−1 and b(x)=∑i=2ps−1−kps−2−2kbi(x−1)i∈Ωps−1−2k.
Let b(x)=∑i=2ps−1−kps−2−2kbi(x−1)i where
bi∈Fpm for all i=2ps−1−k,…,ps−2−2k. Then by Equations (6)
and (20), it follows that
[TABLE]
Moreover, by Theorem 2 we have
∣Sps−1−2k[2ps−1−k]∣=(pm)⌈2ps−1−2k⌉−⌈22ps−1−k⌉.
Hence the number of all self-dual cyclic codes of length ps over Fpm+uFpm
is equal to NE(Fpm+uFpm,ps)=∑k=02ps−1∣Sps−1−2k[2ps−1−k]∣.
Now, we have the following two situations.
(†) Let ps≡3 (mod 4). Then 2ps−1 is odd and 22ps−1+1=4ps+1 is a positive integer. In this situation, we have one of the following two cases:
(†-i) Let k=2ν be even, where 0≤ν≤4ps+1−1. Then
[TABLE]
and ⌈2ps−1−4ν⌉−⌈22ps−1−2ν⌉=2ps−3+1−2ν−(4ps−3+1−ν)=4ps−3−ν.
From these, we deduce the following conclusions:
⋄
Sps−1−2k[2ps−1−k]=Sps−1−4ν[2ps−1−2ν].
⋄
∣Sps−1−4ν[2ps−1−2ν]∣=(pm)⌈2ps−1−4u⌉−⌈22ps−1−2ν⌉=(pm)4ps−3−ν=(pm)4ps+1−1−ν.
Moreover, by Theorem 2 we have
[TABLE]
(†-ii) Let k=2ν+1 be odd, where 0≤ν≤4ps+1−1. Then
[TABLE]
and ⌈2ps−1−4u−2⌉−⌈22ps−1−2ν−1⌉=2ps−3+1−2ν−1−(4ps−3−ν)=4ps−3−ν
From these, we deduce the following conclusions:
⋄
Sps−1−2k[2ps−1−k]=Sps−1−4ν−2[2ps−1−2ν−1].
⋄
∣Sps−1−4ν−2[2ps−1−2ν−1]∣=(pm)⌈2ps−1−4u−2⌉−⌈22ps−1−2ν−1⌉=(pm)4ps+1−1−ν.
Moreover, by Theorem 2 we have
[TABLE]
Therefore, when ps≡3 (mod 4) we have
[TABLE]
(‡) Let ps≡1 (mod 4). Then 2ps−1 is even and 4ps−1 is a positive integer. In this situation, we have one of the following three cases:
(‡-i) Let k=0. Then we have that
[TABLE]
Moreover, by Theorem 2 we have
[TABLE]
(‡-ii) Let k=2ν be even, where 1≤ν≤4ps−1. Then
[TABLE]
and ⌈2ps−1−4u⌉−⌈22ps−1−2ν⌉=2ps−1−2ν−(4ps−1−ν)=4ps−1−ν
From these, we deduce the following conclusions:
⋄
Sps−1−2k[2ps−1−k]=Sps−1−4ν[2ps−1−2ν].
⋄
∣Sps−1−4ν[2ps−1−2ν]∣=(pm)⌈2ps−1−4u⌉−⌈22ps−1−2ν⌉=(pm)4ps−1−ν.
Moreover, by Theorem 2 we have
[TABLE]
(‡-iii) Let k=2ν−1 be odd, where 1≤ν≤4ps−1. Then
[TABLE]
and ⌈2ps−1−4u+2⌉−⌈22ps−1−2ν+1⌉=2ps−1−2ν+1−(4ps−1−ν+1)=4ps−1−ν
From these, we deduce the following conclusions:
⋄
Sps−1−2k[2ps−1−k]=Sps−1−4ν+2[2ps−1−2ν+1].
⋄
∣Sps−1−4ν+2[2ps−1−2ν+1]∣=(pm)⌈2ps−1−4u+2⌉−⌈22ps−1−2ν+1⌉=(pm)4ps−1−ν.
Moreover, by Theorem 2 we have
[TABLE]
Therefore, when ps≡1 (mod 4) we have
[TABLE]
i.e., NE(Fpm+uFpm,ps)=(pm)4ps−1+2(pm−1(pm)4ps−1−1).
□
Remark Let Γ∈{⟨xps−1⟩(Fpm+uFpm)[x],⟨xps+1⟩(Fpm+uFpm)[x]} and define a map
τ:Γ→Γ by:
[TABLE]
where x−1=xps−1 in ⟨xps−1⟩(Fpm+uFpm)[x]
and x−1=−xps−1 in ⟨xps+1⟩(Fpm+uFpm)[x]. Then
τ is a ring automorphism on Γ. As p is odd, the map φ defined by
[TABLE]
is an isomorphism of rings from ⟨xps−1⟩(Fpm+uFpm)[x]
onto ⟨xps+1⟩(Fpm+uFpm)[x] such that the diagram
\begin{array}[]{ccc}\ \ \ \ \frac{(\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}})[x]}{\langle x^{p^{s}}-1\rangle}&\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}&\frac{(\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}})[x]}{\langle x^{p^{s}}+1\rangle}\cr\tau\downarrow&&\ \ \ \downarrow\tau\cr\ \ \ \ \frac{(\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}})[x]}{\langle x^{p^{s}}-1\rangle}&\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}&\frac{(\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}})[x]}{\langle x^{p^{s}}+1\rangle}\end{array}
commutes.
For any ideal C of the ring Γ, its annihilating ideal is defined
as
[TABLE]
Then it is well known that C⊥=τ(Ann(C)) (cf. [12], [13]), and hence C⊥=C if and only if
τ(Ann(C))=C. From these
we deduce that all distinct self-dual negacyclic codes of length ps over Fpm+uFpm are the following:
∙
φ(C)={α(−x)∣α(x)∈C},
where C is an arbitrary self-dual cyclic code of length ps
over Fpm+uFpm.
Then by Theorem 3, one can list all distinct self-dual negacyclic codes of length ps over Fpm+uFpm
explicitly. Here we omitted.
5 Self-dual cyclic codes over F3m+uF3m of length 3s for s=1,2,3
In this section, we show how to list explicitly all distinct
self-dual cyclic codes over Fpm+uFpm of length ps for some specific odd
prime number p and positive integers m,s. Here, we take cases of p=3 and s=1,2,3 as examples.
(I) Let s=1. Then 43+1=1 and ν=0. In this case,
NE(F3m+uF3m,3)=2(3m−13m−1)=2,
and there is no integer j
satisfying one of the following two inequalities:
2=43+1−ν+1≤j≤23+1−2ν−1=1;
1=43+1−ν≤j≤23+1−2ν−2=0.
By Theorem 3(†), all self-dual cyclic codes over F3m+uF3m of length 3 are given by:
[TABLE]
(II) Let s=2. Then 32≡1 (mod 4) and 432−1=2. In this case, the number
of self-dual cyclic codes over F3m+uF3m of length 32 is
[TABLE]
by Theorem 3(‡). It is clear that
G_{3}+I_{3}=\left(\begin{array}[]{ccc}2&0&0\cr-1&0&0\cr 1&-1&2\end{array}\right)
and
[TABLE]
By Theorem 3(‡), we have the following three cases:
Case 1.
k=0. In this case, 32−1=8, 232−1=4 and 3=432−1+1≤j≤232−1=4. Using the notation
in Theorem 2, we have
[TABLE]
and
(b4,b5,b6,b7)=(a4Υ5[4,8)+a6Υ7[4,8))tr=(2a4,a4,2a6,a4+2a6). Hence
there are (3m)2 self-dual cyclic codes over F3m+uF3m of length 32:
∙ ⟨(x−1)b(x)+u⟩, where
[TABLE]
and a4,a6∈F3m arbitrary.
Case 2.
k=2ν where 1≤ν≤432−1=2. If ν=2, there is 1 codes:
∙ ⟨u(x−1)4,(x−1)5⟩.
If ν=1, we have 32−1−4ν=4, 232−1−2ν=2 and
2=432−1−ν+1≤j≤232−1−2ν+1=2. Hence
[TABLE]
In this case, there are 3m codes:
∙ ⟨(x−1)3b(x)+u(x−1)2,(x−1)7⟩, where b(x)=2a2(x−1)2 and a2∈F3m arbitrary.
Case 3.
k=2ν−1 where 1≤ν≤2. If ν=2, there is 1 codes:
∙ ⟨u(x−1)3,(x−1)6⟩.
If ν=1, we have 32−1−4ν+2=6, 232−1−2ν+1=3 and
3=432−1−ν+2≤j≤232−1−2ν+1=3. Hence
[TABLE]
In this case, there are 3m codes:
∙ ⟨(x−1)2b(x)+u(x−1),(x−1)8⟩, where b(x)=2a4(x−1)4+a4(x−1)5 and a4∈F3m
arbitrary.
(III) Let s=3. Then 33≡3 (mod 4) and 433+1=7. In this case, the number
of self-dual cyclic codes over F3m+uF3m of length 33 is
[TABLE]
by Theorem 3(†). From
G_{27}+I_{27}=\left(\begin{array}[]{ccc}G_{9}+I_{9}&0&0\cr-G_{9}&-G_{9}+I_{9}&0\cr G_{9}&-G_{9}&G_{9}+I_{9}\end{array}\right), we list all these codes by the following two cases:
Case 1.
k=2ν where 0≤ν≤6. In this case, there are 36m+35m+34m+33m+32m+3m+1
self-dual cyclic codes over F3m+uF3m of length 33 given by:
∙ ⟨u(x−1)12,(x−1)15⟩.
∙ ⟨(x−1)11b(x)+u(x−1)10,(x−1)17⟩,
where b(x)=b3(x−1)3+b4(x−1)4+b5(x−1)5 with
[TABLE]
and a4∈F3m arbitrary.
∙ ⟨(x−1)9b(x)+u(x−1)8,(x−1)19⟩,
where b(x)=b5(x−1)5+b6(x−1)6+b7(x−1)7+b8(x−1)8+b9(x−1)9 with
[TABLE]
and a6,a8∈F3m arbitrary.
∙ ⟨(x−1)7b(x)+u(x−1)6,(x−1)21⟩,
where b(x)=∑i=713bi(x−1)i with
[TABLE]
and a8,a10,a12∈F3m arbitrary.
∙ ⟨(x−1)5b(x)+u(x−1)4,(x−1)23⟩,
where b(x)=∑i=917bi(x−1)i with
[TABLE]
and a10,a12,a14,a16∈F3m arbitrary.
∙ ⟨(x−1)3b(x)+u(x−1)2,(x−1)25⟩,
where b(x)=∑i=1121bi(x−1)i with
[TABLE]
and a12,a14,a16,a18,a20∈F3m arbitrary.
∙ ⟨(x−1)b(x)+u⟩,
where b(x)=∑i=1325bi(x−1)i with
[TABLE]
and A=(a14,a16,a18,a20,a22,a24)∈F3m6 arbitrary.
Case 2.
k=2ν+1 where 0≤ν≤6. In this case, there are 36m+35m+34m+33m+32m+3m+1
self-dual cyclic codes over F3m+uF3m of length 33 given by:
∙ ⟨u(x−1)13,(x−1)14⟩.
∙ ⟨(x−1)12b(x)+u(x−1)11,(x−1)16⟩, where
b(x)=b2(x−1)2+b3(x−1)3 with
[TABLE]
∙ ⟨(x−1)10b(x)+u(x−1)9,(x−1)18⟩, where
b(x)=b4(x−1)4+b5(x−1)5+b6(x−1)6+b7(x−1)7 with
[TABLE]
and a4,a6∈F3m arbitrary.
∙ ⟨(x−1)8b(x)+u(x−1)7,(x−1)20⟩, where
b(x)=b6(x−1)6+b7(x−1)7+b8(x−1)8+b9(x−1)9+b10(x−1)10+b11(x−1)11 with
[TABLE]
and a6,a8,a10∈F3m arbitrary.
∙ ⟨(x−1)6b(x)+u(x−1)5,(x−1)22⟩, where
b(x)=∑i=815(x−1)i with
[TABLE]
and a8,a10,a12,a14∈F3m arbitrary.
∙ ⟨(x−1)4b(x)+u(x−1)3,(x−1)24⟩, where
b(x)=∑i=1019(x−1)i with
[TABLE]
and a10,a12,a14,a16,a18∈F3m arbitrary.
∙ ⟨(x−1)2b(x)+u(x−1),(x−1)26⟩, where
b(x)=∑i=1223(x−1)i with
[TABLE]
and A=(a12,a14,a16,a18,a20,a22)∈F3m6 arbitrary.
6 Conclusions and further research
For any odd prime number p and positive integers m,s, we have given an explicit representation for all distinct self-dual cyclic codes of length ps
over the finite chain ring Fpm+uFpm (u2=0) by a new way different from that used in [6] and [8]. In particular, we provide an efficient method to construct precisely all distinct self-dual cyclic codes of length ps
over Fpm+uFpm by use of Kronecker products
of matrices over Fp with a specific type.
Giving an explicit
representation and enumeration for self-dual cyclic codes and self-dual negacyclic codes
over Fpm+uFpm for arbitrary length and obtaining some bounds for the minimal distance such as BCH-like of a self-dual cyclic code over the ring Fpm+uFpm by just looking at its representation of such codes are future topics of interest.
Acknowledgments
Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics,
Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is
supported in part by the National Natural Science Foundation of
China (Grant Nos. 11671235, 11801324), the Shandong Provincial Natural Science Foundation, China
(Grant No. ZR2018BA007), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University)
(Grant No. AM201804) and the Scientific Research Fund of Hunan
Provincial Key Laboratory of Mathematical Modeling and Analysis in
Engineering (No. 2018MMAEZD09). H.Q. Dinh is
grateful for the Centre of Excellence in Econometrics, Chiang Mai University,
Thailand, for partial financial support.
