Repeated-root constacyclic codes of length $3lp^{s}$ and their dual codes

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by  Li Liu

Repeated-root constacyclic codes of length 3lps and their dual codes ∗
Li Liu, Lanqiang Li, Xiaoshan Kai, Shixin Zhu
School of Mathematics, Hefei Universityof Technology, Hefei 230009, Anhui, P.R.China
Abstract: Let p 6= 3 be any prime and l 6= 3 be any odd prime with gcd(p,l)= 1. The mul-
tiplicative group F∗ = hξi can be decomposed into mutually disjoint union of gcd(q−1,3lps)
q
cosets over the subgroup hξ3lpsi, where ξ is a primitive (q−1)th root of unity. We classify all
repeated-root constacyclic codes of length 3lps over the finite field F into some equivalence
q
classes by this decomposition, where q =pm, s andm are positive integers. Accordingto these
6
equivalence classes,we explicitly determine the generatorpolynomials ofall repeated-rootcon-
1
0 stacyclic codesoflength3lps overFq andtheir dualcodes. Self-dualcyclic codes oflength3lps
2 over F exist only when p=2. And we give all self-dual cyclic codes of length 3·2sl over F
q 2m
n and their enumeration.
a
J Keywords: Repeated-root constacyclic codes, Cyclic codes, Dual codes, Generator polynomial

2
2
1 Introduction

]
T
Constacyclic codes over finite fields play a very important role in the theory of error-
I
. correcting codes. More importantly, constacyclic codes have practical applications. As these
s
c codes have rich algebraic structures, they can be efficiently encoded and decoded using shift
[ registers. They also have very good error-correcting properties. All of those explain their

preferred role in engineering.
2
v Repeated-rootcyclic codes werefirstinvestigatedin the 1990sby Castagnoliin[1]andVan
1 Lintin[2],whereitwasprovedthatrepeated-rootcycliccodeshaveaconcatenatedconstruction,
2 and are asymptotically bad. However,it is well knownthat there still exist a few optimal such
4
codes (see [12−14]),whichencouragesmanyscholarstostudy the classofcodes. Forexample,
1
Dinh determined the generator polynomials of all constacyclic codes and their dual codes over
0
. Fq of length 2ps,3ps and 6ps in [3 − 5]. Since then, these results have been extended to
1
more general code lengths. In 2012, Bakshi and Raka gave the generator polynomials of all
0
constacyclic codes of length 2tps over F in [6], where q is a power of an odd prime p. In 2014,
6 q
1 Chenet.alstudiedallconstacycliccodesoflengthlps overFq in[7],wherel is aprime different
: from p. In [7], all constacyclic codes of length lps over F and their dual codes were obtained,
v q
and all self-dual and all linear complementary dual constacyclic codes were given. Recently, in
i
X
[8], Sharma explicitly determined the generator polynomials of all repeated-root constacyclic
r codes of length ltps over F and their dual codes. Further, they listed all self-dual cyclic and
a pm
negacyclic codes and also determined all self-orthogonal cyclic and negacyclic codes of length
ltps over F . What’s more, Chen et.al studied all constacyclic codes of length 2lmps over
pm
F of characteristic p in [9], and they gave the characterization and enumeration of all linear
q
complementarydualandself-dualconstacycliccodesoflength2lmps overF . Intheconclusion
q
of their paper, they proposed to study all constacyclic codes of length klmps over F , where p
q
is the characteristic of F , l is an odd prime different from p, and k is a prime different from l
q
and p. However,this is not an easy work.
∗ E-mailaddresses: [email protected](LiLiu),[email protected](Lanqiang Li).
This research is supported by the National Natural Science Foundation of China (No.61572168,No.11401154)
andtheAnhuiprovincialNaturalScienceFoundationunderGrant(No.158085MA13).
1
Inthispaper,westudyallconstacycliccodesoflength3lpsoverF ,wherep6=3isanyprime
q
and l 6=3 is any odd prime with gcd(p,l)=1. The article is organizedas follows. In section 3,
we decompose the multiplicative cyclic group F∗ = hξi into mutually disjoint union of cosets
q
of hξ3lpsi, which are one-to-one correspondence to the equivalence classes of all constacyclic.
Based on this decomposition, Section 4 explicitly determines the generator polynomials of all
λ−constacyclic codes of length 3lps over F and their dual codes, where λ is any none-zero
q
element of F and q = pm is a power of prime. As an application, we give all self-dual cyclic
q
codes of length 3·2sl over F and their enumeration in Section 5.
2m
2 Preliminaries
Let F be the finite field of order q = pm, where p 6= 3 is a prime and the characteristic
q
of the field, and m is a positive integer. Let F∗ = hξi be the multiplicative cyclic group of
q
none-zero elements of F , where ξ is a primitive (q−1)th root of unity.
q
Foranyelementλ∈F∗,λ−constacycliccodesoflengthnoverF areregardedastheideals
q q
hg(x)i of the quotient ring F [x]/(xn −λ), where g(x)|xn −λ. Further, the definition of the
q
dual code of code C is as follows:
C⊥ ={x∈Fn|x·y =0,∀y ∈C},
q
where x·y denotes the Euclidean inner product of x and y in Fn. The code C is called to be
q
a self-orthogonal code if C ⊆ C⊥ and a self-dual code if C = C⊥. Let C be a λ−constacyclic
code oflengthn overF generatedby apolynomialg(x), i.e., C =hg(x)i. As g(x)|xn−λ,then
q
there must be a polynomial h(x) ∈ F [x] such that h(x) = xn−λ. It is clear that h(x) is also
q g(x)
monic if g(x) is monic. The polynomial h(x) is called the parity check polynomial of code C.
It is well known that the dual code C⊥ is generated by h(x)∗, where h(x)∗ is the reciprocal
polynomial of h(x). For any f(x) ∈ F [x], the reciprocal polynomial of f(x) is defined as
q
f(x)∗ =f(0)−1xdeg(f(x))f(1). Obviously, (f f )∗ =f∗f∗, and (f∗)∗ =f , for any polynomials
x 1 2 1 2 1 1
f (x),f (x)∈F [x].
1 2 q
Let n be any positive integer. For any integer s, 0 ≤ s ≤ n−1, we have the definition of
q−cyclotomic coset of s modulo n is as follows:
C ={s,sq,...,sqns−1},
s
where ns is the leastpositive integersuch that sqns ≡s(mod n). Then it is easy to see that ns
is equal to the multiplicative order of q modulo n . If α denotes a primitive nth root of
gcd(s,n)
unity insomeextensionfieldofF ,thenthe polynomialM (x)= (x−αi)is theminimal
polynomial of αs over F , and q s Qi∈Cs
q
xn−1= M (x)
Y s
givesthe factorizationof(xn−1)intoirreduciblefactorsoverF , wheres runsoveracomplete
q
set of representatives from distinct q−cyclotomic coset modulo n.
Obviously, when n = l, where l 6= 3 is an odd prime with gcd(l,p)= 1, we get that all the
distinct q−cyclotomic cosets modulo l are C0 = {0} and Ck = {gk,gkq,...,gkqnk−1}, for any
integer k, 1≤k≤e= φ(l), by [15,Theorem1], where g is a fixed generatorof the cyclic group
f
Z∗, f =ord(q) is the multiplicative order of q in Z∗, and φ is Euler’s phi-function. Therefore,
l l l
the irreducible factorization of xl−1 over F is given by
q
xl−1=M (x)M (x)M (x)...M (x),
0 1 2 e
2
where M (x)= (x−ηi) with η being a primitive lth root of unity.
i Qj∈Ci
Inaddition,wedetermineallthedistinctq3−cyclotomiccosetsmodulol,whichisneededto
prove our main results. There exist two subcases. When gcd(f,3)=1, it is easy to prove that
f = ord(q) = ord(q3), then hqi = hq3i in Z∗. According to the definition of q3−cyclotomic
l l l
coset modulo l, we have that C and C , 1 ≤ k ≤ e = φ(l), are also all of the distinct
0 k f
q3−cyclotomic cosets modulo l. When gcd(f,3)=3, we prove that ord(q3)= f. It is easy to
l 3
verify that A ={0},
0
Ak ={gk,gkq3,...,gkq3(f3−1)},
Akq ={gkq,gkqq3,...,gkqq3(f3−1)},
Akq2 ={gkq2,gkq2q3,...,gkq2q3(f3−1)},
consist of all the distinct q3−cyclotomic cosets modulo l, where 1 ≤ k ≤ e. Then we have the
irreducible factorization of xl−1 in F [x] as follows:
q3
xl−1=A (x)A (x)A (x)A (x)A (x)A (x)A (x)...A (x)A (x)A (x),
0 1 q q2 2 2q 2q2 e eq eq2
where A (x) = (x−1), A (x) = (x−ηs), A (x) = (x−ηt) and A (x) =
0 k Qs∈Ak kq Qt∈Akq kq2
(x−ηj), 1≤k ≤e.
Qj∈Akq2
We also give all the distinct q−cyclotomic cosets modulo 3l. As gcd(q,3) = 1, we have
qφ(3) ≡1(mod 3) by Euler’s Theorem, i.e., q2 ≡1(mod 3). Then, it is simple to verify that
f, q ≡1(mod 3);

ord (q)= f, q ≡2(mod 3) with f even;
3l 
2f, q ≡2(mod 3) with f odd.

From [16,Chapter 8], there exists a primitive root r modulo l such that gcd(rl−1−1,l) = 1.
l
Assume that g = r+(1−r)l2, we have gl−1−1 ≡ (r+(1−r)l2)l−1−1 ≡ rl−1−1(mod l2).
Therefore, gcd(gl−1−1,l) = gcd(rl−1−1,l) = 1. It’s clear that g is a primitive root modulo lt,
l l
t≥1, such that g ≡1(mod 3).
We give all the distinct q−cyclotomic cosets modulo 3l by the following lemma.
Lemma 2.1. (I) If q ≡ 1(mod 3), then we have that all the distinct q−cyclotomic cosets
modulo 3l are given by
B0 ={0},Bl ={l},B−l ={−l},
B ={agk,agkq,...,agkqf−1},
agk
for a∈R={1,−1,3} and 0≤k ≤e−1.
(II) If q ≡2(mod3)and f is even, we havethat allthe distinct q−cyclotomic cosetsmodulo 3l
are given by B ={0},B ={l,lq},
0 l
B ′ ={gk′,gk′q,...,gk′qf−1}, for 0≤k′ ≤2e−1,
gk
B ={3gk,3gkq,...,3gkqf−1}, for 0≤k ≤e−1.
3gk
(III) If q ≡ 2(mod 3) and f is odd, we have that all the distinct q−cyclotomic cosets modulo
3l are given by
B ={0},B ={l,lq},
0 l
3
B ={gk,gkq,...,gkq2f−1},
gk
B ={3gk,3gkq,...,3gkqf−1},
3gk
for 0≤k ≤e−1.
Proof. (I) Firstly, we prove that the cyclotomic cosets B ,0 ≤ k ≤ e−1, are distinct. If
agk
there exist some k ,k , 0≤k ,k ≤e−1, such that B =B , then we have
1 2 1 2 agk1 agk2
a gk1 ≡a gk2qj(mod 3l),
1 2
for some integer j, where a ,a ∈R={1,−1,3}. Therefore, we get
1 2
gcd(a gk1,3l)=gcd(a gk2qj,3l)=gcd(a gk2,3l),
1 2 2
as gcd(q,3l)=1. From this, we can deduce a =a or a =−a =±1.
1 2 1 2
If a =−a =±1, then
1 2
−gk1 ≡gk2qj(mod 3l), i.e., −1≡gk1−k2qj(mod 3l),
for some integer j. Due to g ≡ 1(mod 3) and q ≡ 1(mod 3), we deduce −1 ≡ 1(mod 3). This
is a contradiction.
If a =a , assume that a =a =a, then we have
1 2 1 2
agk1 ≡agk2qj(mod 3l), i.e., gk1−k2 ≡qj(mod l),
for some integer j. Further, we have g(k1−k2)f ≡ qjf ≡ 1(mod l). As g is a primitive root
modulo l, we get φ(l)|(k −k )f, i.e., e= φ(l)|k −k . Since 0≤k ,k ≤e−1, we must have
1 2 f 1 2 1 2
k =k . Secondly, we get
1 2
e−1 e−1
|B0|+|Bl|+|B−l|+XX|Bagk| = 3+XXf
a∈Rk=0 a∈Rk=0
= 3+ ef
X
a∈R
= 3+3φ(l)
= 3l.
So, the conclusion (I) holds. The conclusions (II) and (III) are also established in a similar
way.
Assume that Bo(x),Bl(x),B−l(x) and Bagk(x) are the minimal polynomials of the corre-
sponding cosets Bo,Bl,B−l and Bagk. From the above lemma, we get the following theorem
immediately.
Theory 2.2. The irreducible factorization of x3l−1 over F is given as follows:
q
(I) If q ≡1(mod 3), then
e−1
x3l−1=B0(x)Bl(x)B−l(x) Y YBagk(x),
a∈Rk=0
4
where a∈R={1,−1,3} and 0≤k ≤e−1.
(II) If q ≡2(mod 3) and f is even, then
2e−1 e−1
x3l−1=B0(x)Bl(x) Y Bgk′(x)YB3gk(x),
k′=0 k=0
′
where 0≤k ≤e−1,0≤k ≤2e−1.
(III) If q ≡2(mod 3) and f is odd, then
e−1
x3l−1=B (x)B (x) B (x)B (x),
0 l Y gk 3gk
k=0
where 0≤k ≤e−1.
Thenexttwolemmasgivethenecessaryandsufficientconditionsforjudgingthereducibility
of binomials and trinomial, which were given by Wan in [17].
Lemma 2.3. Suppose that n ≥ 2, Let k = ord(a) be the multiplicative order of a, for any
a∈F∗. Then, the binomial xn−a is irreducible over F if and only if
q q
(i) Every prime divisor of n divides k, but not (q−1);
k
(ii) If 4|n, then 4|(q−1).
Lemma 2.4. Let t be a positive integer, and H(x) ∈ F [x] be irreducible over F with
q q
deg(H(x)) = n, x does not divide H(x), and e denotes the order of any root of H(x). Then
H(xt) is irreducible over F if and only if
q
(i) Each prime divisor of t divides e;
(ii) gcd(t,qn−1)=1;
e
(iii) If 4|t, then 4|(qn−1).
3 A classification of constacyclic codes of length 3lps
Let ξ be a primitive (q−1)th rootof unity, and F∗ =hξi be a cyclic groupof order (q−1)
q
as before. It is easy to verify that hξ3lpsi=hξ3li=hξdi and the index |F∗ :hξ3lpsi|=d, where
q
d=gcd(q−1,3lps). Thus,themultiplicativecyclicgroupF∗ canbedecomposedintomutually
q
disjoint union of cosets over the subgroup hξ3lpsi as follows:
Lemma 3.1. F∗ =hξi=hξdi ξpshξdi ... ξps(d−1)hξdi, where d=gcd(q−1,3lps).
q
S S S
According to the properties of the coset, we obtain the following lemma immediately.
Lemma 3.2. For any two none-zero elements λ and µ of F , there exists some integer j,
q
0≤j ≤d−1 such that λ,µ∈ξjpshξdi if and only if λ−1µ∈hξdi, where d=gcd(q−1,3lps).
Ifλandµareinthesamecoset,webuildaone-to-onecorrespondencebetweenλ−constacyclic
codes and µ−constacyclic codes of length 3lps over F as following theorem, which shows that
q
λ−constacyclic codes and µ−constacyclic codes are equivalent.
5
Theorem 3.3. Let λ,µ∈F∗. Then there exists some integer a∈F∗ such that
q q
ϕ:F [x]/(x3lps −µ)→F [x]/(x3lps −λ),
q q
f(x)7→f(ax),
is an isomorphism if and only if λ,µ∈ξjpshξdi, where 0≤j ≤d−1.
Proof. (⇒) If ϕ is an isomorphism, then we have
µ=ϕ(µ)=ϕ(x3lps)=(ϕ(x))3lps =(ax)3lps =a3lpsx3lps =a3lpsλ,
i.e., λ−1µ=a3lps. As a=ξk ∈F∗, for some positive integer k, then λ−1µ=ξk·3lps ∈hξdi. By
q
Lemma 3.2, we get that there exists j, 0≤j ≤d−1, such that λ,µ∈ξjpshξdi.
(⇐) If there exists j, 0≤j ≤d−1, such that λ,µ∈ξjpshξdi, then we have λ−1µ∈hξdi=
hξ3lpsi, by Lemma 3.2 again. Thus, λ−1µ = ξk·3lps, for some integer k. Set a = ξk, then
λa3lps =µ. Further, it is easy to prove that the following map is an isomorphism:
ϕ:F [x]/(x3lps −µ)→F [x]/(x3lps −λ),
q q
f(x)7→f(ax).
From Theorem 3.3, we get the following obvious corollaries.
Corollary 3.4. For any two elements λ and µ of F∗, a λ−constacyclic code is equiva-
q
lent to an µ−constacyclic code if and only if there exists j, 0 ≤ j ≤ d − 1, such that
λ,µ∈ξjpshξdi. Further,aλ−constacycliccodeandanµ−constacycliccodearebothequivalent
to a ξjps−constacyclic code.
Corollary 3.5. Let λ be any element of F∗, then there exists some integer j, 0 ≤ j ≤ d−1,
q
such that a λ−constacyclic code is equivalent to a ξjps−constacyclic code.
Obviously,theTheorem3.3anditstwocorollariesshowthatallconstacycliccodesoflength
3lps over F are classified into d = gcd(q−1,3lps) mutually disjoint classes. It is enough to
q
consider λ−constacyclic codes, where λ = ξjps, 0 ≤ j ≤ d−1, and d = gcd(q−1,3lps), if we
want to determine all constacyclic codes of length 3lps over F . Therefore, we mainly study
q
λ−constacyclic codes in Section 4.
4 All constacyclic codes of length 3lps over F
q
Let f(x) be any polynomial of F [x] and leading coefficient a 6= 0, we denote f(x) =
q n
a−1f(x). Then, f(x) is said to be the monic polynomial of f(x). b
n
Fromtheabovbediscussioninthe section3,weknowthatthe numberofequivalenceconsta-
cyclic classes is equalto d=gcd(q−1,3lps). It is obvious that the value of d has the following
four cases:
(i) d=gcd(q−1,3lps)=1.
(ii) d=gcd(q−1,3lps)=3.
(iii) d=gcd(q−1,3lps)=l.
(iv) d=gcd(q−1,3lps)=3l.
6
4.1 All constacyclic codes of length 3lps over F when d = 1
q
From Lemma 3.1, we see that F∗ = hξi is the decomposition of coset over the subgroup
q
hξdi, when d=gcd(q−1,3lps)=1. In this case,it is clear thatall constacycliccodes of length
3lps over F are equivalent to the cyclic codes. Therefore, we have the following theorem.
q
Theorem 4.1. Let d= gcd(q−1,3lps)= 1, then λ−constacyclic codes C of length 3lps over
F areequivalent to the cyclic codes, for any λ∈F∗, i.e., there exists a unique element a∈F∗
q q q
such that a3lpsλ=1. Further, we have the map
ϕ :F [x]/(x3lps −1)→F [x]/(x3lps −λ),
a q q
f(x)7→f(ax),
is an isomorphism, and the irreducible factorization of x3lps −λ in F [x] is given by:
q
(I) If f is even, then
2e−1 e−1
x3lps −λ=B0(ax)psBl(ax)ps Y Bgk′(ax)ps YB3gk(ax)ps,
b b k′=0 b k=0 b
′
where 0≤k ≤e−1,0≤k ≤2e−1.
Therefore, we have
2e−1 e−1
C =hB0(ax)εBl(ax)ρ Y Bgk′(ax)τk′ YB3gk(ax)υki,
b b k′=0 b k=0 b
2e−1 e−1
C⊥ =hB0(a−1x)ps−εB−l(a−1x)ps−ρ Y B−gk′(a−1x)ps−τk′ YB−3gk(a−1x)ps−υki,
b b k′=0 b k=0 b
where 0≤ε,ρ,τk′,υk ≤ps, for any k =0,1,2,...,e−1, and k′ =0,1,2,...,2e−1.
(II) If f is odd, then
e−1
x3lps −λ=B (ax)psB (ax)ps B (ax)psB (ax)ps,
0 l Y gk 3gk
b b k=0 b b
where 0≤k ≤e−1.
Therefore, we have
e−1
C =hB (ax)εB (ax)ρ B (ax)τkB (ax)υki,
0 l Y gk 3gk
b b k=0 b b
e−1
C⊥ =hB0(a−1x)ps−εB−l(a−1x)ps−ρYB−gk(a−1x)ps−τkB−3gk(a−1x)ps−υki,
b b k=0 b b
where 0≤ε,ρ,τ ,υ ≤ps, for any k =0,1,2,...,e−1.
k k
Proof. From Theorems 2.2 and 3.3, we see that this theorem is obtained immediately.
4.2 All constacyclic codes of length 3lps over F when d = 3
q
7
In this subsection, we consider the second case, i.e., d = gcd(q − 1,3lps) = 3. In this
case, we have that F∗ = hξi = hξ3i ξpshξ3i ξ2pshξ3i is the decomposition of cosets of
q
F∗ over subgroup hξ3lpsi by Lemma 3.1S. TherefoSre, it is enough to consider the cyclic codes,
q
ξps−constcycliccodesandξ2ps−constcycliccodesifwewanttodetermineallconstacycliccodes
of length 3lps over F , when gcd(q−1,3lps)=3.
q
From the Section 2, we have that the irreducible factorizationof xl−1 is givenby xl−1=
e M (x), where M (x) = (x−ηi) with η is a primitive lth root of unity. According
tQoit=h0is, wi e deduce theifollowinQgj∈leCmima, which gives the irreducible factorizationof xlps−ξj in
F [x], when gcd(q−1,l)=1.
q
Lemma 4.2. Let gcd(q −1,l) = 1. Then, for any ξj ∈ F∗, there exists a unique element
q
b ∈ F∗ such that blpsξj = 1, where j = 0,1,2,...q−1. Thus, the irreducible factorization of
j q j
xlps −ξj is given by
e
xlps −ξj = M (b x)ps,
Y i j
i=0c
where M (x) is the minimal polynomial of the q−cyclotomic coset C modulo l.
i i
Proof. SimilartoLemma3.1,wehavethedecompositionofcosetofF∗ =hξioverthesubgroup
q
hξlpsiisgivenbyF∗ =hξi,whengcd(q−1,l)=1. Next,byTheorem3.3,wehavetheresult.
q
Lemma 4.3. Let gcd(q−1,3lps)= 3. Then the irreducible factorization of x3lps −1 is given
by
e
x3lps −1=YMi(x)psMi(bq−1x)psMi(b2(q−1)x)ps,
3 3
i=0 c c
where Mi(x) is the minimal polynomial of the q−cyclotomic coset Ci modulo l and bq−1,
b2(q−1) ∈Fq∗. 3
3
Proof. As gcd(q−1,3lps)=3, then it is clear that ξq−31 ∈Fq∗ is a primitive 3th root of unity.
Hence, we have
x3lps −1=(xl−1)ps(xl−ξq−31)ps(xl−ξ2(q3−1))ps.
Further, by Lemma 4.2, we get
e
x3lps −1=YMi(x)psMi(bq−1x)psMi(b2(q−1)x)ps,
3 3
i=0 c c
where Mi(x) is the minimal polynomial of the q−cyclotomic coset Ci modulo l and bq−1,
b2(q−1) ∈Fq∗. 3
3
Before determining ξips−constacyclic codes, i = 1,2, we must explicitly decompose the
polynomial x3lps −ξips (i=1,2), into the product of monic irreducible factors. Obviously, we
only need to determine the irreducible factorization of x3l −ξi (i = 1,2). Firstly, we consider
the polynomialx3−ξi (i=1,2). Since, x3−ξi (i=1,2)isirreducibleinF [x],wegetthatF
q q3
is a splitting field for x3−ξi (i = 1,2) over F . Thus, there exists ν ∈ F such that ν3 = ξi
q i q3 i
(i=1,2). Further, it is easy to get that ν ,αν , and α2ν are all the roots of x3−ξi (i=1,2),
i i i
in F , where α is a primitive 3th root of unity. In addition, we see that ν ∈F but ν ∈/ F ,
q3 i q3 i q
and ν is primitive 3(q−1)th root of unity, i=1,2.
i
8
Asgcd(3(q−1),l)=1,wecanfindabijectionθ fromthesetD toitselfsuchthatθ(ν)=νl,
for anyν ∈D, where D consistsof allthe primitive 3(q−1)th rootsofunity ofF∗. Therefore,
q3
there exists a unique element ω ∈D such that ν−1 =θ(ω )=ωl, i.e., ωlν =1, i=1,2.
i i i i i i
From the above discussion, we have the following two lemmas.
Lemma 4.4. The irreducible factorization of x3l−ξ over F is given as follows:
q
(I) If gcd(f,3)=1,
e
x3l−ξ = R (x),
Y i
i=0
where R (x) = M (ω x)M (αω x)M (α2ω x) for any i = 0,1,2,...,e, with ω is a primitive
i i 1 i 1 i 1 1
3(q−1)th root ocf unity acnd α is acprimitive 3th root of unity.
(II) If gcd(f,3)=3,
e
x3l−ξ =P(x) Q (x)U (x)Z (x),
Y i i i
i=1
where P(x) = (x − ω−1)(x − αω−1)(x − α2ω−1), Q (x) = A (ω x)A (αω x)A (α2ω x),
1 1 1 i i 1 iq 1 iq2 1
U (x) = A (αω x)A (α2ω x)A (ω x), and Z (x) = A (α2ω bx)A (ωbx)A (αωbx) for any
i i 1 iq 1 iq2 1 i i 1 iq 1 iq2 1
i=1,2,...b,e, with ωb1 is a primbitive 3(q−1)th root of unbity, α is abprimitiveb3th root of unity.
Proof. (I) Ifgcd(f,3) = 1, we get that C and C ,1 ≤ k ≤ e = φ(l) are all the q3−cyclotomic
0 k f
cosets modulo l. Let ν be a root of x3−ξ, and α be a primitive 3th root of unity. Then we
1
have ν ,αν , and α2ν are all the roots of x3−ξ over F , i.e.,
1 1 1 q3
x3l−ξ =(xl−ν )(xl−αν )(xl−α2ν ).
1 1 1
From the above discussion, we know that there exists ω such that ωlν = 1, where ω is a
1 1 1 1
primitive 3(q−1)th root of unity. Hence, we have ωq = αω or α2ω . Next, we just consider
1 1 1
ωq = αω . However, for the case ωq = α2ω , we see that the result is still right in the same
1 1 1 1
way. As gcd(3,l) = 1, we know l ≡ 1(mod 3) or l ≡ 2(mod 3). When l ≡ 2(mod 3), we
have (αω )lαν = αl+1 = 1 and (α2ω )lα2ν = α2(l+1) = 1. When l ≡ 1(mod 3), we have
1 1 1 1
(α2ω )lαν = α2l+1 = 1 and (αω )lα2ν = αl+2 = 1. Hence, there always exist ω ,αω and
1 1 1 1 1 1
α2ω suchthatωlν =1,(αω )lαν =1and(α2ω )lα2ν =1orωlν =1,(α2ω )lαν =1and
1 1 1 1 1 1 1 1 1 1 1
(αω )lα2ν =1. Further, by Lemma 4.2, we get
1 1
e
x3l−ξ = M (ω x)M (αω x)M (α2ω x),
Y i 1 i 1 i 1
i=0c c c
which is the monic irreducible factorization of x3l − ξ over F . And we have M (ω x) =
q3 i 1
(x−ω−1ηk), M (αω x)= (x−α2ω−1ηk) and M (α2ω x)= (x−cαω−1ηk).
Qk∈Ci 1 i 1 Qk∈Ci 1 i 1 Qk∈Ci 1
Obviously, when k rcuns over C , ω−1ηk gives all the rootcs of M (ω x). As ωq = αω and
i 1 i 1 1 1
kq,kq2 ∈C , we have (ω−1ηk)q =α2ω−1ηkq and (ω−1ηk)q2 =αωc−1ηkq2, which gives a root of
i 1 1 1 1
M (αω x) and M (α2ω x) respectively. Therefore,it’s easyto deduce that M (ω x)M (α2ω x)
i 1 i 1 i 1 i 1
Mc(α2ω x) is acirreducible polynomial over F . c c
i 1 q
(cII)Whengcd(f,3)=3,wehavethatA0,Ak,Akq,Akq2 consistofallthedistinctq3−cyclotomic
cosetsmodulo l,where1≤k ≤e. Then,the irreduciblefactorizationofxl−1overF is given
q3
by
xl−1=A (x)A (x)A (x)A (x)A (x)A (x)A (x)...A (x)A (x)A (x).
0 1 q q2 2 2q 2q2 e eq eq2
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Next, in the same way as (I), we can proved the conclusion (II) holds.
Using arguments similar to the proof in Lemma 4.3, we have the following lemma, and we
omit its proof here.
Lemma 4.5. The irreducible factorization of x3l−ξ2 over F is given as follows:
q
(I) If gcd(f,3)=1,
e
x3l−ξ2 = R′(x),
Y i
i=0
where R′(x) = M (ω x)M (αω x)M (α2ω x) for any i = 0,1,2,...,e, with ω is a primitive
i i 2 i 2 i 2 2
3(q−1)th root ocf unity acnd α is acprimitive 3th root of unity.
(II) If gcd(f,3)=3,
e
x3l−ξ2 =P′(x) Q′(x)U′(x)Z′(x),
Y i i i
i=1
where P′(x) = (x − ω−1)(x − αω−1)(x − α2ω−1), Q′(x) = A (ω x)A (αω x)A (α2ω x),
2 2 2 i i 2 iq 2 iq2 2
U′(x) = A (αω x)A (α2ω x)A (ω x), and Z′(x) = A (α2ω bx)A (ωbx)A (αωb x) for any
i i 2 iq 2 iq2 2 i i 2 iq 2 iq2 2
i=1,2,...b,e,withωb2isaprimitbive3(q−1)throotofunitbyandαisabprimitivbe3throotofunity.
Combining Lemmas 4.3, 4.4 and 4.5, we easily obtain the next result.
Theorem 4.6. Let gcd(q − 1,3lps) = 3 and α = ξq−31. For any element λ of Fq∗ and
λ−constacyclic code C of length 3lps over F , one of the following cases holds:
q
(I) If λ∈hξ3i, then there exists some element c∈F∗ such that c3lpsλ=1, and we have
q
e
C =hYMi(cx)εiMi(cbq−1x)σiMi(cb2(q−1)x)τii,
3 3
i=0c c c
e
C⊥ =hYM−i(c−1x)ps−εiM−i(c−1b−q−11x)ps−σiM−i(c−1b−2(1q−1)x)ps−τii,
i=0c c 3 c 3
where 0≤ε ,σ ,τ ≤ps for any i=0,1,2,...,e.
i i i
(II) If λ ∈ ξpshξ3i, then there exists some element c ∈ F∗ such that c3lpsλ = ξps, and one of
1 q 1
the following holds:
(i) If gcd(f,3)=1,
e
C =h R (c x)εi,
Y i 1
i=0 b
e
C⊥ =hYR−i(c−11x)ps−εi,
i=0 b
where 0≤ε≤ps, R−i(x)=M−i(ω1−1x)M−i(α2ω1−1x)M−i(αω1−1x) for any i=0,1,2,...,e.
(ii) If gcd(f,3)=3, c c c
e
C =hP(c x)ε Q (c x)εiU (c x)σiZ (c x)τii,
1 Y i 1 i 1 i 1
b i=1 b b b
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