On Repeated-Root Constacyclic Codes of Length 2 a m p r over Finite Fields
aa r X i v : . [ c s . I T ] M a y On Repeated-Root Constacyclic Codes of Length 2 a mp r over Finite Fields. Aicha Batoul, Kenza Guenda and T. Aaron Gulliver ∗ Abstract
In this paper we investigate the structure of repeated root constacyclic codes oflength 2 a mp r over F p s with a ≥ m, p ) = 1. We characterize the codes in termsof their generator polynomials. This provides simple conditions on the existence ofself-dual negacyclic codes. Further, we gave cases where the constacyclic codes areequivalent to cyclic codes. Keywords :Repeated-root Constacyclic codes,Negacyclic Codes,Self-dual Codes.
Constacyclic codes over finite fields form a remarkable class of linear codes, as they includethe important family of cyclic codes. Constacyclic codes also have practical applicationsas they can be efficiently encoded using simple shift registers. They have rich algebraicstructures for efficient error detection and correction. This explains their preferred role inengineering. Repeated-root constacyclic codes, were first studied in 1967 by Berman [3],then by several authors such as Falkner et al [11] and Salagean [16]. Repeated-root cycliccodes were first investigated in the most generality in the 1990s by Castagnoli et al [4],and van Lint [17], where they showed that repeated-root cyclic codes have a concatenatedconstruction, and are not asymptotically good. However, it turns out that optimal repeated-root constacyclic codes still exist. These motivate researchers to further study this class ofcodes.Recently, Dinh, in a series of papers [8], [9] and [10], determined the generator polyno-mials of all constacyclic codes over F q , of lengths 2 p r , 3 p r and 6 p r . These results have been ∗ A. Batoul and K. Guenda are with the Faculty of Mathematics USTHB, University of Science andTechnology of Algiers, Algeria. T. Aaron Gulliver is with the Department of Electrical and ComputerEngineering, University of Victoria, PO Box 3055, STN CSC, Victoria, BC, Canada V8W 3P6. email:[email protected], [email protected], [email protected]. t p r over F p s were given in [1]. The generator polynomials of all constacyclic codesof length lp r over F q were characterized in [5], where l is a prime different from p .In this paper, we extend the main results of Batoul et al given in [2] and of Guendaand Gulliver given in [12] to constacyclic codes of length 2 a mp r over F p s where a ≥ m is an odd integer with ( m, p ) = 1. The remainder of the paper is organized as follows.Some preliminary results are given in Section 2. In Section 3, the structure of generatorof constacyclic codes of length mp r is given using the generator polynomial of constacycliccodes of length m . In Section 4, the structure of generator of constacyclic codes of length2 a mp r is given. Further, we gave cases where the constacyclic codes are equivalent to cycliccodes. It is well know that the only self-dual constacyclic codes over finite fields are eithercyclic codes over fields with even characteristic or negacyclic codes. For that in Section 5,we give conditions on the existence of self-dual negacyclic codes of length 2 a mp r over F p s ,where p is odd. Let p be a prime number and F q the finite field with q = p s elements. We note its group ofunits F ∗ q .The order of an element a in the multiplicative group F ∗ q is the least integer b such that a b = 1 in F ∗ q , we note b by ord q ( a ). Let i be an integer with 0 ≤ i < n , the q -cyclotomiccoset of i modulo n is the set C i = { i, iq, . . . , iq l − } ( mod n ) , where l is the smallest positive integer such that iq l ≡ i ( mod n ).The minimal polynomial of β i over F q is M β i ( x ) = Y j ∈ C i ( x − β j ) , where C i is the q -cyclotomic coset modulo n and β is a primitive element of F q .An [ n, k ] linear code C over F p s is a k -dimensional subspace of F np s . For λ in F ∗ q , a linearcode C of length n over F q is said to be constacyclic if it satisfies( λc n − , c , . . . , c n − ) ∈ C, whenever ( c , c , . . . , c n − ) ∈ C. When λ = 1 the code is called cyclic, and when λ = − C ⊥ of C is defined as C ⊥ = { x ∈ F nq ; P ni =1 x i y i = 0 ∀ y ∈ C } . Aninteresting class of codes is the so-called self-dual codes. A code is called Euclidean self-dualif it satisfies C = C ⊥ . Note that the dual of a λ -constacyclic code is a λ − -constacyclic2ode. A monomial linear transformation of F nq is an F q -linear transformation τ such thatthere exists scalars λ , . . . , λ n in F ∗ q and a permutation σ ∈ S n (the group of permutation ofthe set { , , . . . , n } ) such that, for all ( x , x , . . . , x n ) ∈ F nq , we have τ ( x , . . . , x n ) = ( λ x σ (1) , λ x σ (2) , . . . , λ n x σ ( n ) ) . Two linear codes C and C ′ of length n are called monomially equivalent if there existsa monomial transformation of F nq , such that τ ( C ) = C ′ . Here, whenever two codes aresaid to be equivalent it is meant that they are monomially equivalent. Let C be a λ -constacyclic,then C is an ideal of the quotient ring R n = F p s [ x ] / h x n − λ i . It is well knownthat every λ -constacyclic code is generated by a unique polynomial of least degree. Such apolynomial is called the generator of the code and it is a divisor of x n − λ . Therefore, thereis a one-to-one correspondence between constacyclic codes of length n over F q , and divisorsof x n − λ . mp r over F p s Throughout this section p is an odd prime number and n = mp r , with m an integer suchthat ( m, p ) = 1.This section provides the structure of constacyclic codes of length mp r over F q . We givethis important lemma which we need after. Lemma 3.1
Let q = p s be a prime power. Then for all λ in F ∗ q there exist λ in F ∗ q , suchthat λ = λ p r , for r in N . Proof.
Let λ ∈ F ∗ q , q = p s and s be a positive integer. If s < r then there exists integers k and m such that r − m = ks with 0 ≤ m ≤ s − , s − m = s − ( r − ks ) = s ( k + 1) − r . Let λ = λ p s − m , then λ p r = ( λ p s ( k +1) − r ) p r = ( λ p s ( k +1) ) = λ. If r < s , then ( λ p s − r ) p r = λ . (cid:4) It is well know that λ -constacyclic codes over F q are principal ideals generated by factorsof x mp r − λ . Since F p s has characteristic p , and by Lemma 3.1 the polynomial x mp r − λ canbe factored as x mp r − λ = x mp r − λ p r = ( x m − λ ) p r . (1)The polynomial x m − λ is a monic square free polynomial. Hence from [6, Proposition2.7] it factors uniquely as a product of pairwise coprime monic irreducible polynomials f ( x ) , . . . , f l ( x ). Thus from (1) we obtain the following factorization of x mp r − λ . x mp r − λ p r = f ( x ) p r . . . f l ( x ) p r . (2)3 λ -constacyclic code of length n = mp r over F p s is then generated by a polynomial of theform A ( x ) = Y f ik i , (3)where f i ( x ) , ≤ i ≤ l , are the polynomials given in (2) and 0 ≤ k i ≤ p r . a mp r over F p s In this section we first recall the following important result of Batoul et al. given in [2]
Proposition 4.1 [2, Proposition 3.2] Let q be a prime power, n a positive integer and λ an element in F ∗ q . If F ∗ q contains an element δ , where δ is an n -th root of λ , then a λ -constacyclic code of length n is equivalent to a cyclic code of length n . And we give the structure of repeated-root constacyclic codes over F q , q = p s of length 2 a mp r , a ≥
1. But before that and in the goal of using the isomorphism between cyclic codes andconstacyclic codes of the same length, given in Proposition 4.1, we give the structure ofcyclic codes of length 2 a mp r over F q , for that, we need the following Lemma: Lemma 4.2
Let a ≥ and α a primitive a -th root of the unity in F ∗ q ,the following holds:1) α i is a primitive a − i -th root of the unity in F ∗ q for all i, i ≤ a .2) α m is a primitive a -th root of the unity in F ∗ q for all odd integer m .3) Q a k =1 α k = − . Proof.
1) Let i, i ≤ a , in the cyclic group F ∗ q , we have that ord ( α i ) = ord ( α )(2 i ,ord ( α )) = a (2 i , a ) = a i =2 a − i .2) Since (2 a , m ) = 1, so ord ( α m ) = ord ( α )( m,ord ( α )) = a ( m, a ) = 2 a .3) ( x a −
1) = Q a k =1 ( x − α k ) then Q a k =1 α k = ( − a ( − = − (cid:4) Proposition 4.3
Let q be a power of an odd prime p and n = 2 a m a positive integer suchthat m is an odd integer and ( m, p ) = 1 , a ≥ . Then if F ∗ q contains a primitive a -root ofunity α and the f i ( x ) , ≤ i ≤ l are the monic irreducible factors of x m − in F q [ x ] ,then: x a m − a Y k =1 ( l Y i =0 f i ( α − k x )) . (4)4 roof. Assume that x m − Q li =0 f i ( x ) is the factorization of x m − F q . This factorization is unique since it is over a unique factorization domain (UFD).Let α ∈ F ∗ q be a primitive 2 a -th root of unity and let 1 ≤ k ≤ a .( α − k x ) m − Q li =0 f i ( α − k x )( α − k ) m ( x m − ( α k ) m ) = α − k Q li =0 f i ( α − k x )( x m − α km ) = α k ( m − Q li =0 f i ( α − k x )( x m − ( α m ) k ) = α k ( m − Q li =0 f i ( α − k x )Then by Lemma 4.2 α m is also a primitive 2 a -th root of unity, we obtain: Q a k =1 ( x m − ( α m ) k ) = Q a k =1 α k ( m − Q li =0 f i ( α − k x )= Q a k =1 α k ( m − Q a k =1 Q li =0 f i ( α − k x )= (cid:16)Q a k =1 α k ( m − (cid:17) (cid:16)Q a k =1 Q li =0 f i ( α − k x ) (cid:17) = (cid:18)Q a k =1 α km α k (cid:19) (cid:16)Q a k =1 Q li =0 f i ( α − k x ) (cid:17) Since ( x a m −
1) = (( x m ) a − ( α m ) a ) = Q a k =1 ( x m − α km ) we obtain the result: x a m − a Y k =1 ( l Y i =0 f i ( α − k x )) . (cid:4) Corollary 4.4
Let q be a power of an odd prime p and n = 2 a mp r a positive integer suchthat m is an odd integer and ( m, p ) = 1 , a ≥ . Then if F ∗ q contains a primitive a -root ofunity α and the f i ( x ) , ≤ i ≤ l are the monic irreducible factors of x m − in F q then: ( x a mp r −
1) = ( x a m − p r = a Y k =1 l Y i =0 f p r i ( α − k x ) . Proof.
Since the characteristic of F q is p , the proof follows from Proposition 4.3. (cid:4) In the following we give the structure of cyclic codes of length 2 a mp r over F q Corollary 4.5
Let q be a power of an odd prime p , n = 2 a mp r be a positive integer suchthat m is odd integer, with a ≥ and ( m, p ) = 1 . Then if F ∗ q contains a primitive a -root ofunity α and the f i ( x ) , ≤ i ≤ l are the monic irreducible factors of x m − in F q [ x ] then anycyclic code of length n = 2 a mp r is generated by Q a k =1 ( Q li =0 f j i i ( α − k x )) where ≤ j i ≤ p r . Proof.
Since any cyclic code of length n = 2 a mp r is generated by a divisor of ( x a mp r − x a mp r −
1) = ( x a m − p r = a Y k =1 l Y i =0 f p r i ( α − k x )5o we deduce the result. (cid:4) Now we generalize Proposition 4.1.
Theorem 4.6
Let q be a power of an odd prime p and m an odd integer such that ( m, p ) = 1 .Let λ and δ in the multiplicative group F ∗ q such that δ m = λ , if δ = β a in F ∗ q , then thefollowing hold:(i) The λ -constacyclic codes of length a mp r over F q are equivalent to cyclic codes of length a mp r over F q .(ii) If q ≡ a +1 , then − λ -constacyclic codes of length a mp r over F q are equivalentto cyclic codes of length a mp r over F q . Proof.
For the part (i), let λ ∈ F ∗ q such that there exists δ ∈ F ∗ q , δ m = λ and δ = β a in F q then λ = β a m . So by Lemma 3.1 there exists β ∈ F ∗ q such that β = β p r . Then λ = β a mp r ,hence by Proposition 4.1, λ -constacyclic codes of length 2 a mp r over F q are equivalent tocyclic codes over F q . For the part (ii), since q ≡ a +1 and by Lemma 4.2 there existsa primitive 2 a +1 -root of unity α ∈ F ∗ q . So α a = − − λ = ( − mp r λ = ( α a ) mp r β a mp r =( αβ ) a mp r . Then by Proposition 4.1, − λ -constacyclic codes of length 2 a mp r over F q areequivalent to cyclic codes over F q . (cid:4) Corollary 4.7
Let λ = β a mp r , α a primitive a -th root of unity in F ∗ q and let C be a λ -constacyclic code of length a mp r , then: C = h ( a Y k =1 ( l Y i =0 f j i i ( β − α − k x )) i , where ≤ j i ≤ p r . Proof.
By Lemma 4.2 if q ≡ a , then there exists a primitive 2 a -th root α of unityin F ∗ q . Thus by Corollary 4.4( x a mp r −
1) = ( x a m − p r = a Y k =1 l Y i =0 f p r i ( α − k x ) , then (( β − x ) a mp r −
1) = (( β − x ) a m − p r = a Y k =1 l Y i =0 f p r i ( β − α − k x ) , so ( x a mp r − λ ) = λ (( β − x ) a m − p s = a Y k =1 l Y i =0 f p r i ( β − α − k x ) . λ -constacyclic codes of length 2 a mp r is generated by a divisor of ( x a mp r − λ ) thenwe have the result. (cid:4) Corollary 4.8
Let λ = β a m and let C be a − λ -constacyclic code of length a mp r . If q ≡ a +1 then C = h ( a Y k =1 ( l Y i =0 f j i i ( β − α − k +1 x ))) i where ≤ j i ≤ p r . Proof.
By Lemma 4.2 if q ≡ a +1 there exists a primitive 2 a +1 -th root α of unityin F ∗ q . Thus by Corollary 4.4( x a +1 mp r −
1) = ( x a m − p r ( x a m + 1) p r = a Y k =1 l Y i =0 f p r i ( α − k x ) a Y k =1 l Y i =0 f p r i ( α − k +1 x ) . Then (( β − x ) a mp r + 1) = (( β − x ) a m + 1) p r = a Y k =1 ( l Y i =0 f p r i ( β − α − k +1 x ) . Then ( x a mp r + λ ) = λ (( β − x ) a m − p s = a Y k =1 ( l Y i =0 f p r i ( β − α − k +1 x ) . Since any − λ -constacyclic codes of length 2 a mp r is generated by a divisor of ( x a mp r + λ )we obtain the result. (cid:4) Example 4.9
Let n = 2 · · = 1750 , q = 5 and β be a primitive element of F ⋆ . Further,let λ ∈ { β i , ≤ i ≤ } . Since (7 ,
24) = 1 , β i = β i (7 · − · = ( β ) · i .Then in F wehave that x − = ( x + 4)( x + βx + β x + 4)( x + β x + β x + 4) = f ( x ) f ( x ) f ( x ) .So then x − = f ( x ) f ( x ) f ( x ) f ( − x ) f ( − x ) f ( − x ) . Since cyclic codes of length · · over F are ideals of F [ x ] x − which is a principal ideal ring. Then these codes are generatedby h ( f s ( x ) f j ( x ) f k ( x ) f l ( − x ) f m ( − x ) f s ( − x )) i , s, j, k, l, m, s ∈ { , . . . , } . Therefore, λ -constacyclic codes of length · · = 1750 over F are ideals of F [ x ] x − λ whichis a principal ideal ring, and these codes are generated by h ( f s ( β − i x ) f j ( β − i x ) f k ( β − i x ) f l ( − β − i x ) f m ( − β − i x ) f t ( − β − i x )) i ,s, j, k, l, m, t ∈ { , . . . , } , ≤ i ≤ . n F we have = 49 = − , so for λ ∈ { β i , ≤ i ≤ } , − λ ∈ { (7 α ) i , ≤ i ≤ } .Thus − λ -constacyclic codes of length · · = 1750 over F are ideals of F [ x ] x + λ which isa principal ideal ring, and these codes are generated by h f s ( − β − i x ) f j ( − β − i x ) f k ( − α − i x ) f l (7 β − i x ) f m (7 β − i x ) f t (7 β − i x ) i ,s, j, k, l, m, t ∈ { , . . . , } , ≤ i ≤ . Hence we obtain Table 1. a mp r over F p s Let p be an odd prime number and n = mp r , with m an integer (odd or even) such that( m, p ) = 1. This section provides conditions on the existence of self-dual negacyclic codesof length n = 2 a mp r over F p s . It is well known that negacyclic codes over F p s are principalideals generated by the factors of x mp r + 1. Since F p s has characteristic p , and negacycliccodes are a particular case of constacyclic codes, so by results of Section 4, the polynomial x mp r + 1 can be factored as x mp r + 1 = ( x m + 1) p r . (5)The polynomial x m + 1 is a monic square free polynomial. Hence from [6, Proposition2.7] it factors uniquely as a product of pairwise coprime monic irreducible polynomials f ( x ) , . . . , f l ( x ). Thus from (5) we obtain the following factorization of x mp r + 1 x mp r + 1 = f ( x ) p r . . . f l ( x ) p r . (6)A negacyclic code of length n = mp r over F p s is then generated by a polynomial of the form A ( x ) = Y f ik i , (7)where f i ( x ) , ≤ i ≤ l , are the polynomials given in (6) and 0 ≤ k i ≤ p r .For a polynomial f ( x ) = a + a x . . . + a r x r , with a = 0 and degree r (hence a r = 0),the reciprocal of f is the polynomial denoted by f ∗ and defined as f ∗ ( x ) = x r f ( x − ) = a r + a r − x + . . . + a x r . (8)If a polynomial f ( x ) is equal to its reciprocal, then f ( x ) is called self-reciprocal. We caneasily verify the following equalities( f ( x ) ∗ ) ∗ = f ( x ) and ( f g ( x )) ∗ = f ( x ) ∗ g ( x ) ∗ . (9)It is well known (see [8, Proposition 2.4], that the dual of the negacyclic code generatedby A ( x ) is the negacyclic code generated by B ∗ ( x ) where B ( x ) = x n + 1 A ( x ) . (10)Hence we have the following lemma. 8able 1: The Generators Polynomials of λ -Constacyclic Codes of Length 1750 over F . β i λ = ( β i ) · · f s ( − β − i x ) f j ( − β − i x ) f k ( − β − i x ) f l (7 β − i x ) f m (7 β − i x ) f t (7 β − i x ) β ( β ) · · f s ( − β − x ) f j ( − β − x ) f k ( − β − x ) f l (7 β − x ) f m (7 β − x ) f t (7 β − x ) β · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) β · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) β · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 α − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x ) α · ( β · ) · · f s ( − β − · x ) f j ( − β − · x ) f k ( − β − · x ) f l (7 β − · x ) f m (7 β − · x ) f t (7 β − · x )9 emma 5.1 A negacyclic code C of length n generated by a polynomial A ( x ) is self-dual ifand only if A ( x ) = B ∗ ( x ) . Denote the factors f i in the factorization of x m + 1 which are self-reciprocal by g , . . . g s , andthe remaining f j grouped in pairs by h , h ∗ , . . . , h t , h ∗ t . Hence l = s + 2 t and the factorizationgiven in (6) becomes x n + 1 = ( x m + 1) p r = g p r ( x ) . . . g p r s ( x ) × h p r ( x ) h ∗ p r ( x ) . . . h p r t ( x ) h ∗ p r t ( x ) . (11)In the following we give the structure of negacyclic codes over F p s of length 2 a mp r , a ≥ Lemma 5.2
Let q = p s be an odd prime power such that q ≡ a +1 . Then there is aring isomorphism between the ring F q [ x ] x ampr − and the ring F q [ x ] x ampr +1 Proof. If q ≡ a +1 then by Lemma 4.2 there exists a primitive 2 a +1 -th root α ofunity in F ∗ q . So − − mp r = ( α a ) mp r and then by Proposition 4.1, negacyclic codes oflength 2 a mp r over F q are equivalent to cyclic codes of length 2 a mp r over F q . (cid:4) Corollary 5.3
Let q = p s be an odd prime power such that q ≡ a +1 and n = 2 a mp r with m an odd integer such that ( m, p ) = 1 . Then a negacyclic code of length n over F p s is a principal ideal of F p s [ x ] / h x n + 1 i generated by a polynomial of the following form Q a k =1 ( Q li =0 f j i i ( α − k +1 x )) where ≤ j i ≤ p r and f i ( x ) are monic irreducible factors of x m − . Proof.
It suffices to find the factors of x mp r + 1. Since q ≡ a +1 andfrom Lemma 4.2, there exist α ∈ F ∗ q , a primitive 2 a +1 -th root of unity.So x mp r + 1 canbe decomposed as ( x m + 1) p r = ( Q a k =1 Q li =0 f i ( α − k +1 x )) p r . The result then follows fromthe isomorphisms given in Lemma 5.2.We recall the most important result of [12]. Theorem 5.4 ( [12, Theorem 2.2]) There exists a self-dual negacyclic code of length mp r over F p s if and only if there is no g i (self-reciprocal polynomial) in the factorization of x mp r +1 given in (11). Furthermore, a self-dual negacyclic code C is generated by a polynomial ofthe following form h b ( x ) h ∗ p r − b ( x ) . . . h b t t ( x ) h ∗ p r − b t t ( x ) . (12) (cid:4) In the following we generalize [12, Theorem 3.7] for the length 2 a mp r . But before thatwe need the following lemmas. 10 emma 5.5 ( [12, Lemma 3.5]) Let m be an odd integer and Cl m ( i ) the p s cyclotomic classof i modulo m . The polynomial f i ( x ) is the minimal polynomial associated with Cl m ( i ) ,hence we have Cl m ( i ) = Cl m ( − i ) if and only if f i ( x ) = f ∗ i ( x ) . Lemma 5.6 ( [12, Lemma 3.6]) Let m be an odd integer and p a prime number. Then ord m ( p s ) is even if and only if there exists a cyclotomic class Cl m ( i ) which satisfies Cl m ( i ) = Cl m ( − i ) . Theorem 5.7
Let q = p s be an odd prime power such that q ≡ a +1 , and n = 2 a mp r be an integer with ( m, p ) = 1 and a > . Then there exists a negacyclic self-dual code oflength a mp r over F p s if and only if ord m ( q ) is odd. Proof.
Under the hypothesis on q , and m we have from Corollary 5.3 that the polyno-mial x mp r + 1 = Q a k =1 Q li =0 f j i i ( α − k +1 x )) , where f i ( x ) are the monic irreducible factors of x m − F p s . By Lemma 5.6, ord m ( p s ) is odd if and only if there is no cyclotomic classsuch that Cl m ( i ) = Cl m ( − i ). From Lemma 5.5, this is equivalent to saying that there areno irreducible nontrivial factors of x m − f i ( x ) = f ∗ i ( x ). From corollary Corol-lary 5.3, we obtain that f i ( x ) = f ∗ i ( x ) for all i = 0 ( f ( x ) = ( x − f i ( α k x ) = f ∗ i ( α k x ) are true for all 1 ≤ k ≤ a +1 . Then from Theorem 5.4 self-dual negacycliccodes exist. (cid:4) Example 5.8
A self-dual negacyclic code of length over F does not exist. There is noself-dual negacyclic code of length 30 over F , but there is a self-dual code over F of length . References [1] G. K. Bakshi, M. Raka,
A class of constacyclic codes over a finite field,
Finite FieldsAppl., 18, 362-377, 2012.[2] A. Batoul, K. Guenda and T. A. Gulliver,
Some constacyclic codes over finite chainrings ,to appear in AMC.[3] S.D. Berman,
Semisimple cyclic and abelian codes II,
Cybernetics 3 1723, 1967.[4] G. Castagnoli, J.L. Massey, P.A. Schoeller, N. Von Seemann,
On repeated-root cycliccodes , IEEE Trans. Inform. Theory 37, 337-342, 1991.[5] B. Chen, Y. Fan, L. Lin and H. Liu,
Constacyclic codes over finite fields,
Finite FieldsAppl., 18, 1217-1231, 2012. 116] H.Q. Dinh and S.R. L´opez-Permouth,
Cyclic and negacyclic codes over finite chainrings,
IEEE Trans. Inform. Theory, vol. 50, no. 8, pp. 1728–1744, Aug. 2004.[7] H.Q. Dinh
On the linear ordering of some classes of negacyclic and cyclic codes andtheir distance distributions,
Finite Fields Appl., 14, 22-40, 2008.[8] H.Q. Dinh
Repeated-root constacyclic codes of length p s , Finite Fields Appl., 18, 133-143, 2012.[9] H.Q. Dinh
Structure of repeated-root constacyclic codes of length p s and their duals ,Discrete Math., 313, 983-991, 2013.[10] H.Q. Dinh Structure of repeated-root cyclic and negacyclic codes of length p s and theirduals, AMS Contemporary Mathematics, 609 , 69-87, 2014.[11] G. Falkner, B. Kowol and W. Heise, E. Zehendner,
On the existence of cyclic optimalcodes,
Atti Sem. Mat. Fis. Univ. Modena 28, 326-341. 1979.[12] K. Guenda and T. A. Gulliver,
Self-dual Repeated-Root Cyclic and Negacyclic Codesover Finite Fields , Proc. IEEE Int. Symp. Inform. Theory, pp. 2904-2908, 2012.[13] W.C. Huffman and V. Pless,
Fundamentals of Error-Correcting Codes,
Cambridge Univ.Press, New York, 2003.[14] Y. Jia, S. Ling, and C. Xing,
On Self-dual cyclic codes over finite fields,
IEEE Trans.Inform. Theory, vol. 57, no. 4, Apr. 2011.[15] J.L. Massey, D.J. Costello, J. Justesen,
Polynomial weights and code constructions,
IEEE Trans. Inform. Theory 19, 101-110, 1973.[16] A. Salagean,
Repeated-root cyclic and negacyclic codes over a finite chain ring , DiscreteAppl. Math., 154, 413-419, 2006.[17] J.H. Van Lint,