Repeated-root constacyclic codes of length 4ℓmpn

Finite Fields and Their Applications 40 (2016) 163–200

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Finite Fields and Their Applications www.elsevier.com/locate/ffa

Repeated-root constacyclic codes of length 4m pn Anuradha Sharma ∗,1 , Saroj Rani Department of Mathematics, Indian Institute of Technology Delhi, New Delhi 110016, India

a r t i c l e

i n f o

Article history: Received 31 July 2015 Received in revised form 23 January 2016 Accepted 7 April 2016 Available online 3 May 2016 Communicated by W. Cary Huffman MSC: 94B15

a b s t r a c t Constacyclic codes form a well-known class of linear codes, and are generalizations of cyclic and negacyclic codes. In this paper, we determine generator polynomials of all constacyclic codes of length 4m pn over the finite field Fq with q elements, where p,  are distinct odd primes, q is a power of p and m, n are positive integers. We also determine their dual codes, and list all self-dual constacyclic codes of length 4m pn over Fq . © 2016 Elsevier Inc. All rights reserved.

Keywords: Cyclic codes Negacyclic codes Cyclotomic cosets

1. Introduction Constacyclic codes form an algebraically-rich family of error-correcting codes, and have good error-correcting properties. These codes can be effectively encoded and decoded using linear shift registers, which justify their preferred role from engineering perspective.

* Corresponding author. 1

E-mail address: [email protected] (A. Sharma). Research support by DST India, under the grant no. SERB/F/3551/2012-13, is gratefully acknowledged.

http://dx.doi.org/10.1016/j.ffa.2016.04.001 1071-5797/© 2016 Elsevier Inc. All rights reserved.

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In [4], constacyclic codes are first introduced as generalizations of cyclic and negacyclic codes. Since then, the problem of determination of the algebraic structure of constacyclic codes is of great interest. However, it is known only for some special classes of constacyclic codes. Below we provide a brief survey of some of the recent results in this direction. In a series of papers, Dinh [11–15] determined all repeated-root constacyclic codes of length aps over Fpr (p is a prime) in terms of their generator polynomials, where a ∈ {1, 2, 3, 4, 6}, p is a prime with gcd(a, p) = 1 and s, r are positive integers. Later, Bakshi and Raka [2] determined all Λ-constacyclic codes of length 2t ps (t ≥ 1, s ≥ 0 are integers) over Fpr , where p is an odd prime and Λ is a non-zero element in Fpr whose multiplicative order is 2a with a ≥ 0 an integer. Chen et al. [9] defined an equivalence relation, called isometry, to characterize generator polynomials of all constacyclic codes of length t ps over Fpr , where , p are distinct primes and s, t, r are positive integers. In another work, Chen et al. [7] characterized all constacyclic codes of length ps over Fpr and their dual codes, where , p are distinct primes and s, r are positive integers. They also determined all self-dual and complimentary-dual constacyclic codes of length ps over Fpr . Sharma [19] determined all constacyclic codes of length t ps over Fpr and their dual codes, where , p are distinct primes,  is odd and s, t, r are positive integers. She also determined all self-dual and self-orthogonal constacyclic codes of length t ps over Fpr . Recently, Chen et al. [8] determined the algebraic structure of all constacyclic codes of length 2t ps over Fpr and their dual codes in terms of their generator polynomials, where , p are distinct odd primes and s, t, r are positive integers. They also determined all complimentary-dual and self-dual constacyclic codes of length 2t ps over Fpr . In another recent work, Batoul et al. [3] investigated the algebraic structure of constacyclic codes of length a2t ps over Fpr provided pr ≡ 1 (mod 2t ), where p is an odd prime, a is an odd positive integer with gcd(a, p) = 1 and r, s, t are positive integers. They also provided certain sufficient conditions under which these codes are equivalent to cyclic codes of length a2t ps over Fpr . In the same work, they derived a necessary and sufficient condition for the existence of a self-dual negacyclic code of length a2t ps over Fpr . In another direction, Blackford [6] studied a class of simple-root constacyclic codes over finite fields that are isometric to their dual via a multiplier. Throughout this paper, let p,  be distinct odd primes, q be a power of the prime p, Fq be the finite field of order q and m, n be positive integers. In this paper, we determine the algebraic structure of all constacyclic codes of length 4m pn over Fq and their dual codes in terms of their generator polynomials. Besides this, we determine all self-dual constacyclic codes of length 4m pn over Fq . This paper is organized as follows: In Section 2, we state some preliminaries that are needed to derive our main results. In Section 3, we determine all repeated-root constacyclic codes of length 4m pn over Fq and their dual codes by considering the following four cases separately: (i) q ≡ 1 (mod 4) and gcd(, q − 1) = 1; (ii) q ≡ 3 (mod 4) and gcd(, q − 1) = 1; (iii) q ≡ 1 (mod 4) and gcd(, q − 1) = ; and (iv) q ≡ 3 (mod 4) and gcd(, q − 1) =  (Theorems 3.2–3.5). Finally, in Section 4, we determine all self-dual constacyclic codes of length 4m pn over Fq (Theorem 4.1).

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2. Some preliminaries In this section, we state some preliminaries that are needed to derive our main results. Let Fq be the finite field of prime power order q and N be a positive integer. For a non-zero element Λ ∈ Fq , a Λ-constacyclic code C of length N over Fq is an ideal   in the principal ideal ring Fq [X]/ X N − Λ . There exists a unique monic polynomial g(X) ∈ C that generates the code C and is a factor of X N − Λ in Fq [X], called the generator polynomial of C and we write C = g(X). On the other hand, each factor of X N −Λ in Fq [X] generates a Λ-constacyclic code of length N over Fq . The dual code of C,   denoted by C ⊥ , is given by C ⊥ = {a(X) ∈ Fq [X]/ X N − Λ−1 : a(X)c∗ (X) = 0 for all c(X) ∈ C}, where c∗ (X) = X deg c(X) c(X −1 ) for all c(X) ∈ C. Note that the dual code C ⊥ is a Λ−1 -constacyclic code of the same length N over Fq and has generator polynomial N −1 −Λ  . Furthermore, the code C is said h(X) = h(0) X deg h(x) h(X −1 ), where h(X) = Xg(X) ⊥ to be self-dual if it satisfies C = C . It is clear that if C is self-dual, then we must have Λ = Λ−1 , which implies that Λ ∈ {1, −1}. From the above discussion, we see that to write down all Λ-constacyclic codes of length N over Fq more explicitly, we need to factorize the polynomial X N − Λ ∈ Fq [X] into monic irreducible polynomials over Fq . For this, we study q-cyclotomic cosets, which are as discussed below: Let Q be a power of q and K be any positive integer coprime to Q. For any (Q) integer s, the Q-cyclotomic coset of s modulo K is defined as the set Cs = 2 Ks −1 Ks {s, sQ, sQ , · · · , sQ }, where Ks is the least positive integer satisfying sQ ≡ (Q) s (mod K). Note that the cardinality of Cs equals the multiplicative order of Q modulo K gcd(K,s) . The Q-cyclotomic cosets modulo K are useful in describing the factorization of X K − 1 in FQ [X], as follows: If β is a primitive Kth root of unity in some extension field of FQ , then for each (Q) integer s, the polynomial Ms (X) = (X − β j ) is the minimal polynomial of (Q)

β s over FQ . Moreover, if modulo K, then

j∈Cs (Q) (Q) (Q) Cs1 , Cs2 , · · · , Cst

XK − 1 =

t 

are all the distinct Q-cyclotomic cosets

Ms(Q) (X) i

(1)

i=1

is the factorization of X K − 1 into monic irreducible polynomials over FQ . From now onwards, we focus our attention on constacyclic codes of length N = 4m pn over Fq , where p,  are distinct odd primes, q is a power of p and m, n are positive integers. In order to determine all constacyclic codes of length 4m pn over Fq , we need to factorize n m the polynomial X 4p  − Λ ∈ Fq [X] into monic irreducible polynomials over Fq for each m Λ ∈ F∗q , which we shall express in terms of irreducible factorizations of X  −1 over Fq or Fq2 or Fq4 (see Section 3). For this, we will first determine all the distinct q b -cyclotomic cosets modulo m for b ∈ {1, 2, 4}. To do so, let f be the multiplicative order of q modulo

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 and let us write q f = 1 + t e for some integer e, where t ≥ 1 is an integer and  does not divide e. Now using Lemma 1 of Sharma et al. [18], we see that the multiplicative order of q modulo d , denoted by Od (q), is given by Od (q) = f max{0,d−t} = λ(d) (say) d ) for 1 ≤ d ≤ m. Let δ(d) = φ( λ(d) for 1 ≤ d ≤ m, where φ denotes the Euler’s phi function. m  −1 Let δ = δ(d). Let g be a primitive root modulo  satisfying gcd( g  −1 , ) = 1. By d=1

Theorem 10.6 of [1], we see that the integer g is a primitive root modulo d for each integer d ≥ 1. Then in the following proposition, for each b ∈ {1, 2, 4}, we determine all the distinct q b -cyclotomic cosets modulo m and write the factorization of the polynomial m X  − 1 over Fqb . Proposition 2.1. Let b ∈ {1, 2, 4} be fixed and let νb = gcd(b, f ). Then all the distinct (q b )

q b -cyclotomic cosets modulo m are given by C0 (q b ) Cm−d gk qr

(q νb )

= {0} = C0



λ(d) m−d k r+b νb −1

 =

m−d k r



m−d k r+b

g q ,

g q

m−d k r+2b

,

g q

and

,··· ,

g q

(q νb )

= Cm−d gk qr for 0 ≤ k ≤ δ(d) − 1, 0 ≤ r ≤ νb − 1 and 1 ≤ d ≤ m. Furthermore, the irreducible m factorization of the polynomial X  − 1 over Fqb is given by

m

X  − 1 = (X − 1)

m δ(d)−1 b −1   ν d=1 k=0



(q νb )

with Mm−d gk qr (X) = j∈C

(q νb )

(q νb )

Mm−d gk qr (X)

r=0

(X − η j ) as the minimal polynomial of η 

m−d k r

g q

over

m−d g k q r

Fqb for 0 ≤ k ≤ δ(d) − 1, 0 ≤ r ≤ νb − 1 and 1 ≤ d ≤ m, where η is a primitive m th root of unity over Fq . Proof. To prove this, we first observe that the cyclic subgroups of the unit group of Zd generated by q b and q νb are equal for each integer d ≥ 1. Now working in a similar way as in Theorem 2 of Sharma et al. [18] and using (1), the result follows. 2 The following lemma plays an important role in studying dual codes of constacyclic codes of length 4m pn over Fq . Lemma 2.1. For a fixed b ∈ {1, 2, 4}, let νb = gcd(b, f ). Then for 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, we have

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⎧ (qνb ) (q νb ) M ⎪ δ(d) (X) = M−m−d g k (X) if f is odd and νb = 1; ⎪ ⎪ g k+ 2 ⎨ m−d (q νb )  (q νb ) (X) if 2νb |f ; M Mm−d gk (X) = m−d g k ⎪ ⎪ νb ⎪ (q ) ⎩ if νb  f and νb is even. M νb (X) m−d g k q

2

Proof. First let 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1 be fixed. If f is odd and νb = 1, then the result follows immediately from Lemma 4.3 of Chen et al. [8] and Lemma 1 of Sharma [20]. Next suppose that either f is even or νb ∈⎧ {2, 4}. Then in view of Lemma 1 of Sharma νb ) ⎨ C (qm−d if 2νb |f ; gk  (q νb ) νb [20], it is enough to prove that C−m−d gk = (q ) ⎩C νb if νb  f and νb is even. m−d g k q 2 (q νb ) (q νb ) C0 , Cm−d gk qj νb

To prove this, by Proposition 2.1, we see that with 0 ≤ j ≤ νb −1, 0 ≤ k ≤ δ(d) − 1 and 1 ≤ d ≤ m, are all the distinct q -cyclotomic cosets modulo m . From this, we see that there exist integers k (0 ≤ k ≤ δ(d) −1) and j (0 ≤ j ≤ νb −1) satisfying (q νb ) (q νb ) C−m−d gk = Cm−d gk qj . That is, there exists an integer u (0 ≤ u ≤ λ(d) νb − 1) satisfying 



m−d g k q j+νb u ≡ −m−d g k (mod m ), which implies that g k −k q j+νb u ≡ −1 (mod d ). Now working as in the proof of Lemma 8 of Sharma [20], we get k = k , which gives q j+νb u ≡ −1 (mod d ). This implies that j + νb u ≡ λ(d) 2 (mod λ(d)), which holds if and λ(d) only if gcd(νb , λ(d)) = νb divides 2 − j. Now using the fact that 0 ≤ j ≤ νb − 1 and  νb ∈ {1, 2, 4}, we obtain j =

0 if 2νb |f ; This proves the lemma. 2 νb 2 if νb  f and νb is even.

From now onwards, we will follow the same notations as in Section 2. 3. Constacyclic codes of length 4m pn over Fq and their dual codes In this section, we will determine generator polynomials of all constacyclic codes of length 4m pn over Fq and their dual codes, where , p are distinct odd primes, q is a power of p and m, n are positive integers. To do this, we assume, throughout this paper, that ξ n is a primitive element of Fq . It is easy to see that ξ p is a generator of the cyclic group m n n F∗q = Fq \ {0}. If d = gcd(4m , q − 1), then the sets Ai = {ξ 4j p +ip : 1 ≤ j ≤ q−1 for d } m n 0 ≤ i ≤ d − 1, are all the distinct cosets in F∗q of the cyclic group A0 generated by ξ 4 p . From this, we see that if Λ ∈ F∗q , then Λ ∈ Ai for some i (0 ≤ i ≤ d − 1). Accordingly, we classify all Λ-constacyclic codes of length 4m pn (m, n ≥ 1) over Fq into d disjoint classes as follows: Theorem 3.1. Let i (0 ≤ i ≤ d − 1) be fixed. For every Λi ∈ Ai , there exists   ˆ i ∈ Fq satisfying Λ ˆ 4m pn = Λ−1 ξ ipn . Then the map Φi : Fq [X]/ X 4m pn − ξ ipn −→ Λ i i  4m pn    4m pn  n ˆ i X) + X 4m pn − Λi Fq [X]/ X − Λi , defined by f (X) + X − ξ ip → f (Λ   m n n m n n for every f (X) + X 4 p − ξ ip ∈ Fq [X]/ X 4 p − ξ ip , is a ring isomorphism. As a consequence, for each i (0 ≤ i ≤ d − 1), all the Λi -constacyclic codes of length 4m pn

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n

over Fq are given by Φi (C), where C runs over all ξ ip -constacyclic codes of length 4m pn over Fq . Proof. Working as in Proposition 4 of Sharma [19], the result follows. 2 In view of the above theorem, we see that to determine all constacyclic codes of length n 4 p over Fq , we need to determine all ξ ip -constacyclic codes of length 4m pn over Fq for 0 ≤ i ≤ d − 1, where d = gcd(4m , q − 1). To do this, we shall distinguish the following two cases: I. gcd(, q − 1) = 1 and II. gcd(, q − 1) = . m n

3.1. gcd(, q − 1) = 1 Let us assume that gcd(, q − 1) = 1. Here we have  d = gcd(4 , q − 1) = m

4 if q ≡ 1 (mod 4); 2 if q ≡ 3 (mod 4).

First of all, we shall consider the case q ≡ 1 (mod 4) for which d = 4. In view of n Theorem 3.1, we need to determine all ξ ip -constacyclic codes of length 4m pn over Fq for 0 ≤ i ≤ 3. To do this, throughout this paper, let θ be a primitive element of Fq4 q 4 −1

satisfying θ q−1 = ξ and α be a primitive element of Fq2 satisfying αq+1 = ξ, (such primitive elements θ and α always exist). Let η be a primitive m th root of unity in an m  extension field of Fq . Let δ = δ(d). For any polynomial f (X) ∈ Fq [X] of degree t with d=1

the leading coefficient as ft , we denote the corresponding monic polynomial ft−1 f (X) by f(X) and the monic polynomial f (0)−1 X deg f (X) f (X −1 ) by f(X). Note that for f (X), g(X) ∈ Fq [X], we have fg(X) = f(X) g (X),

(2)

where f g(X) = f (X)g(X). In the following theorem, we determine generator polynon mials of all ξ ip -constacyclic codes of length 4m pn over Fq and their dual codes for 0 ≤ i ≤ 3, provided q ≡ 1 (mod 4) and gcd(, q − 1) = 1. Theorem 3.2. Let p,  be distinct odd primes, q be a power of the prime p satisfying q ≡ 1 (mod 4) and gcd(, q − 1) = 1, and m, n be positive integers. Let the inte(d) (d) (d) (d) gers υ, τ, σ, , ϑk , ρk , ςk , ωk run through the set {0, 1, 2, · · · , pn } for each relevant k and d. Then we have the following: m

−q+1

I. Let i = 0. There exists an element b1 ∈ Fq satisfying b1 = ξ 4 and there are precisely (pn + 1)4+4δ distinct cyclic codes of length 4m pn over Fq , given by

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169

 (X − 1)υ (X + 1)τ (X + b1 )σ (X − b1 )

×

m δ(d)−1  

(q) (q) (q) Mm−d gk (X)ϑk M (−X)ρk M (b X)ςk m−d g k m−d g k 1 (d)

d=1 k=0

 m−d k (−b1 X)ωk ×M g 

(d)

(q)

(d)

(d)



with the corresponding dual code as 

n

(X − 1)p ×

−υ

n

(X + 1)p

m δ(d)−1  

−τ

n

(X − b1 )p n

(q)

M−m−d gk (X)p

d=1 k=0

 m−d k (−b−1 X)p ×M 1 − g

n

(q)

(d)

−ωk

(d)

−ϑk

−σ

n

(X + b1 )p

−

(q)m−d k (−X)pn −ρk M (q)m−d k (b−1 X)pn −ςk M 1 − g − g (d)

(d)

 . i(−q 4 +1)

m

II. For each i ∈ {1, 3}, there exists an element bi+1 ∈ Fq4 satisfying bi+1 = θ 4(q−1) . n (a) When f is odd, there are precisely (pn + 1)1+δ distinct ξ ip -constacyclic codes of length 4m pn over Fq , given by 

Z0 (X)τ

m δ(d)−1  

(d)

(d)

Zk (X)ωk



d=1 k=0

with the dual code as m δ(d)−1      (d) n (d) pn −τ  Zk (X)p −ωk , Z0 (X) d=1 k=0

where for each k and d, −q −q −q Z0 (X) = (X − b−1 i+1 )(X − bi+1 )(X − bi+1 )(X − bi+1 ), 2

3

(d) (q) (q) (q) (q) Zk (X) = M (b X)M (bq X)M (bq X)M (bq X). m−d g k i+1 m−d g k i+1 m−d g k i+1 m−d g k i+1 2

3

n

(b) When f ≡ 2 (mod 4), there are precisely (pn + 1)1+2δ distinct ξ ip -constacyclic codes of length 4m pn over Fq , given by 

τ

Z0 (X)

m δ(d)−1   d=1 k=0

with the corresponding dual code as

(d)

(d)

(d)

(d)

Bk (X)ρk Dk (X)ωk



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0 (X)pn −τ Z

m δ(d)−1     (d)  (d) n n (d) (d) Bk (X)p −ρk Dk (X)p −ωk , d=1 k=0

where for each k and d, −q −q −q Z0 (X) = (X − b−1 i+1 )(X − bi+1 )(X − bi+1 )(X − bi+1 ), 2

3

 m−d k (bi+1 X)M  m−d k (bq X)M  m−d k (bq X) Bk (X) = M i+1  g  g q i+1  g (q 2 )

(d)

(q 2 )

(q 2 )

2

) (qm−d (bq X), ×M g k q i+1  2

3

 m−d k (bq X)M  m−d k (bq X)M  m−d k (bq X) Dk (X) = M i+1 i+1  g  g q i+1  g (q 2 )

(d)

(q 2 )

2

(q 2 )

3

) (qm−d (b X). ×M  g k q i+1 2

n

(c) When f ≡ 0 (mod 4), there are precisely (pn + 1)1+4δ distinct ξ ip -constacyclic codes of length 4m pn over Fq , given by m δ(d)−1     (d) (d) (d) (d) (d) (d) (d) (d) τ Pk (X)ϑk Qk (X)ρk Rk (X)ςk Sk (X)ωk Z0 (X) d=1 k=0

with the corresponding dual code as 

0 (X)pn −τ Z

m δ(d)−1    (d)  (d)  (d) n n n (d) (d) (d) Pk (X)p −ϑk Qk (X)p −ρk Rk (X)p −ςk d=1

 (d) × Sk (X)p

n

k=0

(d) −ωk



,

where for each k and d, −q −q −q Z0 (X) = (X − b−1 i+1 )(X − bi+1 )(X − bi+1 )(X − bi+1 ), 2

3

(d) ) ) ) (qm−d (qm−d (qm−d (b X)M (bq X)M (bq X) Pk (X) = M  g k i+1  g k q i+1  g k q 2 i+1 4

4

4

2

) (qm−d ×M (bq X),  g k q 3 i+1 4

3

 m−d k (bq X)M  m−d k (bq X)M  m−d k 2 (bq X) Qk (X) = M i+1 i+1  g  g q i+1  g q (d)

(q 4 )

(q 4 )

2

(q 4 )

3

) (qm−d ×M (b X),  g k q 3 i+1 4

 m−d k (bq X)M  m−d k (bq X)M  m−d k 2 (bi+1 X) Rk (X) = M i+1 g g q i+1 g q    (d)

(q 4 )

(q 4 )

2

3

(q 4 )

) (qm−d ×M (bq X),  g k q 3 i+1 4

 m−d k (bq X)M  m−d k (bi+1 X)M  m−d k 2 (bq X) Sk (X) = M i+1 i+1  g  g q  g q (d)

(q 4 )

(q 4 )

3

) (qm−d ×M (bq X).  g k q 3 i+1 4

2

(q 4 )

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171

−(q+1)

m

III. Let i = 2. There exists an element b3 ∈ Fq2 satisfying b3 = α 2 . n (a) When f is odd, there are precisely (pn + 1)2+2δ distinct ξ 2p -constacyclic codes of length 4m pn over Fq , given by m δ(d)−1     (d) (d) (d) (d) W0 (X)τ E0 (X)σ Vk (X)ςk Tk (X)ωk d=1 k=0

with the corresponding dual code as 

0 (X)pn −τ E 0 (X)pn −σ W

m δ(d)−1     (d)  (d) n n (d) (d) Vk (X)p −ςk Tk (X)p −ωk , d=1 k=0

where for each k and d, −q W0 (X) = (X − b−1 3 )(X − b3 ), −(q+1) 2

E0 (X) = (X − b3 (d) Vk (X)

=

−q(q+1) 2

)(X − b3

),

(q) (q) M (b X)M (bq X), m−d g k 3 m−d g k 3 q(q+1) 2

q+1

(d) (q) (q) Tk (X) = M (b 2 X)M (b m−d g k 3 m−d g k 3

X).

n

(b) When f is even, there are precisely (pn + 1)2+4δ distinct ξ 2p -constacyclic codes of length 4m pn over Fq , given by m δ(d)−1     (d) (d) (d) (d) (d) (d) (d) (d) Fk (X)ϑk Gk (X)ρk Hk (X)ςk Jk (X)ωk W0 (X)τ E0 (X)σ d=1 k=0

with the dual code as 

0 (X)pn −τ E 0 (X)pn −σ W

m δ(d)−1    (d)  (d) n n (d) (d) Fk (X)p −ϑk Gk (X)p −ρk d=1 k=0

 (d)  (d) n n  (d) (d) × Hk (X)p −ςk Jk (X)p −ωk , where for each k and d, −q W0 (X) = (X − b−1 3 )(X − b3 ), − (q+1) 2

E0 (X) = (X − b3

− q(q+1) 2

)(X − b3

),

 m−d k (b3 X)M  m−d k (bq X), Fk (X) = M g g q 3   (d)

2

(q )

2

(q )

(d) ) ) (qm−d (qm−d Gk (X) = M (bq X)M (b X),   gk 3 gk q 3 2

2

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A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200 q+1

q(q+1) 2

 m−d k (b 2 X)M  m−d k (b Hk (X) = M 3 g g q 3   (q 2 )

(d)

(q 2 )

q(q+1) 2

 m−d k (b Jk (X) = M 3 g  (q 2 )

(d)

X),

q+1

 m−d k (b 2 X). X)M g q 3  (q 2 )

n

Proof. To determine generator polynomials of all ξ ip -constacyclic codes of length 4m pn m n n over Fq and their dual codes, we need to factorize the polynomial X 4 p − ξ ip into monic irreducible polynomials over Fq for 0 ≤ i ≤ 3. For this, we proceed as follows:  I. Let us choose u1 =

−(q−1) 4 −3(q−1) 4

if m ≡ 1 (mod 4); if m ≡ 3 (mod 4). m

Note that b1 = ξ u1 ∈ Fq satisfies b1 = ξ q ≡ 1 (mod 4), we have X 4

m n

p

−q+1 4

. Further, as b−2 1

m

=ξ

q−1 2

= −1 and

 pn m − 1 = X 4 − 1  m pn  m pn  m n m pn m p −m  = X − 1 + b , X + 1 X  − b− X 1 1

which can be rewritten as  pn m n m X 4 p − 1 = X 4 − 1  m pn  m pn  pn m = − X − 1 X + 1 (b1 X) − 1  pn m × (b1 X) + 1 .

(3)

Now by using Proposition 2.1, we have X 4

m n

p

n

(q)

n

(q)

n

(q)

n

(q)

− 1 = −M0 (X)p M0 (−X)p M0 (b1 X)p M0 (−b1 X)p ×

m δ(d)−1  

n

(q)

n

(q)

n

(q)

Mm−d gk (X)p Mm−d gk (−X)p Mm−d gk (b1 X)p

d=1 k=0 n

(q)

× Mm−d gk (−b1 X)p .  m−d k (−X), M m−d k (b1 X) = Further, we observe that Mm−d gk (−X) = (−1)λ(d) M  g  g (q) λ(d) (q) (q) ξ u1 λ(d) M M m−d k (−b1 X) for 1 ≤ d ≤ m−d k (b1 X) and M m−d k (−b1 X) = (−b1 ) (q)



g



(q)

g

(q)



g

m and 0 ≤ k ≤ δ(d) − 1. From this and using b21 = −1, we see that the irreducible m n factorization of X 4 p − 1 over Fq is given by X 4

m n

p

n

n

n

n

− 1 = (X − 1)p (X + 1)p (X − b1 )p (X + b1 )p ×

m δ(d)−1  

n n n (q) (q) (q) Mm−d gk (X)p M (−X)p M (b X)p m−d g k m−d g k 1

d=1 k=0 n (q) ×M (−b1 X)p . m−d g k

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

II. Let i ∈ {1, 3} be fixed. As gcd(m , q 4 − 1) divides −i(q 4 −1) 4(q−1)

ui+1 satisfying m ui+1 ≡

−i(q 4 −1) 4(q−1) ,

there exists an integer

(mod q 4 − 1). From this, we see that bi+1 = θui+1

−i(q 4 −1) 4(q−1)

m

satisfies bi+1 = θ . As θ is a primitive element of Fq4 satisfying θ q ≡ 1 (mod 4), we have  X

4m pn

−ξ

ipn

=

X 

=

X

q 4 −1 q−1

= ξ and

4 pn  i(q 4 −1) − θ 4(q−1)

4m

m

173

−θ

i(q 4 −1) 4(q−1)

pn  X

m

+θ

i(q 4 −1) 4(q−1)

pn  pn i(q 4 −1) 2m 2(q−1) X +θ ,

which can be rewritten as X 4

m n

p

n

− ξ ip = ξ ip

n



m

pn 

(bi+1 X) − 1

q 4 −1

m

2m (q−1)

Since θ 2 = −1 and i is odd, we observe that bi+1 this, we obtain X 4

m n

p

pn 

(bi+1 X) + 1

n

− ξ ip = −ξ ip

n

=θ

pn

m

(bi+1 X)2 + 1 −i(q 4 −1) 2

.

= −1. From

pn  pn  pn  m m m (bi+1 X) − 1 (bi+1 X) + 1 (bqi+1 X) − 1

 pn m × (bqi+1 X) + 1 .

(4)

Now as −m ui+1 q 2 ≡ (q −1)(q+1) − m ui+1 (mod q 4 − 1), we have b− i+1 4 −m −m ui+1 q 2 −m ui+1 = −θ = −bi+1 , which gives θ 4

m 2

q

=

pn  2 pn  m m = − (bqi+1 X) − 1 . (bi+1 X) + 1 (q 4 −1)(q 2 +q) − m ui+1 q (mod q 4 − 1) 4 m m − q −θ− ui+1 q = bi+1 , from which we obtain

Further, we observe that −m ui+1 q 3 ≡ that

m 3 q b− i+1

=θ

−m ui+1 q 3

=

implies

pn  3 pn  m m = − (bqi+1 X) − 1 . (bqi+1 X) + 1 In view of this, (4) can be rewritten as X 4

m n

p

n

− ξ ip = −ξ ip

n

pn  pn  2 pn  m m m (bi+1 X) − 1 (bqi+1 X) − 1 (bqi+1 X) − 1

 3 pn m × (bqi+1 X) − 1 .

(5)

174

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

(a) When f is odd, by (5) and using Proposition 2.1, we see that X 4 factorizes over Fq4 as follows: X 4

m n

p

m n

p

− ξ ip

n

n (q) (bi+1 X)pn M (q) (bq X)pn M (q) (bq X)pn M (q) (bq X)pn − ξ ip = M 0 0 0 0 i+1 i+1 i+1 2

×

m δ(d)−1  

3

n n (q) (q) M (b X)p M (bq X)p m−d g k i+1 m−d g k i+1

d=1 k=0 n n (q) (q) ×M (bq X)p M (bq X)p . m−d g k i+1 m−d g k i+1 2

3

j Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then b−1 i+1 η (q) (q) −q q −1  m−d k (bi+1 X), (b η j )q = b η jq is a zero of M  m−d k (b X), is a zero of M i+1 i+1 i+1 g g   2 2 2 2 (q) −q jq q −q jq q −q jq 2 q −q 3 jq 3  (b η ) = b η is a zero of M m−d k (b X) and (b η ) = b η is (q)

i+1

i+1

i+1

g



i+1

i+1

j (q) a zero of M (bq X), which implies that the minimal polynomial of b−1 i+1 η m−d g k i+1 over Fq is given by 3

 m−d k (bi+1 X)M  m−d k (bq X)M  m−d k (bq X)M  m−d k (bq X). Zk (X) = M i+1 i+1 i+1 g g g g     (d)

(q)

(q)

2

(q)

(q)

3

Next it is easy to observe that the minimal polynomial of b−1 i+1 over Fq is given by −q −q −q Z0 (X) = (X − b−1 i+1 )(X − bi+1 )(X − bi+1 )(X − bi+1 ). 2

3

From this, we see that

X

4m pn

−ξ

ipn

pn

= Z0 (X)

m δ(d)−1  

n

(d)

Zk (X)p

d=1 k=0 m n

n

is the irreducible factorization of the polynomial X 4 p − ξ ip over Fq . (b) When f ≡ 2 (mod 4), by (5) and using Proposition 2.1, we see that the polynom n n mial X 4 p − ξ ip factorizes over Fq4 as follows: X 4

m n

p

n (q ) (bi+1 X)pn M (q ) (bq X)pn M (q ) (bq X)pn − ξ ip = M 0 0 0 i+1 i+1 2

2

 ×M 0

(q 2 )

3

n

(bqi+1 X)p

2

m δ(d)−1 1   

2

n ) (qm−d M (b X)p  g k q h i+1 2

d=1 k=0 h=0 n n ) ) (qm−d (qm−d ×M (bq X)p M (bq X)p g k q h i+1 g k q h i+1   2

2

n ) (qm−d ×M (bq X)p .  g k q h i+1 2

3

2

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

175

(q 2 )

Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then working as in −q j j part II(a), we see that the minimal polynomials of b−1 i+1 η , bi+1 η over Fq are given by

 m−d k (bi+1 X)M  m−d k (bq X)M  m−d k (bq X) Bk (X) = M i+1 g g q i+1 g    (q 2 )

(d)

(q 2 )

(q 2 )

2

) (qm−d (bq X), ×M g k q i+1  2

3

(d) ) ) ) (qm−d (qm−d (qm−d (bq X)M (bq X)M (bq X) Dk (X) = M  g k i+1  g k q i+1  g k i+1 2

2

2

2

3

) (qm−d (b X), ×M  g k q i+1 2

respectively. Further, we note that the minimal polynomial of b−1 i+1 over Fq is given by −q −q −q Z0 (X) = (X − b−1 i+1 )(X − bi+1 )(X − bi+1 )(X − bi+1 ). 2

3

From this, it follows that

X 4

m n

p

n

n

− ξ ip = Z0 (X)p

m δ(d)−1  

n

(d)

n

(d)

Bk (X)p Dk (X)p

d=1 k=0 m n

n

is the irreducible factorization of the polynomial X 4 p − ξ ip over Fq . m n n (c) When f ≡ 0 (mod 4), by (5) and using Proposition 2.1, we see that X 4 p −ξ ip factorizes over Fq4 as follows: X 4

m n

p

 − ξ ip = M 0

(q 4 )

n

n (q ) (bq X)pn M (q ) (bq X)pn (bi+1 X)p M 0 0 i+1 i+1 4

3 m δ(d)−1   

(q ) (bq X)pn ×M 0 i+1 4

4

3

2

n ) (qm−d M (b X)p  g k q h i+1 4

d=1 k=0 h=0

×

4 4 2 n n ) ) (qm−d (qm−d M (bq X)p M (bq X)p  g k q h i+1  g k q h i+1

 m−d k h (bq X)p . ×M i+1  g q (q 4 )

3

n

(q 4 )

Arguing as in part II(a), we see that if j ∈ Cm−d gk with 1 ≤ d ≤ m and −q j −q j j 0 ≤ k ≤ δ(d) − 1, then the minimal polynomials of b−1 i+1 η , bi+1 η , bi+1 η , 2

j b−q i+1 η over Fq are given by 3

(d) ) ) ) (qm−d (qm−d (qm−d Pk (X) = M (b X)M (bq X)M (bq X)  g k i+1  g k q i+1  g k q 2 i+1 4

4

) (qm−d ×M (bq X),  g k q 3 i+1 4

3

4

2

176

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

 m−d k (bq X)M  m−d k (bq X)M  m−d k 2 (bq X) Qk (X) = M i+1 i+1 g g q i+1 g q    (q 4 )

(d)

(q 4 )

(q 4 )

2

3

 m−d k 3 (bi+1 X), ×M g q  (q 4 )

 m−d k (bq X)M  m−d k (bq X)M  m−d k 2 (bi+1 X) Rk (X) = M i+1  g  g q i+1  g q (q 4 )

(d)

(q 4 )

2

(q 4 )

3

 m−d k 3 (bq X), ×M i+1  g q (q 4 )

(d) ) ) ) (qm−d (qm−d (qm−d Sk (X) = M (bq X)M (b X)M (bq X) g k i+1 g k q i+1 g k q 2 i+1    4

4

3

4

) (qm−d ×M (bq X), g k q 3 i+1  4

2

respectively. From this, we see that

pn

Z0 (X)

m δ(d)−1  

(d)

n

(d)

n

(d)

n

(d)

n

Pk (X)p Qk (X)p Rk (X)p Sk (X)p

d=1 k=0 m n

n

is the irreducible factorization of the polynomial X 4 p − ξ ip over Fq . III. Since q ≡ 1 (mod 4) and α is a primitive element of Fq2 satisfying αq+1 = ξ, we have X 4

m n

p

 pn n m − ξ 2p = X 4 − (α2(q+1) )  m pn  m pn  pn q+1 q+1 m = X − α 2 X + α 2 X 2 + αq+1 .

m Now as gcd(m , q 2 − 1) divides −q−1 2 , there exists an integer u3 satisfying  u3 ≡ −q−1 −q−1 2 u3 m (mod q − 1). From this, we see that b3 = α satisfies b3 = α 2 , which 2 implies that

X 4

m n

p

−q 2 +1 2

As α obtain X 4

m n

p

n

n

− ξ 2p = ξ 2p m (q−1)

= b3

n

pn  pn  pn  m m m . (b3 X) − 1 (b3 X) + 1 (b3 X)2 + 1

and α

n

− ξ 2p = −ξ 2p



q 2 −1 2

−m (q−1)

= −1, we have b3

m

(b3 X) − 1

pn 

m

(b3 X) + 1

= −1. From this, we

pn 

q+1

m

(b3 2 X) − 1

pn

 q+1 pn m × (b3 2 X) + 1 .

(6)

− m u3 (mod q 2 − 1), we have b3− q = α− u3 q = −α− u3 =  pn  pn m m m m −b− , which gives (b3 X) + 1 = − (bq3 X) − 1 . Further, − u32q(q+1) ≡ 3 Since −m u3 q ≡

q 2 −1 2

m

m

m

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

(q 2 −1)(q+1) 4

−α

−

−m u3 (q+1) 2

−m q(q+1) 2

m u3 (q+1) 2 m −

= −b3 

(mod q 2 − 1) implies that b3

(q+1) 2

q+1 2

(b3

= α

177

−m u3 q(q+1) 2

=

, from which we obtain

X)

m

 q(q+1) pn pn m 2 +1 = − (b3 X) − 1 .

In view of this, (6) can be rewritten as X 4

m n

p

n

− ξ 2p = −ξ 2p

n



pn 

m

(b3 X) − 1

pn  q+1 pn m (b3 2 X) − 1

m

(bq3 X) − 1

 q(q+1) pn m 2 × (b3 X) − 1 .

(7)

(a) When f is odd, by (7) and using Proposition 2.1, we see that the polynomial m n n X 4 p − ξ 2p factorizes over Fq2 as follows: X 4

m n

p

q(q+1) 2

q+1

n (q) (b3 X)pn M (q) (bq X)pn M (q) (b 2 X)pn M (q) (b − ξ 2p = M 3 3 0 0 0 0 3

×

m δ(d)−1  

n

X)p

n n (q) (q) (b X)p M (bq X)p M m−d g k 3 m−d g k 3

d=1 k=0 q(q+1) 2

q+1

n (q) (q) ×M (b 2 X)p M (b m−d g k 3 m−d g k 3

n

X)p .

(q)

Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then working −(q+1)

j 2 as in part II(a), we see that the minimal polynomials of b−1 ηj 3 η and b3 (d) (q) (q) (d) q  m−d k (b3 X)M  m−d k (b X) and T (X) = over Fq are given by Vk (X) = M 3 k  g  g q(q+1) 2

q+1

(q) (q) (b 2 X)M (b M m−d g k 3 m−d g k 3

X), respectively. We also observe that the min-

−(q+1) 2

imal polynomials of b−1 3 and b3

− (q+1) 2

b−q 3 ) and E0 (X) = (X − b3 lows that X 4

m n

p

over Fq are given by W0 (X) = (X−b−1 3 )(X− − q(q+1) 2

)(X − b3

n

n

n

− ξ 2p = W0 (X)p E0 (X)p

), respectively. From this, it folm δ(d)−1   n n (d) (d) Vk (X)p Tk (X)p is

d=1 k=0 m n

n

the irreducible factorization of the polynomial X 4 p − ξ 2p over Fq . (b) When f is even, by (7) and using Proposition 2.1, we see that the polynomial m n n X 4 p − ξ 2p factorizes over Fq2 as follows: X 4

m n

p

q+1

q(q+1) 2

n (q ) (b3 X)pn M (q ) (bq X)pn M (q ) (b 2 X)pn M (q ) (b − ξ 2p = M 3 3 0 0 3 0 0 2

×

2

m δ(d)−1 1   

2

2

n n ) ) (qm−d (qm−d (b X)p M (bq X)p M  gk qh 3  gk qh 3 2

2

d=1 k=0 h=0 q+1

q(q+1) 2

n ) ) (qm−d (qm−d ×M (b 2 X)p M (b   gk qh 3 gk qh 3 2

2

X).

X)

178

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

(q 2 )

Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then working as in −(q+1) 2

−q j j part II(a), the minimal polynomials of b−1 3 η , b3 η , b3 over Fq are given by

−q(q+1) 2

η j and b3

ηj

(d) ) ) (qm−d (qm−d Fk (X) = M (b X)M (bq X),  gk 3  gk q 3 2

2

(d) ) ) (qm−d (qm−d Gk (X) = M (bq X)M (b X), gk 3 gk q 3   2

2

q+1

q(q+1) 2

(d) ) ) (qm−d (qm−d Hk (X) = M (b 2 X)M (b  gk 3  gk q 3 2

2

q(q+1) 2

(d) ) (qm−d Jk (X) = M (b  gk 3 2

q+1 2

 m−d k (b X)M  g q 3 (q 2 )

X) X),

respectively. From this, we see that n

n

W0 (X)p E0 (X)p

m δ(d)−1  

(d)

n

(d)

n

(d)

n

n

(d)

Fk (X)p Gk (X)p Hk (X)p Jk (X)p

d=1 k=0

is the irreducible factorization of the polynomial X 4

m n

p

n

− ξ 2p over Fq .

Now the desired result follows from the fact that for 0 ≤ i ≤ 3, the generator n polynomial g(X) of a ξ ip -constacyclic code C of length 4m pn over Fq is a factor of m n n X 4 p − ξ ip in Fq [X] with the generator polynomial of the corresponding dual code  m n n C ⊥ as  h(X) = h(0)−1 X deg h(X) h(X −1 ), where h(X) = X 4 p − ξ ip /g(X). 2 Next we shall consider the case q ≡ 3 (mod 4) for which d = gcd(4m , q − 1) = gcd(4, q − 1) = 2. In view of Theorem 3.1, we next determine generator polynomials of n all ξ ip -constacyclic codes of length 4m pn over Fq and their dual codes for 0 ≤ i ≤ 1. Theorem 3.3. Let p,  be distinct odd primes, q be a power of the prime p satisfying q ≡ 3 (mod 4) and gcd(, q − 1) = 1, and m, n be positive integers. Let the integers (d) (d) (d) (d) υ, τ, σ, ϑk , ρk , ςk , ωk run through the set {0, 1, 2, · · · , pn } for each relevant k and d. Then we have the following: −q 2 +1

m

I. Let i = 0. There exists an element c1 ∈ Fq2 satisfying c1 = α 4 . (a) When f is odd, there are precisely (pn + 1)3+3δ distinct cyclic codes of length 4m pn over Fq , given by  (X − 1)υ (X + 1)τ T0 (X)σ ×

m δ(d)−1   d=1 k=0

with the dual code as

(q) (d) (q) Mm−d gk (X)ϑk M (−X)ρk Yk (X)ςk m−d g k (d)

(d)

(d)



A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

179

 n n 0 (X)pn −σ (X − 1)p −υ (X + 1)p −τ T ×

m δ(d)−1     (d) (d)  (d) n n n  (q) (q) (d) Mm−d gk (X)p −ϑk Mm−d gk (−X)p −ρk Yk (X)p −ςk , d=1 k=0

(d) (q) (q) where T0 (X) = (X − c1 )(X − cq1 ) and Yk (X) = M (c X)M (cq X) m−d g k 1 m−d g k 1 for each k and d. (b) When f is even, there are precisely (pn + 1)3+4δ distinct cyclic codes of length 4m pn over Fq , given by



(X − 1)υ (X + 1)τ T0 (X)σ ×

m δ(d)−1  

(d)

(d)

(d)

(d)

(d)

(d)

(d)

(d)

Lk (X)ϑk Ak (X)ρk Hk (X)ςk Mk (X)ωk



d=1 k=0

with the dual code as  n n 0 (X)pn −σ (X − 1)p −υ (X + 1)p −τ T m δ(d)−1     (d)  (d)  (d)  (d) n n n n (d) (d) (d) (d) × Lk (X)p −ϑk Ak (X)p −ρk Hk (X)p −ςk Mk (X)p −ωk , d=1 k=0

where for each k and d, T0 (X) = (X − c1 )(X − cq1 ), (d)

(q 2 )

(q 2 )

Lk (X) = Mm−d gk (X)Mm−d gk q (X),  m−d k (−X)M  m−d k (−X), Ak (X) = M g g q   (d)

(q 2 )

(q 2 )

(d) ) ) (qm−d (qm−d Hk (X) = M (c X)M (cq X),  gk 1  gk q 1 2

2

(d) ) ) (qm−d (qm−d Mk (X) = M (cq X)M (c X).  gk 1  gk q 1 2

2

m

−q−1

II. Let i = 1. There exists an element c2 ∈ Fq2 satisfying c2 = α 4 . n (a) When f is odd, there are precisely (pn + 1)2+2δ distinct ξ p -constacyclic codes of length 4m pn over Fq , given by m δ(d)−1     (d) (d) (d) (d) υ τ S0 (X) K0 (X) Jk (X)ϑk Uk (X)ρk d=1 k=0

with the dual code as

180

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200 m δ(d)−1      (d)  (d) n n (d) (d) pn −υ  pn −τ  S0 (X) K0 (X) Jk (X)p −ϑk Uk (X)p −ρk , d=1 k=0

where for each k and d, −q S0 (X) = (X − c−1 2 )(X − c2 ), −q K0 (X) = (X + c−1 2 )(X + c2 ), (d) (q) (q) (c X)M (cq X), Jk (X) = M m−d g k 2 m−d g k 2 (d) (q) (q) Uk (X) = M (−c2 X)M (−cq2 X). m−d g k m−d g k n

(b) When f is even, there are precisely (pn + 1)2+4δ distinct ξ p -constacyclic codes of length 4m pn over Fq , given by 

S0 (X)υ K0 (X)τ

m δ(d)−1  

(d)

(d)

(d)

(d)

(d)

(d)

(d)

(d)

Xk (X)ϑk Vk (X)ρk Pk (X)ςk Ok (X)ωk



d=1 k=0

with the corresponding dual code as 

0 (X)pn −υ K 0 (X)pn −τ S m δ(d)−1     (d)  (d)  (d)  (d) n n n n (d) (d) (d) (d) × Xk (X)p −ϑk Vk (X)p −ρk Pk (X)p −ςk Ok (X)p −ωk , d=1 k=0

where for each k and d, −q S0 (X) = (X − c−1 2 )(X − c2 ), −q K0 (X) = (X + c−1 2 )(X + c2 ),

 m−d k (c2 X)M  m−d k (cq X), Xk (X) = M  g  g q 2 (d)

(q 2 )

(q 2 )

 m−d k (cq X)M  m−d k (c2 X), Vk (X) = M 2   g g q (d)

(q 2 )

(q 2 )

 m−d k (−c2 X)M  m−d k (−cq X), Pk (X) = M 2  g  g q (d)

(q 2 )

(q 2 )

(d) ) ) (qm−d (qm−d Ok (X) = M (−cq2 X)M (−c2 X).   gk gk q 2

2

Proof. To prove this, we shall first factorize the polynomials X 4 into monic irreducible polynomials over Fq .

m n

p

n

− ξ ip , i ∈ {0, 1},

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

181

q 2 −1

I. As q ≡ 3 (mod 4) and α 2 = −1, we have  pn m n m X 4 p − 1 = X 4 − 1  m pn  m pn  m pn  m pn q 2 −1 q 2 −1 = X − 1 . X + 1 X − α 4 X + α 4 Now as gcd(m , q 2 − 1) divides

−q 2 +1 , 4

there exists an integer v1 satisfying m v1 ≡

−q 2 +1 4

m

−q 2 +1 4

(mod q 2 − 1). From this, we see that c1 = αv1 satisfies c1 = α implies that  pn m n m X 4 p − 1 = X 4 − 1  m pn  m pn  pn m = − X − 1 X + 1 (c1 X) − 1  pn m × (c1 X) + 1 . (q 2 −1)(q−1) 4

, which

(8)

q − m v1 (mod q 2 − 1), we have c− = α− v1 q = 1 n     m m m m p pn , which gives (c1 X) + 1 = − (cq1 X) − 1 . In view of −α− v1 = −c− 1 this, (8) can be rewritten as  m pn  m pn  pn  pn m n m m X 4 p − 1 = X  − 1 . (9) X + 1 (c1 X) − 1 (cq1 X) − 1

Since −m v1 q ≡

m

m

(a) When f is odd, by (9) and using Proposition 2.1, we see that the polynomial m n X 4 p − 1 factorizes over Fq2 as follows: X 4

m n

p

n (q) (q) (−X)pn M (q) (c1 X)pn M (q) (cq X)pn − 1 = M0 (X)p M 0 0 0 1

×

m δ(d)−1  

n n n (q) (q) (q) Mm−d gk (X)p M (−X)p M (c X)p m−d g k m−d g k 1

d=1 k=0 n (q) ×M (cq X)p . m−d g k 1

(q)

Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then working j as in Theorem 3.2 II(a), we see that the minimal polynomial of c−1 1 η over (d) (q) (q) q  m−d k (c1 X)M  m−d k (c X). Working in a similar Fq is given by Yk (X) = M 1  g  g manner, as cq+1 = 1, we see that the minimal polynomial of c−1 1 1 over Fq is given q by T0 (X) = (X − c1 )(X − c1 ). From this, it follows that X 4

m n

p

n

n

n

− 1 = (X − 1)p (X + 1)p T0 (X)p ×

m δ(d)−1  

n n n (q) (d) (q) Mm−d gk (X)p M (−X)p Yk (X)p m−d g k

d=1 k=0

is the irreducible factorization of the polynomial X 4

m n

p

− 1 over Fq .

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A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

(b) When f is even, by (9) and using Proposition 2.1, we see that the polynomial m n X 4 p − 1 factorizes over Fq2 as follows: X 4

m n

p

(q 2 )

− 1 = M0 ×

n (q ) (−X)pn M (q ) (c1 X)pn M (q ) (cq X)pn (X)p M 1 0 0 0 2

1 m δ(d)−1   

2

2

n n (q ) ) (qm−d Mm−d gk qh (X)p M (−X)p gk qh  2

2

d=1 k=0 h=0 n n ) ) (qm−d (qm−d ×M (c X)p M (cq X)p .   gk qh 1 gk qh 1 2

2

(q 2 )

Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then working as in j j j Theorem 3.2 II(a), we see that the minimal polynomials of c−1 1 η , η , −η and −q j c1 η over Fq are given by (d) ) ) (qm−d (qm−d Hk (X) = M (c X)M (cq X),  gk 1  gk q 1 2

2

(q 2 )

(d)

(q 2 )

Lk (X) = Mm−d gk (X)Mm−d gk q (X),  m−d k (−X)M  m−d k (−X) Ak (X) = M  g  g q (q 2 )

(d)

(q 2 )

 m−d k (cq X)M  m−d k (c1 X), Mk (X) = M 1 g g q   (q 2 )

(d)

(q 2 )

respectively. Further, it is easy to see that the minimal polynomial of c−1 1 over −q q+1 −1 v1 (q+1) Fq is given by T0 (X) = (X − c1 )(X − c1 ), as c1 = α = 1. From this, it follows that pn

pn

pn

(X − 1) (X + 1) T0 (X)

m δ(d)−1  

n

(d)

n

(d)

(d)

n

(d)

n

Lk (X)p Ak (X)p Hk (X)p Mk (X)p

d=1 k=0 m n

is the irreducible factorization of the polynomial X 4 p − 1 over Fq . II. Since q ≡ 3 (mod 4) and α is a primitive element of Fq2 satisfying αq+1 = ξ, we have X 4

m n

p

 pn  m pn  m pn q+1 q+1 n m − ξ p = X 4 − αq+1 = X − α 4 X + α 4  pn q+1 m × X 2 + α 2 .

m Now as gcd(m , q 2 − 1) divides −q−1 4 , there exists an integer v2 satisfying  v2 ≡ −q−1 m −q−1 (mod q 2 − 1). From this, it follows that c2 = αv2 ∈ Fq2 satisfies c2 = α 4 , 4 which implies that

X 4

m n

p

n

n

− ξp = ξp



m

pn 

(c2 X) − 1

m

pn 

(c2 X) + 1

pn m . (c2 X)2 + 1

(10)

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200 −q 2 +1

2m (q−1)

m

As α 2 = α2 v2 (q−1) = c2 and α view of this, (10) can be rewritten as X 4

m n

p

n

n

− ξ p = −ξ p



q 2 −1 2

pn 

m

(c2 X) − 1

−2m (q−1)

= −1, we have c2 pn 

m

(c2 X) + 1

m

183

= −1. In pn

(cq2 X) − 1

 pn m × (cq2 X) + 1 .

(11)

(a) When f is odd, by (11) and using Proposition 2.1, we see that the irreducible m n n factorization of X 4 p − ξ p over Fq2 is given by the following: X 4

m n

p

n (q) (c2 X)pn M (q) (cq X)pn M (q) (−c2 X)pn M (q) (−cq X)pn − ξp = M 2 2 0 0 0 0

×

m δ(d)−1  

n n (q) (q) (c X)p M (cq X)p M m−d g k 2 m−d g k 2

d=1 k=0 n n (q) (q) ×M (−c2 X)p M (−cq2 X)p . m−d g k m−d g k

(q)

Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then working as −1 j j in Theorem 3.2 II(a), we see that the minimal polynomials of c−1 2 η , −c2 η , (d) (q) (q)  m−d k (c2 X)M  m−d k (cq X), c−1 and −c−1 over Fq are given by Jk (X) = M 2 2 2  g  g (d) (q) (q) q −q −1   U (X) = M m−d k (−c2 X)M m−d k (−c X), S0 (X) = (X − c )(X − c ) and k



g

2

g



2

2

−q 4 p K0 (X) = (X +c−1 −ξ p = 2 )(X +c2 ), respectively. From this, we see that X δ(d)−1 m  (d) n n  n n (d) Jk (X)p Uk (X)p is the irreducible factorization S0 (X)p K0 (X)p d=1 k=0 m n

m n

n

n

of the polynomial X 4 p − ξ p over Fq . m n n (b) When f is even, by (11) and using Proposition 2.1, we see that X 4 p − ξ p factorizes over Fq2 as follows: X 4

m n

p

n (q ) (c2 X)pn M (q ) (cq X)pn M (q ) (−c2 X)pn M (q ) (−cq X)pn − ξp = M 0 0 2 0 0 2 2

×

2

1 m δ(d)−1   

2

2

n n ) ) (qm−d (qm−d M (c X)p M (cq X)p gk qh 2 gk qh 2   2

2

d=1 k=0 h=0 n n ) ) (qm−d (qm−d ×M (−c2 X)p M (−cq2 X)p . gk qh gk qh   2

2

(q 2 )

Now if j ∈ Cm−d gk with 1 ≤ d ≤ m and 0 ≤ k ≤ δ(d) − 1, then the minimal

−q j −q j −1 j −1 −1 j polynomials of c−1 2 η , c2 η , −c2 η , −c2 η , c2 and −c2 over Fq are given by

 m−d k (c2 X)M  m−d k (cq X), Xk (X) = M  g  g q 2 (d)

(q 2 )

(q 2 )

(d) ) ) (qm−d (qm−d Vk (X) = M (cq X)M (c X),   gk 2 gk q 2 2

2

184

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

 m−d k (−c2 X)M  m−d k (−cq X), Pk (X) = M 2 g g q   (q 2 )

(d)

(q 2 )

(d) ) ) (qm−d (qm−d (−cq2 X)M (−c2 X), Ok (X) = M gk gk q   2

2

−q S0 (X) = (X − c−1 2 )(X − c2 ), −q K0 (X) = (X + c−1 2 )(X + c2 ),

respectively. From this, it follows that X 4

m n

p

n

n

n

− ξ p = S0 (X)p K0 (X)p ×

m δ(d)−1  

(d)

n

(d)

n

n

(d)

(d)

n

Xk (X)p Vk (X)p Pk (X)p Ok (X)p

d=1 k=0

is the irreducible factorization of the polynomial X 4

m n

p

n

− ξ p over Fq .

Now the desired result follows from the fact that for 0 ≤ i ≤ 1, the generator n polynomial g(X) of a ξ ip -constacyclic code C of length 4m pn over Fq is a factor of m n n X 4 p − ξ ip in Fq [X] with the generator polynomial of the corresponding dual code  m n n C ⊥ as  h(X) = h(0)−1 X deg h(X) h(X −1 ), where h(X) = X 4 p − ξ ip /g(X). 2 Remark 3.1. The generator polynomials of dual codes, determined in Theorems 3.2 and 3.3, can be explicitly evaluated in terms of cyclotomic cosets by using (2) and applying Lemma 2.1. 3.2. gcd(, q − 1) =  Throughout this section, let us assume gcd(, q − 1) = . Let r be a positive integer q−1 satisfying r  (q − 1), and let y = min{m, r}. Let β = ξ y , which is a primitive y th root of unity in Fq . Then we have  d = gcd(4 , q − 1) = gcd(4 , q − 1) = m

y

4y if q ≡ 1 (mod 4); 2y if q ≡ 3 (mod 4).

First of all, we shall consider the case q ≡ 1 (mod 4) for which d = 4y . In view of n Theorem 3.1, we determine generator polynomials of all ξ ip -constacyclic codes of length 4m pn over Fq and their dual codes for 0 ≤ i ≤ 4y − 1 in the following theorem. Theorem 3.4. Let p,  be distinct odd primes, q be a power of the prime p satisfying q ≡ 1 (mod 4) and gcd(, q − 1) = , and m, n be positive integers. Let the integers (j) (j) (j) (j) k , τk , ωk , ρk , σh , ςh , ϑh , h run through the set {0, 1, 2, · · · , pn } for each relevant k, j and h. Then we have the following:

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

I. For i = 0, there are precisely (pn + 1)4( 4m pn over Fq , given by y −1  

y

+(m−r)φ(y ))

185

distinct cyclic codes of length

−1 k ρk k ωk (X − β k )k (X + β k )τk (X − b−1 1 β ) (X + b1 β )

k=0

×

y m−r  

(j)

(j)

(j)

(j)

h ϑh h h (X  − β h )σh (X  + β h )ςh (X  − b− (X  + b− 1 β ) 1 β ) j

j

j

j

j

j



j=1 h=1 h

with the corresponding dual code as y −1  

(X − β −k )p

n

−k

(X + β −k )p

n

−τk

(X − b1 β −k )p

n

−ωk

(X + b1 β −k )p

n

−ρk

k=0

×

m−r y  

(X  − β −h )p j

n

j=1 h=1 h

× (X  + b1 β −h )p j

j

n

(j)

− h

(j)

−σh

(X  + β −h )p j

n

(j)

−ςh

(X  − b1 β −h )p j

j

n

(j)

−ϑh

 , −q+1

m

where b1 is an element in Fq satisfying b1 = ξ 4 , (such an element b1 always exists). II. For i = ay with a ∈ {1, 3}, we have the following: m y n (a) When m ≤ r, there are precisely (pn + 1) distinct ξ a p -constacyclic codes  m−1  of length 4m pn over Fq , given by (X 4 − ξ a β k )k with the corresponding k=0

 m−1  n dual code as (X 4 − ξ −a β −k )p −k . k=0

r

(b) When m > r, there are precisely (pn + 1) +(m−r)φ( cyclic codes of length 4m pn over Fq , given by

r

)

y n

distinct ξ a

p

-consta-

r r −1 m−r      (j) j j 4 k k (X − γa β ) (X 4 − γa β h )σh

j=1 h=1 h

k=0

with the corresponding dual code as r −1 m−r r      (j) j j n 4 −1 −k pn −k (X − γa β ) (X 4 − γa− β −h )p −σh ,

k=0

j=1 h=1 h

 r m r where γa is an element in ξ  satisfying γa− ξ a = 1 for a ∈ {1, 3}, (such  r  an element γa always exists in ξ ).

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A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

III. For i = 2y , we have the following: m m (a) When m ≤ r, there exists an element c3 ∈ Fq satisfying c3 = −ξ  . Then m y n there are precisely (pn + 1)2 distinct ξ 2 p -constacyclic codes of length 4m pn over Fq , given by −1  m 

(X 2 − ξβ k )k (X 2 − c3 β k )ωk



k=0

with the corresponding dual code as −1  m 

(X 2 − ξ −1 β −k )p

n

−k

−k p (X 2 − c−1 ) 3 β

n

−ωk

 .

k=0 m

r

(b) When m > r, there exists an element c4 ∈ Fq satisfying c4 = −ξ  . Then there r r y n are precisely (pn + 1)2( +(m−r)φ( )) distinct ξ 2 p -constacyclic codes of length 4m pn over Fq , given by r −1  

(X 2 − γ1 β k )k (X 2 − c4 β k )ωk

k=0

×

m−r r  

j

(j)

j

j

(j)

j

(X 2 − γ1 β h )σh (X 2 − c4 β h )ϑh



j=1 h=1 h

with the corresponding dual code as −1   n −k pn −ωk (X 2 − γ1−1 β −k )p −k (X 2 − c−1 ) 4 β r

k=0

×

m−r r  

(X 2 − γ1− β −h )p j

j

n

(j)

−σh

−h p (X 2 − c− ) 4 β j

j

n

(j)

−ϑh

 ,

j=1 h=1 h

where γ1 is as chosen in part II(b). IV. For i = wz with 0 < i < 4y , gcd(w, ) = 1 and 0 ≤ z ≤ y − 1, we have the following: z z n (a) When 2  w, there are precisely (pn + 1)2 distinct ξ w p -constacyclic codes of length 4m pn over Fq , given by z −1   

X 2

m−z

−ξ

i+k(q−1) 2z

k=0

with the corresponding dual code as

k 

X 2

m−z

+ξ

i+k(q−1) 2z

ωk 

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200 z −1   

X 2

m−z

−i−k(q−1) 2z

−ξ

187

pn −k  pn −ωk  −i−k(q−1) m−z + ξ 2z X 2 .

k=0 z

(b) When 4|w, there are precisely (pn + 1)4 distinct ξ w length 4m pn over Fq , given by z −1   

X

m−z

−ξ

i+k(q−1) 4z

k 

X

m−z

+ξ

z n

p

i+k(q−1) 4z

-constacyclic codes of

τk

k=0

 m−z ρk  m−z k  i+(k+z )(q−1) i+(k+z )(q−1) 4z 4z × X −ξ +ξ X

with the corresponding dual code as z −1    pn −k  m−z pn −τk −i−k(q−1) −i−k(q−1) m−z − ξ 4z + ξ 4z X X

k=0

 m−z pn −ρk  m−z pn − k  −i−(k+z )(q−1) −i−(k+z )(q−1) 4z 4z × X −ξ +ξ X . z

z n

(c) When w is odd, there are precisely (pn + 1) distinct ξ w p -constacyclic codes  z k  −1  i+k(q−1) m−z of length 4m pn over Fq , given by − ξ z X 4 with the dual k=0

 z pn −k  −1  −i−k(q−1) m−z z code as −ξ X 4 . k=0 n

Proof. In order to determine generator polynomials of all ξ ip -constacyclic codes of length 4m pn over Fq and their dual codes for 0 ≤ i ≤ 4y − 1, we will first factorize the m n polynomials X 4 p − ξ i , 0 ≤ i ≤ 4y − 1, into monic irreducible polynomials over Fq . I. For i = 0, we need to factorize the polynomial X 4 polynomials over Fq . By (3), we have X 4

m n

p

m

n

m

n

m n

p

− 1 into monic irreducible

m

n

m

n

− 1 = −(X  − 1)p (X  + 1)p ((b1 X) − 1)p ((b1 X) + 1)p .

(12)

Now by Theorem 3.1 of Chen et al. [10], we see that the irreducible factorization of m X  − 1 over Fq is given by X

m

−1=

y  −1

(X − β ) k

m−r y  

j

(X  − β h ).

(13)

j=1 h=1 h

k=0

Next using (13) and by applying Theorem 3.75 of [17, Ch. 3], we observe that y  −1

k=0

(X + β k )

m−r y   j=1 h=1 h

j

(X  + β h ),

y  −1

m−r y  

k=0

j=1 h=1 h

(b1 X − β k )

j

((b1 X) − β h )

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

188

and y  −1

k

(b1 X + β )

m−r y  

j

((b1 X) + β h )

j=1 h=1 h

k=0

m

m

m

are irreducible factorizations of the polynomials X  +1, (b1 X) −1 and (b1 X) +1 over Fq , respectively. On substituting these factorizations in (12) and using the fact −q+1 m m that b2 = ξ 2 u1 = ξ 2 = −1, we see that 1 X

4m pn

−1=

y  −1

−1 k p k p (X − β k )p (X + β k )p (X − b−1 1 β ) (X + b1 β ) n

k=0 m−r y  

×

n

n

n

h p (X  − β h )p (X  + β h )p (X  − b− 1 β ) j

n

j

n

j

j

n

j=1 h=1 h h p × (X  + b− 1 β ) j

j

n

m n

is the irreducible factorization of X 4 p − 1 over Fq . m n y n II. For i = ay with a ∈ {1, 3}, we need to factorize the polynomial X 4 p − ξ a p into monic irreducible polynomials over Fq . (a) Since m ≤ r, we have y = m. It is clear that X 4

m n

p

y n

− ξ a

p

m

m

n

= (X 4 − ξ a )p .

Further, for 0 ≤ k ≤ m − 1, we observe that X

4m

−ξ

am

=ξ

=

am

m −1 



X4 ξa



m −1

=ξ

am

m −1   k=0

X4 − βk ξa



 4  X − ξaβk ,

k=0

as β is a primitive m th root of unity in Fq . From this and using Theorem 3.75 of m pn −1  4 m n m n [17, Ch. 3], we see that X 4 p −ξ a p = X − ξaβk is the irreducible m n

k=0

y n

factorization of the polynomial X 4 p − ξ a p over Fq . (b) Since m > r, we have y = r. Now as gcd(r+m , q − 1) divides ar , there exists an integer za satisfying m+r za ≡ −ar (mod q − 1). From this, it follows that  r  m r there exists an element γ ∈ ξ satisfying γa− ξ a = 1. Next we observe that a  γa is a primitive

q−1 gcd(ar ,q−1)

th root of unity in Fq . Now using Theorem 3.1 m

y

of Chen et al. [10], we note that the irreducible factorization of X  − ξ a r −1 m−r r   m y j j over Fq is given by X  − ξ a = (X − γa β k ) (X  − γa β h ). From k=0

j=1 h=1 h

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

189

m n

y n

this and using Theorem 3.75 of [17, Ch. 3], we see that X 4 p − ξ a p = r −1 m−r  r n  j j n (X 4 − γa β k )p (X 4 − γa β h )p is the irreducible factorization of j=1 h=1 h

k=0 m n

y n

X 4 p − ξ a p over Fq . m n y n III. For i = 2y , we need to factorize X 4 p − ξ 2 p into monic irreducible polynomials over Fq . (a) Since m ≤ r, we have y = m. It is clear that X 4

m n

p

− ξ 2

y n

p

m

m

n

m

m

n

m

m

n

= (X 4 − ξ 2 )p = (X 2 − ξ  )p (X 2 + ξ  )p .

As β is a primitive m th root of unity in Fq , we have X

2m

−ξ

m

=

m −1 

(X 2 − ξβ k ).

(14)

k=0 q−1

m

m

m

q−1

m

Since ξ 2 = −1, we write X 2 +ξ  = X 2 −ξ  + 2 . Now as gcd(m , q − 1) q−1 m m divides m + q−1 2 , there exists an integer v3 satisfying  v3 ≡  + 2 (mod q − q−1 m m m 1). From this, it follows that c3 = ξ v3 ∈ Fq satisfies c3 = ξ  + 2 = −ξ  . From this and using the fact that β is a primitive m th root of unity in Fq , we get X

2m

+ξ

m

=

m −1 

(X 2 − c3 β k ).

(15)

k=0 m n

Now by using (14), (15) and Theorem 3.75 of [17, Ch. 3], we see that X 4 p − m −1 2 y n n n (X − ξβ k )p (X 2 − c3 β k )p is the irreducible factorization of ξ 2 p = m n

k=0

y n

X 4 p − ξ 2 p over Fq . (b) Since m > r, we have y = r. Working as in part II(b), we see that there exists  r m r an element γ1 ∈ ξ  satisfying γ1− ξ  = 1. This gives X 4

m n

p

r

m

r

m

m

n

m

r

n

− ξ 2 = (X 2 − ξ  )p (X 2 + ξ  )p n

m

m

n

= (X 2 − γ1 )p (X 2 + γ1 )p .

(16)

Now by Theorem 3.1 of Chen et al. [10], we see that the irreducible factorization m m of X  − γ1 over Fq is given by X

m

−

m γ1

=

r  −1

k=0

which gives

(X − γ1 β ) k

m−r r   j=1 h=1 h

j

j

(X  − γ1 β h ),

190

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

m

m

X 2 − γ1 =

r  −1

(X 2 − γ1 β k )

m−r r  

j

j

(X 2 − γ1 β h ).

(17)

j=1 h=1 h

k=0

Now by using Theorem 3.75 of [17, Ch. 3], we see that equation (17) provides m m the irreducible factorization of X 2 − γ1 over Fq . q−1 q−1 m m m r On the other hand, as ξ 2 = −1, we have X 2 + γ1 = X 2 − ξ  + 2 . r Now as gcd(m , q − 1) divides q−1 2 +  , there exists an integer v4 satisfying q−1 m r  v4 ≡  + 2 (mod q − 1). From this, we see that c4 = ξ v4 ∈ Fq satisfies m r c4 = −ξ  . This gives X

2m

+

m γ1

m c4

=



X2 c4



m

−1 .

Now by Theorem 3.1 of Chen et al. [10], we see that m

m

X 2 + γ1 =

r  −1

(X 2 − c4 β k )

m−r r  

j

j

(X 2 − c4 β h ).

(18)

j=1 h=1 h

k=0

Further, by (16)–(18) and using Theorem 3.75 of [17, Ch. 3], we see that r  −1

(X − γ1 β )(X − c4 β ) 2

k

2

k

m−r r  

j

j

j

j

(X 2 − γ1 β h )(X 2 − c4 β h )

j=1 h=1 h

k=0

m n

r

is the irreducible factorization of X 4 p − ξ 2 over Fq . m n z n IV. For i = wz with gcd(w, z ) = 1, we need to factorize the polynomial X 4 p −ξ w p q−1 into monic irreducible polynomials over Fq . To do this, as z < m and ξ z is a primitive z th root of unity, we write ⎛ X 4

m n

p

− ξ w

z n

p

= ξ w

=

z n

p

⎝ X

z  −1

(X 4

m−z

4m−z

⎞pn

z

− 1⎠

ξw − ξ w+

k(q−1) z

n

)p .

(19)

k=0

(a) When 2  w, by (19), we get X

4m pn

−ξ

wz pn

=

z  −1

(X 2

m−z

w

−ξ2+

k(q−1) 2z

n

)p (X 2

m−z

w

+ξ2+

k(q−1) 2z

n

)p . (20)

k=0

Now using Theorem 3.75 of [17, Ch. 3], we see that equation (20) provides the m n z n irreducible factorization of X 4 p − ξ w p over Fq .

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

(b) Here 4|w, q ≡ 1 (mod 4) and ξ

X

4m pn

−ξ

wz pn

=

z  −1

q−1 2

(X 

191

= −1. In this case, we see that (19) gives

m−z

w

−ξ4+

k(q−1) 4z )

n

)p (X 

m−z

w

+ξ4+

k(q−1) 4z )

n

)p

k=0 w

k(q−1) q−1 4z + 4

)p

w

k(q−1) q−1 4z + 4

)p .

× (X 

m−z

−ξ4+

× (X 

m−z

+ξ4+

n

n

(21)

Now by applying Theorem 3.75 of [17, Ch. 3], we see that equation (21) provides m n z n the irreducible factorization of X 4 p − ξ w p over Fq . m n z n (c) When w is odd, by Theorem 3.75 of [17, Ch. 3], we see that X 4 p − ξ w p = z  −1 k(q−1) m−z m n z n (X 4 − ξ w+ z ) is the irreducible factorization of X 4 p − ξ w p k=0

over Fq . Now the desired result follows from the fact that for 0 ≤ i ≤ 4y − 1, the generator n polynomial g(X) of a ξ ip -constacyclic code C of length 4m pn over Fq is a factor of m n n X 4 p −ξ ip in Fq [X] with the generator polynomial of the corresponding dual code  m n n C ⊥ as  h(X) = h(0)−1 X deg h(X) h(X −1 ), where h(X) = X 4 p − ξ ip /g(X). 2 Next we shall consider the case q ≡ 3 (mod 4) for which d = 2y . We also observe that when q ≡ 3 (mod 4), exactly one of 2 or −2 is a square in Fq . In view of Theorem 3.1, n we determine generator polynomials of all ξ ip -constacyclic codes of length 4m pn over Fq and their dual codes in the following theorem. Theorem 3.5. Let p,  be distinct odd primes, q be a power of the prime p satisfying q ≡ 3 (mod 4) and gcd(, q − 1) = , and m, n be positive integers. Let the integers (j) (j) (j) (j) κ, ε, τj , ρj , ωj , ςj , ζk , ℘k , τk , ωk , ρk , σh , ςh , ϑh , h run through the set {0, 1, 2, · · · , pn } for each relevant k, j and h. Then we have the following: y

I. For i = 0, there are precisely (pn + 1)3( 4m pn over Fq , given by

+(m−r)φ(y ))

distinct cyclic codes of length

y −1   (X − β k )ζk (X + β k )τk (X 2 + β k )ωk

k=0

×

m−r y  

j

(j)

j

(j)

j

(j)

(X  − β h )σh (X  + β h )ςh (X 2 + β h )ϑh

j=1 h=1 h

with the corresponding dual code as



192

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200 y −1  

(X − β −k )p

n

−ζk

(X + β −k )p

n

−τk

(X 2 + β −k )p

n

−ωk

k=0

×

m−r y  

(X  − β −h )p j

n

(j)

−σh

(X  + β −h )p j

n

(j)

−ςh

(X 2 + β −h )p j

n

(j)

−ϑh

 .

j=1 h=1 h

II. For i = y , we have the following: (2k+m )(q−1) 2m (a) When m ≤ r, there exists an element ak ∈ F∗q satisfying a4k ξ 1+ =1 m for 0 ≤ k ≤  − 1. • When 2 is a square in Fq , i.e., 2 = g2 for some g ∈ Fq , there are precisely m y n (pn +1)2 distinct ξ  p -constacyclic codes of length 4m pn over Fq , given by  m−1  −2 ζk −1 −2 ρk 2 (X 2 + ga−1 with the corresponding k X + ak ) (X − gak X + ak ) k=0

dual code as

 m−1 k=0

n

(X 2 + gak X + a2k )p

−ζk

n

(X 2 − gak X + a2k )p

−ρk

 .

• When −2 is a square in Fq , i.e., −2 = h2 for some h ∈ Fq , there are precisely m y n (pn +1)2 distinct ξ  p -constacyclic codes of length 4m pn over Fq , given by m   −1  −2 ζk −1 −2 ρk 2 with the corresponding (X 2 + ha−1 k X − ak ) (X − hak X − ak ) k=0

dual code as

 m−1 k=0

n

(X 2 − hak X − a2k )p

−ζk

n

(X 2 + hak X − a2k )p

−ρk

 .

y n

n 2 (b) When m > r, there are precisely ξ  p -constacyclic codes of  (p m+ 1) distinct  m m m length 4m pn over Fq , given by (X 2 + 2cX  + 2c2 )ε (X 2 − 2cX  + 2c2 )κ with the corresponding dual code as



X 2 + c−1 X  + 2−1 c−2 m

m

pn −ε 

X 2 − c−1 X  + 2−1 c−2 m

m

pn −κ  ,

 q+1 (q−1)+2r 4 4 where c = 2−1 ξ is an element in F∗q . z y III. For i = w with 0 < i < 2 , gcd(w, ) = 1 and 0 ≤ z ≤ y − 1, we have the following: (a) When 2  w, there are precisely (pn + 1) of length 4m pn over Fq , given by z −1   

X 2

m−z

−ξ

i+j(q−1) 2z

5z −1 2

τj 

distinct ξ w

X 2

m−z

+ξ

z n

p

-constacyclic codes

i+j(q−1) 2z

ωj

j=0 j even

×

z  −1



X

m−z

−ξ

i+k(q−1) 4z

k=0 k odd



× X 2

m−z

+ξ

i+k(q−1) 2z

℘k  m−z ζk i+k(q−1) X + ξ 4z

ρk 

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

193

with the corresponding dual code as z −1    pn −τj  pn −ωj −i−j(q−1) −i−j(q−1) m−z m−z X 2 X 2 − ξ 2z + ξ 2z

j=0 j even

×



z  −1

X

m−z

−ξ

−i−k(q−1) 4z

pn −℘k 

X

m−z

+ξ

−i−k(q−1) 4z

pn −ζk

k=0 k odd

 pn −ρk  −i−k(q−1) m−z × X 2 + ξ 2z . (b) When 4|w, there are precisely (pn + 1) of length 4m pn over Fq , given by

5z +1 2

distinct ξ w

z n

p

-constacyclic codes

z −1    ωk  τk i+k(q−1) i+k(q−1) m−z m−z X 2 X 2 − ξ 2z + ξ 2z

k=0 k odd



z  −1

×

X

m−z

−ξ

i+j(q−1) 4z

j=0 j even



× X 2

m−z

+ξ

i+j(q−1) 2z

ρj 

X

m−z

+ξ

i+j(q−1) 4z

ςj

ωj 

with the corresponding dual code as z −1   

X 2

m−z

−ξ

−i−k(q−1) 2z

pn −ωk 

X 2

m−z

+ξ

−i−k(q−1) 2z

pn −τk

k=0 k odd

×

z  −1



X

m−z

−ξ

−i−j(q−1) 4z

pn −ρj 

X

m−z

+ξ

−i−j(q−1) 4z

pn −ςj

j=0 j even

 pn −ωj  −i−j(q−1) m−z × X 2 + ξ 2z . z n

n 2 (c) When w is odd, there are precisely ξ w p -constacyclic codes of  (p m+1) distinct  m m m length 4m pn over Fq , given by (X 2 +2X  +22 )κ (X 2 −2X  +22 )ε  m m n m with the corresponding dual code as (X 2 + −1 X  + 2−1 −2 )p −κ (X 2 −   q+1 (q−1)+2wz m n 4 4 −1 X  + 2−1 −2 )p −ε , where  = 2−1 ξ is an element in F∗q . n

Proof. In order to determine generator polynomials of all ξ ip -constacyclic codes of length 4m pn over Fq and their dual codes for 0 ≤ i ≤ 2y − 1, we shall first factorize the m n polynomials X 4 p − ξ i , 0 ≤ i ≤ 2y − 1, into monic irreducible polynomials over Fq .

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

194

I. For i = 0, we need to factorize the polynomial X 4 polynomials over Fq . To do so, we first note that X 4

m n

p

m

n

m

m n

− 1 into monic irreducible

p

n

m

n

− 1 = (X  − 1)p (X  + 1)p (X 2 + 1)p .

(22)

Now by Theorem 3.1 of Chen et al. [10], we see that the irreducible factorization of m X  − 1 over Fq is given by X

m

−1=

y  −1

(X − β ) k

m−r y  

j

(X  − β h ).

(23)

j=1 h=1 h

k=0

y −1

Further, using (23) and applying Theorem 3.75 of [17, Ch. 3], we see that βk )

m−r y  

j

(X  + β h ) and

y −1

j=1 h=1 h

(X 2 + β k )

(X +

k=0

m−r y  

j

(X 2 + β h ) are irreducible fac-

j=1 h=1 h 2m

k=0 m

+ 1 over Fq , respectively. On torizations of the polynomials X  + 1 and X substituting these factorizations in (22), we see that X 4

m n

p

−1=

y  −1

n

n

n

(X − β k )p (X + β k )p (X 2 + β k )p

k=0

×

m−r y  

j

n

j

n

j

n

(X  − β h )p (X  + β h )p (X 2 + β h )p

j=1 h=1 h m n

is the irreducible factorization of X 4 p − 1 over Fq . m n y n II. For i = y , we need to factorize the polynomial X 4 p −ξ  p into monic irreducible polynomials over Fq . m n y n 4m pn m pn (a) When m ≤ r, we have y = m. Now X 4 p − ξ  p = X  −m ξ  =  m m n m m m X4 (X 4 − ξ  )p . Further, we see that X 4 − ξ  = ξ  −1 = ξ ξ

m

m −1  k=0

X4 ξ

− βk

=

m −1 

 X 4 − ξβ k , as β is a primitive m th root of

k=0

unity in Fq . From this, it follows that X 4

m n

p

− ξ

m n

p

=

m −1 

X 4 − ξβ k

k=0

pn

.

 −1 m

q−1

Since β = ξ m and ξ n

ξβ k )p

=

m −1

q−1 2

(X 4 + ξ 1+

k=0

= −1, we have X 4

m n

p

− ξ

y n

p

=

(X 4 −

k=0 (m +2k)(q−1) 2m

n

)p . Now for 0 ≤ k ≤ m − 1, as

m

gcd(4, q − 1) divides 1 + ( +2k)(q−1) , there exists an integer tk satisfying 2m (m +2k)(q−1) 4tk ≡ −1 − (mod q − 1), which implies that there exists an ele2m ment ak ∈ F∗q satisfying a4k ξ 1+

(m +2k)(q−1) 2m

= 1. In view of this, we see that for

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

195

 4  0 ≤ k ≤ m − 1, there exist F q -algebra isomorphisms Ψk : Fq [X]/ X + 1 →    (m +2k)(q−1) 2m Fq [X]/ X 4 + ξ 1+ , defined as f (X) + X 4 + 1 → f (ak X) +       (m +2k)(q−1) 2m X 4 + ξ 1+ for every f (X) + X 4 + 1 ∈ Fq [X]/ X 4 + 1 . As q ≡ 3 (mod 4), exactly one of the elements 2 or −2 is a square in Fq . Accordingly, we consider the following two cases separately: • When 2 is a square in Fq , i.e. 2 = g2 for some g ∈ Fq , then the irreducible factorization of X 4 +1 over Fq is given by X 4 +1 = (X 2 +gX+1)(X 2 −gX+1). From this, we see that

X

4m pn

−ξ

y pn

=

m −1  

−ξ 1+

k(q−1) m

pn 

pn (ak X)2 + gak X + 1

k=0

 pn × (ak X)2 − gak X + 1 =

m −1 

−2 p −1 −2 p 2 (X 2 + ga−1 k X + ak ) (X − gak X + ak ) n

n

k=0 m n

y n

is the irreducible factorization of X 4 p − ξ  p over Fq . • When −2 is a square in Fq , i.e. −2 = h2 for some h ∈ Fq , then the irreducible factorization of X 4 +1 over Fq is given by X 4 +1 = (X 2 +hX−1)(X 2 −hX−1). From this, we see that

X

4m pn

−ξ

y pn

=

m −1  

−ξ 1+

k(q−1) m

pn  pn (ak X)2 + hak X − 1

k=0

 pn × (ak X)2 − hak X − 1 =

m −1 

−2 p −1 −2 p 2 (X 2 + ha−1 k X − ak ) (X − hak X − ak ) n

n

k=0 m n

y n

is the irreducible factorization of X 4 p − ξ  p over Fq . q−1 m n y n (b) Since m > r, we have y = r. As ξ 2 = −1 and y = r, we have X 4 p −ξ  p = q−1 m n r n m r n X 4 p − ξ  p = (X 4 + ξ  + 2 )p .  −1 u 2 u Let u = r + q−1 ξ2 = 2 . As u is even and q+1 ≡ 0 (mod 4), we write ξ = 4 2 q+1  r  −1 u q+1 (q−1)+2 4 4 = 4c4 (say), where c = 2−1 ξ . We further note that 4 2 ξ2 c ∈ F∗q , as r + q−1 2 is an even integer and 4 divides q+1. From this, it follows that q−1 r m m m m m 4m + X +ξ 2 = X 4 +4c4 = (X 2 +2cX  +2c2 )(X 2 −2cX  +2c2 ). Now m n y n using Theorems 3.37 and 3.76 of [17, Ch. 3], we observe that X 4 p − ξ  p = m m n m m n (X 2 + 2cX  + 2c2 )p (X 2 − 2cX  + 2c2 )p is the irreducible factorization m n y n of X 4 p − ξ  p over Fq .

196

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

III. For i = wz with gcd(w, ) = 1 and 0 ≤ z ≤ y − 1, we need to factorize the m n n polynomial X 4 p − ξ ip into monic irreducible polynomials over Fq . To do this, q−1 as ξ z is a primitive z th root of unity in Fq and z < m, we have

X

4m pn

−ξ

wz pn

z  −1

(X 4

=

m−z

− ξ w+

k(q−1) z

n

)p .

(24)

k=0

(a) When 2  w, by (24), we have

X

4m pn

−ξ

wz pn

=

z  −1

(X 2

m−z

−ξ

i+k(q−1) 2z

n

)p (X 2

m−z

+ξ

i+k(q−1) 2z

n

)p .

k=0

Further, when k is an orem 3.75 of [17, Ch. i+k(q−1) m−z X 2 + ξ 2z are Next when k is an odd

even integer satisfying 0 ≤ k ≤ z − 1, by using Thei+k(q−1) m−z 3], we see that the binomials X 2 − ξ 2z and irreducible over Fq . integer satisfying 0 ≤ k ≤ z − 1, we see that i+k(q−1) is 2z i+k(q−1)

m−z

i+k(q−1) 4z

m−z

an even integer, which gives (X 2 −ξ 2z ) = (X  −ξ i+k(q−1) z ). Now using Theorem 3.75 of [17, Ch. 3], we see that ξ 4 

z  −1

X 2

m−z

−ξ

i+j(q−1) 2z

pn 

X 2

m−z

+ξ

i+j(q−1) 2z

)(X 

m−z

+

pn

j=0 j even

×

z  −1

 m−z pn  m−z pn i+k(q−1) i+k(q−1) − ξ 4z + ξ 4z X X

k=0 k odd

 pn i+k(q−1) m−z × X 2 + ξ 2z is the irreducible factorization of X 4 (b) When 4|w, by (24), we have

X 4

m n

p

− ξ w

z n

p

=

z  −1

(X 2

m−z

m n

p

− ξ w

z n

−ξ

i+k(q−1) 2z

p

over Fq .

n

)p (X 2

m−z

+ξ

i+k(q−1) 2z

n

)p .

k=0

Now when k is an odd integer satisfying 0 ≤ k ≤ z − 1, applying Theorem 3.75 i+k(q−1) m−z m−z of [17, Ch. 3], we see that both the binomials X 2 − ξ 2z and X 2 + i+k(q−1) ξ 2z are irreducible over Fq for every odd integer k satisfying 0 ≤ k ≤ z −1. m−z Next when k is an even integer satisfying 0 ≤ k ≤ z − 1, we have (X 2 − i+k(q−1) i+k(q−1) i+k(q−1) m−z m−z   z z z 2 4 4 ) = (X −ξ )(X +ξ ). Further, applying Theoξ rem 3.75 of [17, Ch. 3], we see that

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200 z  −1

197

 pn  pn i+k(q−1) i+k(q−1) m−z m−z − ξ 2z + ξ 2z X 2 X 2

k=0 k odd

×

z  −1



X

m−z

−ξ

i+j(q−1) 4z

pn 

X

m−z

+ξ

i+j(q−1) 4z

pn  pn i+j(q−1) m−z + ξ 2z X 2

j=0 j even

m n

z n

is the irreducible factorization of X 4 p − ξ w p over Fq . q−1 m n z n m (c) Here w is odd. Now as ξ 2 = −1, we have X 4 p − ξ w p = (X 4 + q−1 z n ξ w + 2 )p .  −1 v 2  v q+1 v Let v = wz + q−1 ξ 2 = 4 2−1 ξ 2 = 2 . As v is even, we have ξ = 4 2 q+1  4 (q−1)+2wz 4 . We further note that  ∈ F∗q , as 44 (say), where  = 2−1 ξ m

4 wz + q−1 + 2 is an even integer and 4 divides q + 1. From this, we see that X q−1 z m m m m m +w 4 4 2  2 2  2 ξ 2 =X +4 = (X +2X +2 )(X −2X +2 ). Further, m n z n using Theorems 3.37 and 3.76 of [17, Ch. 3], we see that X 4 p − ξ w p = m m n m m n (X 2 + 2X  + 22 )p (X 2 − 2X  + 22 )p is the irreducible factorization m n z n of X 4 p − ξ w p over Fq . Now the desired result follows from the fact that for 0 ≤ i ≤ 2y − 1, the n generator polynomial g(X) of a ξ ip -constacyclic code C of length 4m pn over m n n Fq is a factor of X 4 p − ξ ip in Fq [X] with the generator polynomial of the corresponding dual code C ⊥ as  h(X) = h(0)−1 X deg h(X) h(X −1 ), where h(X) =   4m pn ipn −ξ /g(X). 2 X

4. Self-dual constacyclic codes of length 4m pn over Fq In this section, we shall determine all self-dual constacyclic codes of length 4m pn over Fq . By Theorem 1 of Jia et al. [16], we note that there does not exist any self-dual cyclic code of length 4m pn over Fq . Moreover, by Theorem 3 of Blackford [5], we see that there does not exist any self-dual negacyclic code of length 4m pn over Fq if and only if q ≡ 3 (mod 4). In the following theorem, we determine all the self-dual negacyclic codes of length 4m pn over Fq when q ≡ 1 (mod 4). Theorem 4.1. Let p,  be distinct odd primes, q be a power of the prime p and m, n be positive integers. I. Let gcd(, q − 1) = 1. −q+1 m (i) For q ≡ 1 (mod 8), there exists an element d1 ∈ Fq satisfying d1 = ξ 8 . There are precisely (pn + 1)2+2δ distinct self-dual negacyclic codes of length 4m pn over Fq , given by

198

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200



p (X − d1 )υ (X − d−1 1 )

n

×

m δ(d)−1  

−υ

p (X + d1 )σ (X + d−1 1 )

n

−σ

n (q) (q) (q) M (d X)ϑk M (d X)p −ϑk M (−d1 X)ςk m−d g k 1 m−d g k 1 m−d g k (d)

d=1 k=0

 m−d k (−d1 X)p ×M  g

n

(q)

(d)

(d)

where 0 ≤ υ, σ, ϑk , ςk

(d)

−ςk

(d)

(d)

 ,

≤ pn for each relevant k and d. m

(ii) For q ≡ 5 (mod 8), there exists an element d2 ∈ Fq satisfying d2 = α (a) When f is odd, let us define

−q 2 +1 8

.

−q A0 (X) = (X − d−1 2 )(X − d2 ),

B0 (X) = (X − d2 )(X − dq2 ), (d) (q) (q) Dk (X) = M (d X)M (dq X), m−d g k 2 m−d g k 2

 m−d k (d−1 X)M  m−d k (d−q X) Zk (X) = M 2 2 g g   (d)

(q)

(q)

n 1+δ for 0 ≤ k ≤ δ(d) 2 − 1 with 1 ≤ d ≤ m. Then there are precisely (p + 1) m n distinct self-dual negacyclic codes of length 4 p over Fq , given by δ(d)

m 2   −1 (d) (d)  (d) n n (d) Dk (X)ϑk Dk (X)p −ϑk A0 (X)υ B0 (X)p −υ d=1 k=0

 (d)  (d) n (d) (d) × Zk (X)ςk Zk (X)p −ςk , (d)

(d)

where 0 ≤ υ, ϑk , ςk ≤ pn for each relevant k and d. (b) When f is even, for 0 ≤ k ≤ δ(d) − 1 with 1 ≤ d ≤ m, let us define −q A0 (X) = (X − d−1 2 )(X − d2 ),

B0 (X) = (X − d2 )(X − dq2 ),  m−d k (d2 X)M  m−d k (dq X), Pk (X) = M   g g q 2 (d)

(q 2 )

(q 2 )

 m−d k (d−1 X)M  m−d k (d−q X), Rk (X) = M 2  g  g q 2 (d)

(q 2 )

(q 2 )

(d) ) ) (qm−d (qm−d Sk (X) = M (d−q X)M (d−1 X),  gk 2  gk q 2 2

2

(d) ) ) (qm−d (qm−d Vk (X) = M (dq X)M (d X).  gk 2  gk q 2 2

2

• For f ≡ 2 (mod 4), there are precisely (pn + 1)1+2δ distinct self-dual negacyclic codes of length 4m pn over Fq , given by

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

199

 n A0 (X)υ B0 (X)p −υ ×

m δ(d)−1  

(d)

(d)

n

(d)

Pk (X)ϑk Sk (X)p

(d)

−ϑk

(d)

(d)

n

(d)

Rk (X)ςk Vk (X)p

(d)

−ςk

 ,

d=1 k=0 (d)

(d)

where 0 ≤ υ, ϑk , ςk ≤ pn for each relevant k and d. • For f ≡ 0 (mod 4), there are (pn + 1)1+2δ distinct self-dual negacyclic codes of length 4m pn over Fq , given by 

n

A0 (X)υ B0 (X)p 

−υ

δ(d)−1

×

(d)

(d)

n

(d)

Pk (X)ϑk Rk (X)p

(d)

−ϑk

(d)

(d)

n

(d)

Sk (X)ρk Vk (X)p

(d)

−ρk

 ,

k=0 (d)

(d)

where 0 ≤ υ, ϑk , ρk ≤ pn for each relevant k and d. II. Let gcd(, q − 1) = . With the same notations as in Section 3.2, the following hold: −q+1 m (i) For q ≡ 1 (mod 8), there exists an element d3 ∈ Fq satisfying d3 = ξ 8 . y y There are precisely (pn + 1)2 +2(m−r)φ( ) distinct self-dual negacyclic codes of length 4m pn over Fq , given by 

p (X − d3 )s1 (X − d−1 3 )

n

×

y  −1

−s1

p (X + d3 )s3 (X + d−1 3 )

n

 (X − d3 β k )k (X − d−1 3 β

y

−k pn −k

)

−s3

 (X + d3 β k )τk (X + d−1 3 β

y

−k pn −τk

)

k=1

×

m−r y  

(j)

 (X  − d3 β h )ζh (X  − d− 3 β j

j

j=1 h=1 h  × (X  + d− 3 β j

j

y

(j)

−h pn −ρh

)

(j)

j

j

y

(j)

−h pn −ζh

)

j

j

(j)

(X  + d3 β h )ρh

 ,

(j)

where 0 ≤ s1 , s3 , k , τk , ζh , ρh ≤ pn for each relevant k, h and j. m

(ii) For q ≡ 5 (mod 8) there exists an element d4 ∈ Fq satisfying d4 For each j, k and h, let −q A0 (X) = (X − d−1 4 )(X − d4 ),

B0 (X) = (X − d4 )(X − dq4 ), −q k k Nk (X) = (X − d−1 4 β )(X − d4 β ),

Uk (X) = (X − d4 β 

y

−k

)(X − dq4 β 

y

−k

),

h  Wh (X) = (X  − d− − d−q β h ), 4 β )(X 4 (j)

j

(j)

j

j

j

Yh (X) = (X  − d4 β 

j

j

y

−h

j

j

 )(X  − dq 4 β

y

−h

).

= α

−q 2 +1 8

.

200

A. Sharma, S. Rani / Finite Fields and Their Applications 40 (2016) 163–200

y

Then there are precisely (pn + 1) +(m−r)φ( of length 4m pn over Fq , given by

y

)

distinct self-dual negacyclic codes

y  −1  n n Nk (X)τk Uk (X)p −τk A0 (X)ε B0 (X)p −ε

k=1 y

×

m−r   

(j)

(j)

(j)

n

Wh (X)ρh Yh (X)p

(j)

−ρh

 ,

j=1 h=1 h (j)

where 0 ≤ ε, τk , ρh ≤ pn for each relevant k, h and j. Proof. Working in a similar way as in Theorems 3.2 and 3.3, using Lemma 2.1 and applying Proposition 1 of Sharma [20], the desired result follows. 2 References [1] T.M. Apostol, Introduction to Analytic Number Theory, Springer, 1976. [2] G.K. Bakshi, M. Raka, A class of constacyclic codes over a finite field, Finite Fields Appl. 18 (2) (2012) 362–377. [3] A. Batoul, K. Guenda, T. Aaron Gulliver, On repeated-root constacyclic codes of length 2a mpr over finite field, arXiv:1505.00356v1 [cs.IT], 2015. [4] E.R. Berlekamp, Algebraic Coding Theory, McGraw-Hill Book Company, New York, 1968. [5] T. Blackford, Negacyclic duadic codes, Finite Fields Appl. 14 (4) (2008) 930–943. [6] T. Blackford, Isodual constacyclic codes, Finite Fields Appl. 24 (2013) 29–44. [7] B. Chen, H.Q. Dinh, H. Liu, Repeated-root constacyclic codes of length ps and their duals, Discrete Appl. Math. 177 (2014) 60–70. [8] B. Chen, H.Q. Dinh, H. Liu, Repeated-root constacyclic codes of length 2m pn , Finite Fields Appl. 33 (2015) 137–159. [9] B. Chen, Y. Fan, L. Lin, H. Liu, Constacyclic codes over finite fields, Finite Fields Appl. 18 (6) (2012) 1217–1231. [10] B. Chen, H. Liu, G. Zhang, A class of minimal cyclic codes over finite fields, Des. Codes Cryptogr. 74 (2) (2015) 285–300. [11] H.Q. Dinh, Constacyclic codes of length ps over Fpm + uFpm , J. Algebra 324 (2010) 940–950. [12] H.Q. Dinh, Repeated-root constacyclic codes of length 2ps , Finite Fields Appl. 18 (1) (2012) 133–143. [13] H.Q. Dinh, Structure of repeated-root constacyclic codes of length 3ps and their duals, Discrete Math. 313 (9) (2013) 983–991. [14] H.Q. Dinh, On repeated-root constacyclic codes of length 4ps , Asian-Eur. J. Math. 6 (2) (2013), http://dx.doi.org/10.1142/S1793557113500204. [15] H.Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length 6ps and their duals, AMS Contemp. Math. 609 (2014) 69–87. [16] Y. Jia, S. Ling, C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory 57 (4) (2011) 2243–2251. [17] R. Lidl, H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 2008. [18] A. Sharma, G.K. Bakshi, M. Raka, Polyadic codes of prime power length, Finite Fields Appl. 13 (4) (2007) 1071–1085. [19] A. Sharma, Repeated-root constacyclic codes of length t ps and their dual codes, Cryptogr. Commun. 7 (2) (2015) 229–255. [20] A. Sharma, Self-dual and self-orthogonal negacyclic codes of length 2m pn over a finite field, Discrete Math. 338 (4) (2015) 576–592.