On a class of (δ+α u 2 ) -constacyclic codes over F q [u]/⟨ u 4 ⟩
aa r X i v : . [ c s . I T ] N ov On a class of ( δ + αu )-constacyclic codes over F q [ u ] / h u i Yuan Cao a ∗ , Yonglin Cao b , Jian Gao b a College of Mathematics and Econometrics, Hunan University, Changsha 410082, China b School of Sciences, Shandong University of Technology, Zibo, Shandong 255091, China
Abstract
Let F q be a finite field of cardinality q , R = F q [ u ] / h u i = F q + u F q + u F q + u F q ( u = 0) which is a finite chain ring, and n be a positive integer satisfyinggcd( q, n ) = 1. For any δ, α ∈ F × q , an explicit representation for all distinct( δ + αu )-constacyclic codes over R of length n is given, and the dual codefor each of these codes is determined. For the case of q = 2 m and δ = 1, allself-dual (1 + αu )-constacyclic codes over R of odd length n are provided. Keywords:
Constacyclic code; Dual code; Self-dual code; Finite chain ring
Mathematics Subject Classification (2000)
1. Introduction
Algebraic coding theory deals with the design of error-correcting and error-detecting codes for the reliable transmission of information across noisy chan-nel. The class of constacyclic codes play a very significant role in the theoryof error-correcting codes.Let Γ be a commutative finite ring with identity 1 = 0, and Γ × be themultiplicative group of units of Γ. For any a ∈ Γ, we denote the ideal of Γgenerated by a as h a i Γ , or h a i for simplicity, i.e., h a i Γ = a Γ = { ab | b ∈ Γ } .For any ideal I of Γ, we will identify the element a + I of the residue classring Γ /I with a (mod I ) for any a ∈ Γ in this paper. ∗ corresponding author.E-mail addresses: yuan − [email protected] (Yuan Cao), [email protected] (Yonglin Cao),[email protected] (J. Gao). Preprint submitted to *** May 14, 2018 code over Γ of length N is a nonempty subset C of Γ N = { ( a , a , . . . , a N − ) | a j ∈ Γ , j = 0 , , . . . , N − } . The code C is said to be linear if C is anΓ-submodule of Γ N . All codes in this paper are assumed to be linear. Theambient space Γ N is equipped with the usual Euclidian inner product, i.e.,[ a, b ] E = P N − j =0 a j b j , where a = ( a , a , . . . , a N − ) , b = ( b , b , . . . , b N − ) ∈ Γ N , and the dual code is defined by C ⊥ E = { a ∈ Γ N | [ a, b ] E = 0 , ∀ b ∈ C} . If C ⊥ E = C , C is called a self-dual code over Γ.Let γ ∈ Γ × and C be a linear code over Γ of length N . C is called a γ - constacyclic code if ( γc N − , c , c , . . . , c N − ) ∈ C for all ( c , c , . . . , c N − ) ∈ C .Particularly, C is called a negacyclic code if γ = −
1, and C is called a cycliccode if γ = 1. For any a = ( a , a , . . . , a N − ) ∈ Γ N , let a ( x ) = P N − i =0 a i x i ∈ Γ[ x ] / h x N − γ i . We will identify a with a ( x ) in this paper. By [5] Propositions2.2 and 2.4, we have Lemma 1.1
Let γ ∈ Γ × . Then C is a γ -constacyclic code of length N over Γ if and only if C is an ideal of the residue class ring Γ[ x ] / h x N − γ i . Lemma 1.2
The dual code of a γ -constacyclic code of length N over Γ is a γ − -constacyclic code of length N over Γ , i.e., an ideal of Γ[ x ] / h x N − γ − i .In this paper, let F q be a finite field of cardinality q , where q is power ofa prime, and denote R = F q [ u ] / h u e i = F q + u F q + . . . + u e F q ( u e = 0) where e ≥
2. Let e = k ≥
3. For the case of p = 2 and m = 1 Abualrub andSiap [1] studied cyclic codes over the ring Z + u Z and Z + u Z + u Z forarbitrary length N , then Al-Ashker and Hamoudeh [2] extended some of theresults in [1], and studied cyclic codes of an arbitrary length over the ring Z + uZ + u Z + . . . + u k − Z ( u k = 0). For the case of m = 1, Han et al.[7] studied cyclic codes over R = F p + uF p + . . . + u k − F p with length p s n using discrete Fourier transform. Singh et al. [9] studied cyclic code over thering Z p [ u ] / h u k i = Z p + uZ p + u Z p + . . . + u k − Z p for any prime integer p andpositive integer N . Kai et al. [8] investigated (1 + λu )-constacyclic codesof arbitrary length over F p [ u ] / h u m i , where λ is a unit in F p [ u ] / h u m i , andCao [3] generalized these results to (1 + wγ )-constacyclic codes of arbitrarylength over an arbitrary finite chain ring R , where w is a unit of R and γ generates the unique maximal ideal of R . Sobhani et al. [10] showed that theGray image of a (1 − u e − )-constacyclic code of length n is a length p m ( e − n quasi-cyclic code of index p m ( e − − .Recently, Sobhani [11] determined the structure of ( δ + αu )-constacycliccodes of length p k over F p m [ u ] / h u i , characterized and enumerated self-dual2odes among these codes, where δ, α ∈ F × p m . Moreover, Sobhani proposedsome open problems and further researches in this area: characterize ( δ + αu ) -constacyclic codes of length p k over the finite chain ring F p m [ u ] / h u e i for e ≥ δ + αu )-constacyclic codes over the finite chain ring F p m [ u ] / h u e i for e = 4. Notation 1.3
Let δ, α ∈ F × q and n be a positive integer satisfying gcd( q, n ) =1. We denote • R = F q [ u ] / h u i = F q + u F q + u F q + u F q ( u = 0). • A = F q [ x ] / h ( x n − δ ) i . • A [ v ] / h v − α − ( x n − δ ) i = A + v A ( v = α − ( x n − δ )).The present paper is organized as follows. In Section 2, we provide anexplicit representation for each ( δ + αu )-contacyclic code over R of length n and obtain a formula to count the number of codewords in each code from itsrepresentation. Then we give the dual code for each of such codes in Section3. In Section 4, we determine all self-dual (1 + αu )-contacyclic code over R of odd length n for the case of q = 2 m . Finally, we list all 125 distinct(1 + u )-contacyclic codes over F [ u ] / h u i of length 7 in Section 5.
2. Representation for ( δ + αu )-constacyclic codes over R of length n In this section, we will construct a specific ring isomorphism from A + v A onto R [ x ] / h x n − ( δ + αu ) i . Hence we obtain a one-to-one correspondence betweenthe set of ideals of A + v A onto the set of ideas of R [ x ] / h x n − ( δ + αu ) i , i.e.,the set of ( δ + αu )-constacyclic codes over R of length n .By Notation 1.3, A + v A = { ξ + vξ | ξ , ξ ∈ A} and the addition andmultiplication are defined by( ξ + vξ ) + ( η + vη ) = ( ξ + η ) + v ( ξ + η ),( ξ + vξ )( η + vη ) = ( ξ η + α − ( x n − δ ) ξ η ) + v ( ξ η + ξ η ),for all ξ , ξ , η , η ∈ A .Let ξ + vξ ∈ A + v A where ξ , ξ ∈ A . It is clear that ξ can be uniquelyexpressed as ξ = ξ ( x ) where ξ ( x ) ∈ F q [ x ] satisfying deg( ξ ( x )) < n (we3ill write deg(0) = −∞ for convenience). Dividing ξ ( x ) by α − ( x n − δ ), weobtain a unique pair ( a ( x ) , a ( x )) of polynomials in F q [ x ] such that ξ = ξ ( x ) = a ( x ) + α − ( x n − δ ) a ( x ) , deg( a j ( x )) < n for j = 0 ,
2. Similarly, there is a unique pair ( a ( x ) , a ( x )) of polynomials in F q [ x ] such that ξ = ξ ( x ) = a ( x ) + α − ( x n − δ ) a ( x ) , deg( a j ( x )) < n for j = 1 ,
3. Denote a k ( x ) = P n − i =0 a i,k x i where a i,k ∈ F q for all i =0 , , . . . , n − k = 0 , , ,
3. Then ξ + vξ can be uniquely writtenas a product of matrices: ξ + vξ = (1 , x, . . . , x n − ) M vα − ( x n − δ ) vα − ( x n − δ ) , where M = ( a i,k ) ≤ i ≤ n − , ≤ k ≤ is an n × F q . Now, we defineΨ( ξ + vξ ) = (1 , x, . . . , x n − ) M uu u = n − X i =0 β i x i , where β i = P k =0 u k a i,k ∈ R for all i = 0 , , . . . , n −
1. Then it is clear thatΨ is a bijection from A + v A onto R [ x ] / h x n − ( δ + αu ) i . Furthermore, by v = α − ( x n − δ ), ( x n − δ ) = 0 in A + v A and x n − ( δ + αu ) = 0 in R [ x ] / h x n − ( δ + αu ) i one can easily verify the following conclustions. Theorem 2.1
Using the notations above, Ψ is a ring isomorphism from A + v A onto R [ x ] / h x n − ( δ + αu ) i . Remark
It is clear that both A + v A and R [ x ] / h x n − ( δ + αu ) i are F q -algebras of dimension 4 n . Specifically, { , x, . . . , x n − , v, vx, . . . , vx n − } isan F q -basis of A + v A , ∪ k =0 { u k , u k x, . . . , u k x n − } is an F q -basis of R [ x ] / h x n − ( δ + αu ) i and Ψ is an F q -algebra isomorphism from A + v A onto R [ x ] / h x n − ( δ + αu ) i determined by:Ψ( x i ) = x i if 0 ≤ i ≤ n − , Ψ( x n ) = δ + αu and Ψ( v ) = u.
4y Theorem 2.1, in order to determine all distinct ( δ + αu )-constacycliccodes over R of length n it is sufficient to list all distinct ideals of A + v A .First, we study the structures of A and A + v A in the following.Since δ ∈ F × q and gcd( q, n ) = 1, there are pairwise coprime monic irre-ducible polynomials f ( x ) , . . . , f r ( x ) in F q [ x ] such that x n − δ = f ( x ) . . . f r ( x ) (1)and ( x n − δ ) = ( x n − δ ) = f ( x ) . . . f r ( x ) . For any integer j , 1 ≤ j ≤ r ,we assume deg( f j ( x )) = d j and denote F j ( x ) = x n − δf j ( x ) . Then F j ( x ) = ( x n − δ ) f j ( x ) and gcd( F j ( x ) , f j ( x ) ) = 1. Hence there exist g j ( x ) , h j ( x ) ∈ F q [ x ] such that g j ( x ) F j ( x ) + h j ( x ) f j ( x ) = 1 . (2)From now on, we adopt the following notations. Notation 2.2
For any 1 ≤ j ≤ r , let ε j ( x ) ∈ A be defined by ε j ( x ) ≡ g j ( x ) F j ( x ) = 1 − h j ( x ) f j ( x ) (mod ( x n − δ ) )and denote K j = F q [ x ] / h f j ( x ) i .By the Chinese remainder theorem for commutative rings, we give thestructure and properties of the ring A . Lemma 2.3
Using the notations above, we have the following :(i) ε ( x ) + . . . + ε r ( x ) = 1 , ε j ( x ) = ε j ( x ) and ε j ( x ) ε l ( x ) = 0 in the ring A for all ≤ j = l ≤ r .(ii) A = A ⊕ . . . ⊕ A r where A j = A ε j ( x ) with ε j ( x ) as its multiplicativeidentity and satisfies A j A l = { } for all ≤ j = l ≤ r .(iii) For any integer j , ≤ j ≤ r , and a ( x ) ∈ K j we define ϕ j : a ( x ) ε j ( x ) a ( x ) (mod ( x n − δ ) ) . Then ϕ j is a ring isomorphism from K j onto A j .(iv) For any a j ( x ) ∈ K j for j = 1 , . . . , r , define ϕ ( a ( x ) , . . . , a r ( x )) = P rj =1 ϕ j ( a j ( x )) = P rj =1 ε j ( x ) a j ( x )(mod ( x n − δ ) ) . Then ϕ is a ring isomorphism from K × . . . × K r onto A .In order to describe the structure of A + v A ( v = α − ( x n − δ )), we needthe following lemma. 5 emma 2.4 Let ≤ j ≤ r and denote ω j = α − F j ( x ) (mod f j ( x ) ) . Then ω j is an invertible element of K j and satisfies α − ( x n − δ ) = ω j f j ( x ) in K j . Proof.
Since ω j ∈ K j satisfying ω j ≡ α − F j ( x ) (mod f j ( x ) ), by Equation(2) it follows that( αg j ( x ) F j ( x )) ω j ≡ ( αg j ( x ) F j ( x )) (cid:0) α − F j ( x ) (cid:1) = 1 − h j ( x ) f j ( x ) ≡ f j ( x ) ) , which implies that ( αg j ( x ) F j ( x )) ω j = 1 in the ring K j . Hence ω j ∈ K × j and ω − j = αg j ( x ) F j ( x ) (mod f j ( x ) ). By Equation (1) and F j ( x ) = x n − δf j ( x ) , wededuce that α − ( x n − δ ) = α − f ( x ) . . . f r ( x ) = α − F j ( x ) f j ( x ) = ω j f j ( x ) . (cid:3) Now, we can provide the structure of A + v A . Lemma 2.5
Let ≤ j ≤ r . Using the notations in Lemma 2.4, we denote K j [ v ] / h v − ω j f j ( x ) i = K j + v K j ( v = ω j f j ( x )) , A j + v A j = ε j ( x )( A + v A ) ( v = α − ( x n − δ )) . Then we have the following conclusions :(i) A + v A = ( A + v A ) ⊕ . . . ⊕ ( A r + v A r ) , where ε j ( x ) is the multiplicativeidentity of the ring A j + v A j and this decomposition satisfies ( A j + v A j )( A l + v A l ) = { } for all ≤ j = l ≤ r .(ii) For any ≤ j ≤ r and a ( x ) , b ( x ) ∈ K j , we define Φ j : a ( x ) + vb ( x ) ε j ( x )( a ( x ) + vb ( x )) (mod ( x n − δ ) ) .Then Φ j is a ring isomorphism from K j + v K j onto A j + v A j .(iii) For any β j , γ j ∈ K j , j = 1 , . . . , r , define Φ( β + vγ , . . . , β r + vγ r ) = r X j =1 Φ j ( β j + vγ j ) = r X j =1 ε j ( x )( β j + vγ j ) . Then Φ is a ring isomorphism from ( K + v K ) × . . . × ( K r + v K r ) onto A + v A . 6 roof. (i) Since ε j ( x ) is an element of A and A is a subring of A + v A , ε j ( x ) is also an idempotent of the ring A + v A for all j = 1 , . . . , r . Then theconclusions follow from Lemma 2.3(i) and classical ring theory.(ii) For any a ( x ) , b ( x ) ∈ K j , by the definition of ϕ j in Lemma 2.3(iii) itfollows that Φ j ( a ( x ) + vb ( x )) = ( ε j ( x ) a ( x )) + v ( ε j ( x ) b ( x ))= ϕ j ( a ( x )) + vϕ j ( b ( x )) . Hence Φ j is a bijection from K j + v K j onto A j + v A j by Lemma 2.3(iii).Let a ( x ) , b ( x ) , a ( x ) , b ( x ) ∈ K j , and denote ξ i = a i ( x ) + vb i ( x ) ∈ K j + v K j for i = 1 ,
2. Since ϕ j is a ring isomorphism from K j onto A j , by Lemmas2.4 and 2.5 we have that Φ j ( ξ + ξ ) = Φ j ( ξ ) + Φ j ( ξ ) andΦ j ( ξ ξ ) = Φ j (( a ( x ) b ( x ) + ω j f j ( x ) a ( x ) b ( x ))+ v ( a ( x ) b ( x ) + a ( x ) b ( x )))= ( ε j ( x ) a ( x ))( ε j ( x ) b ( x ))+ α − ( x n − δ )( ε j ( x ) a ( x ))( ε j ( x ) b ( x ))+ v (( ε j ( x ) a ( x ))( ε j ( x ) b ( x ))+( ε j ( x ) a ( x ))( ε j ( x ) b ( x )))= Φ j ( ξ )Φ j ( ξ ) . Therefore, Φ j is a ring isomorphism from K j + v K j onto A j + v A j .(iii) It follows from (i) and (ii) immediately. (cid:3) In order to determine all ideals of A + v A , by Lemma 2.5(iii) and classicalring theory it is sufficient to list all distinct ideals of K j + v K j ( v = ω j f j ( x ))for all j = 1 , . . . , r . To do this, we need the following lemma. Lemma 2.6 (cf. [4] Example 2.1)
Let ≤ j ≤ r . Then we have the following :(i) K j is a finite chain ring, f j ( x ) generates the unique maximal ideal h f j ( x ) i = f j ( x ) K j of K j , the nilpotency index of f j ( x ) is equal to and theresidue class field of K j modulo h f j ( x ) i is K j / h f j ( x ) i ∼ = F q [ x ] / h f j ( x ) i , where F q [ x ] / h f j ( x ) i is an extension field of F q with q d j elements .(ii) Let T j = { P d j − i =0 t i x i | t , t , . . . , t d j − ∈ F q } . Then |T j | = q d j and ev-ery element ξ of K j has a unique f j ( x ) -adic expansion: ξ = b ( x )+ f j ( x ) b ( x ) , b ( x ) , b ( x ) ∈ T j . Hence |K j | = |T j | = q d j . Moreover, ξ ∈ K × j if and only if b ( x ) = 0 . K j + v K j ( v = ω j f j ( x )). Lemma 2.7
Let ≤ j ≤ r . Then all distinct ideals of K j + v K j are givenby: h v l i , l = 0 , , , , . Moreover, the number of elements contained in h v l i is equal to |h v l i| = q (4 − l ) d j . Proof.
Let ξ + vξ ∈ K j + v K j where ξ , ξ ∈ K j . By Lemma 2.6(ii), each ξ i has a unique f j ( x )-expansion: ξ i = b i, ( x ) + f j ( x ) b i, ( x ), b i, ( x ) , b i, ( x ) ∈ T j ,where i = 0 ,
1. By f j ( x ) = v ω − j in the ring K j + v K j , it follows that ξ + vξ = b , ( x ) + v ω − j b , ( x )+ v (cid:0) b , ( x ) + v ω − j b , ( x ) (cid:1) . (3)By the proof of Lemma 2.4, we see that ω − j = αg j ( x ) F j ( x ) (mod f j ( x ) ).Dividing ω − j b i, ( x ) by f j ( x ) we obtain a unique polynomial h i ( x ) ∈ T j suchthat ω − j b i, ( x ) = f j ( x ) a i ( x ) + h i ( x )for some a i ( x ) ∈ F q [ x ]. From this and by f j ( x ) = 0 in K j , we deduce that v ω − j b i, ( x ) = v h i ( x ) + f j ( x ) · f j ( x ) a i ( x ) = v h i ( x ) , i = 0 , . Then by (3) we obtain the following v -expansion for ξ + vξ : ξ + vξ = b , ( x ) + vb , ( x ) + v h ( x ) + v h ( x ) . Obviously, v = ( v ) = ω j f j ( x ) = 0 and v = vω j ( x ) f j ( x ) = 0. Hence thenilpotency index of v is equal to 4 in K j + v K j . Moreover, by Lemma 2.6(ii)we see that ξ + vξ is invertible if and only if b , ( x ) = 0.As stated above, we conclude that v generates the unique maximal ideal h v i of K j + v K j and the residue class field is ( K j + v K j ) / h v i = { b , ( x ) + h v i | b , ( x ) ∈ T j } satisfying | ( K j + v K j ) / h v i| = |T j | = q d j . Therefore, all distinctideals of K j + v K j are given by: { } = h v i ⊂ h v i ⊂ h v i ⊂ h v i ⊂ h v i = K j + v K j . Furthermore, for any 0 ≤ l ≤ h v l i = { X k = l v k t k ( x ) | t k ( x ) ∈ T j , k = l, . . . , } , |h v l i| = |T j | − l = q (4 − l ) d j . (cid:3) Since Ψ is a ring isomorphism from A + v A onto R [ x ] / h x n − ( δ + αu ) i ,by Lemma 2.3(i) and the definition of Ψ we deduce the following corollary. Corollary 2.8
For any integer j , ≤ j ≤ r , denote e j ( x ) = Ψ( ε j ( x )) ∈ R [ x ] / h x n − ( δ + αu ) i . Then (i) e ( x ) + . . . + e r ( x ) = 1 , e j ( x ) = e j ( x ) and e j ( x ) e l ( x ) = 0 in the ring R [ x ] / h x n − ( δ + αu ) i for all ≤ j = l ≤ r .(ii) If ε j ( x ) = e j, ( x ) + α − ( x n − δ ) e j, ( x ) where e j,i ( x ) ∈ F q [ x ] satisfying deg( e j,i ( x )) ≤ n − for i = 0 , , then e j ( x ) = e j, ( x ) + u e j, ( x ).Finally, we give a precise representation for any ( δ + αu )-constacycliccode over R of length n . Theorem 2.9
Using the notations above, all distinct ( δ + αu ) -constacycliccodes over R of length n are given by : C ( l ,...,l r ) = * r X j =1 u l j e j ( x ) + , ≤ l , . . . , l r ≤ . Moreover, the number of codewords contained in C ( l ,...,l r ) is equal to |C ( l ,...,l r ) | = q P rj =1 (4 − l j ) d j .Therefore, the number of ( δ + αu ) -constacyclic codes over R of length n is equal to r . Proof.
By Theorem 2.1 and Lemma 2.5(iii), we see that Ψ ◦ Φ is a ringisomorphism from ( K + v K ) × . . . × ( K r + v K r ) onto R [ x ] / h x n − ( δ + αu ) i .Let C be an ideal of R [ x ] / h x n − ( δ + αu ) i . By classical ring theory andLemma 2.7, for any integer j , 1 ≤ j ≤ r , there is a unique ideal h v l j i of K j + v K j , where 0 ≤ l j ≤
4, such that C = (Ψ ◦ Φ) (cid:0) h v l i × . . . × h v l r i (cid:1) = Ψ (cid:0) Φ { ( ξ , . . . , ξ r ) | ξ j ∈ h v l j i , j = 1 , . . . , r } (cid:1) = Ψ ( r X j =1 ε j ( x ) ξ j | ξ j ∈ h v l j i , j = 1 , . . . , r )! = Ψ r M j =1 (cid:10) ε j ( x ) v l j (cid:11)! = r M j =1 (cid:10) Ψ( ε j ( x ) v l j ) (cid:11) . ◦ Φ is a ring isomorphism from ( K + v K ) × . . . × ( K r + v K r ) onto R [ x ] / h x n − ( δ + αu ) i , by Lemma 2.7 we have |C| = |h v l i × . . . × h v l r i| = Q rj =1 |h v l j i| = q P rj =1 (4 − l j ) d j .By Corollary 2.8 and the remark after Theorem 2.1, we deduce thatΨ( ε j ( x ) v l j ) = Ψ( ε j ( x ))Ψ( v l j ) = u l j e j ( x ) for all j = 1 , . . . , r , which implies C = r M j =1 (cid:10) u l j e j ( x ) (cid:11) = h u l j e ( x ) , . . . , u l r e r ( x ) i . From this and by Corollary 2.8(i), one can easily verify that C = h u l e ( x ) + . . . + u l r e r ( x ) i .As stated above, we conclude that the number of ( δ + αu )-constacycliccodes over R of length n is equal to 5 r by Lemma 2.7. (cid:3)
3. Dual codes of ( δ + αu )-constacyclic codes C over R of length n In this section, we give the dual code C ⊥ E of any ( δ + αu )-constacyclic code C over R = F q [ u ] / h u i of length n , where δ, α ∈ F × q and gcd( q, n ) = 1. ByLemma 1.2, we know that C ⊥ E is a ( δ + αu ) − -constacyclic code over R oflength n , i.e., C ⊥ E is an ideal of the ring R [ x ] / h x n − ( δ + αu ) − i .In the ring R [ x ] / h x n − ( δ + αu ) − i , we have x n = ( δ + αu ) − , i.e., x − n = δ + αu or ( δ + αu ) x n = 1, which implies x − = ( δ + αu ) x n − in R [ x ] / h x n − ( δ + αu ) − i . (4) Lemma 3.1
Define a map τ : R [ x ] / h x n − ( δ + αu ) i → R [ x ] / h x n − ( δ + αu ) − i by the rule that τ ( a ( x )) = a ( x − ) = n − X i =0 a i x − i = ( δ + αu ) n − X i =0 a i x n − i , (5) for all a ( x ) = P n − i =0 a i x i ∈ R [ x ] / h x n − ( δ + αu ) i with a , a , . . . , a n − ∈ R .Then τ is a ring isomorphism from R [ x ] / h x n − ( δ + αu ) i onto R [ x ] / h x n − ( δ + αu ) − i . Proof.
For any g ( x ) ∈ R [ x ], we define τ ( g ( x )) = g (( δ + αu ) x n − ) = g ( x − ) (mod x n − ( δ + αu ) − ) . τ is a well-defined ring homomorphismfrom R [ x ] to R [ x ] / h x n − ( δ + αu ) − i . For any h ( x ) = P n − i =0 h i x i ∈ R [ x ] / h x n − ( δ + αu ) − i , we select g ( x ) = ( δ + αu ) − P n − i =0 h i x n − i ∈ R [ x ]. Then by (4)and the definition of τ , it follows that τ ( g ( x )) = ( δ + αu ) − n − X i =0 h i (( δ + αu ) x n − ) n − i = x n n − X i =0 h i x i − n = h ( x ) . Hence τ is surjective. Then from τ ( x n − ( δ + αu )) = x − n − ( δ + αu ) = ( δ + αu ) − ( δ + αu ) = 0in R [ x ] / h x n − ( δ + αu ) − i and by classical ring theory, we deduce that the map τ induced by τ , which is defined by (5), is a surjective ring homomorphismfrom R [ x ] / h x n − ( δ + αu ) i onto R [ x ] / h x n − ( δ + αu ) − i . Moreover, it is clearthat | R [ x ] / h x n − ( δ + αu ) i| = | R | n = | R [ x ] / h x n − ( δ + αu ) − i| . Therefore, τ is a bijection and hence a ring isomorphism. (cid:3) Lemma 3.2
For any a = ( a , a , . . . , a n − ) ∈ R n and b = ( b , b , . . . , b n − ) ∈ R n , denote a ( x ) = n − X i =0 a i x i ∈ R [ x ] / h x n − ( δ + αu ) i ,b ( x ) = n − X i =0 b i x i ∈ R [ x ] / h x n − ( δ + αu ) − i . Then [ a, b ] E = P n − i =0 a i b i = 0 if τ ( a ( x )) · b ( x ) = 0 in R [ x ] / h x n − ( δ + αu ) − i . Proof.
By Equation (5) and x n = ( δ + αu ) − in R [ x ] / h x n − ( δ + αu ) − i ,it follows that τ ( a ( x )) · b ( x ) = [ a, b ] E + P n − i =1 c i x i for some c , . . . , c n − ∈ R .Hence [ a, b ] E = 0 if τ ( a ( x )) · b ( x ) = 0 in R [ x ] / h x n − ( δ + αu ) − i . (cid:3) Remark
For any ( δ + αu )-constacyclic code C over R of length n , by Lemma3.2 it follows that { b ( x ) ∈ R [ x ] / h x n − ( δ + αu ) − i | τ ( a ( x )) · b ( x ) = 0 , ∀ a ( x ) ∈C} ⊆ C ⊥ E . Now, we determine the dual code of each ( δ + αu )-constacyclic code over R of length n . 11 heorem 3.3 Let C = h P rj =1 u l j e j ( x ) i be a ( δ + αu ) -constacyclic code over R of length n given by Theorem 2.9. Then the dual code of C is given by : C ⊥ E = * r X j =1 u − l j e j ( x − ) + , which is an ideal of the ring R [ x ] / h x n − ( δ + αu ) − i . Proof.
Let D = h P rj =1 u − l j e j ( x ) i be the ideal of R [ x ] / h x n − ( δ + αu ) i generated by P rj =1 u − l j e j ( x ). Since τ is a ring isomorphism from R [ x ] / h x n − ( δ + αu ) i onto R [ x ] / h x n − ( δ + αu ) − i , τ ( D ) is an ideal of R [ x ] / h x n − ( δ + αu ) − i . From this and by Theorem 2.9, we deduce that | τ ( D ) | = |D| = q P rj =1 (4 − (4 − l j )) d j = q P rj =1 l j d j . (6)Moreover, by Corollary 2.8(i) it follows that τ ( C ) · τ ( D ) = τ ( C · D )= τ h r X j =1 u l j e j ( x ) i · h r X j =1 u − l j e j ( x ) i ! = τ * ( r X j =1 u l j e j ( x ))( r X j =1 u − l j e j ( x )) +! = τ * r X j =1 ( u l j u − l j ) e j ( x ) +! = τ ( { } ) = { } . From this and by Lemma 3.2, we deduce that τ ( D ) ⊆ C ⊥ E . Furthermore, by(6) and Theorem 2.9 it follows that |C|| τ ( D ) | = q P rj =1 (4 − l j ) d j q P rj =1 l j d j = q P rj =1 d j = q n = | R | n . Hence we conclude that C ⊥ E = τ ( D ) since R is a finite chain ring (cf. [6]).Finally, since τ is a ring isomorphism defined in Lemma 3.1, we see that C ⊥ E = h τ ( P rj =1 u − l j e j ( x )) i = h P rj =1 u − l j τ ( e j ( x )) i = h P rj =1 u − l j e j ( x − ) i . (cid:3) . Self-dual (1 + αu )-constacyclic codes over F m + u F m + u F m + u F m of odd length In this section, let q = 2 m where m is a positive integer, R = F m [ u ] / h u i , n be an odd positive integer and α ∈ F × m . As (1 + αu ) − = 1 + αu in R , thedual code of every (1 + αu )-constacyclic code over R of length n is also a(1 + αu )-constacyclic code over R of length n by Lemma 1.2.Morover, by (1 + αu ) − = 1 + αu in R and Lemma 3.1, we see that themap τ defined by τ ( a ( x )) = a ( x − ) ( ∀ a ( x ) ∈ R [ x ] / h x n − (1 + αu ) i ) is a ringautomorphism on R [ x ] / h x n − (1 + αu ) i satisfying τ − = τ . From this andby Corollary 2.8(i), we deduce that for each integer j , 1 ≤ j ≤ r , there is aunique integer j ′ , 1 ≤ j ≤ r , such that τ ( e j ( x )) = e j ( x − ) = e j ′ ( x ) (mod x n − (1 + αu )).Hence the ring automorphism τ on R [ x ] / h x n − (1 + αu ) i induces a permu-tation j j ′ on the set { , . . . , r } . In order to simplify notations, we stilldenote this bijection by τ , i.e., τ ( e j ( x )) = e j ( x − ) = e τ ( j ) ( x ) (mod x n − (1 + αu )) . (7)Since the permutation τ on { , . . . , r } satisfies τ − = τ , After a suitablerearrangement of e ( x ) , . . . , e r ( x ), there are nonnegative integers ρ, ǫ suchthat ρ + 2 ǫ = r and • τ ( j ) = j , 1 ≤ j ≤ ρ ; • τ ( ρ + i ) = ρ + ǫ + i and τ ( ρ + ǫ + i ) = ρ + i , 1 ≤ i ≤ ǫ .Now, by Theorem 3.3 and Equation (7), we obtain the following corollaryimmediately. Corollary 4.1
Let C = h P rj =1 u l j e j ( x ) i be a (1 + αu ) -constacyclic code over R of length n given by Theorem 2.9. Then C ⊥ E = DP rj =1 u − l j e τ ( j ) ( x ) E . Finally, we list all self-dual (1 + αu )-constacyclic codes over R of length n as follows. Theorem 4.2
All distinct self-dual (1 + αu ) -constacyclic codes of length n over R are given by : * ρ X j =1 u e j ( x ) + ǫ X i =1 (cid:0) u l ρ + i e ρ + i ( x ) + u − l ρ + i e ρ + i + ǫ ( x ) (cid:1)+ . herefore, the number of self-dual (1 + αu ) -constacyclic codes over R oflength n is equal to ǫ . Proof.
Let C be any (1 + αu )-constacyclic code over R of length n . ByTheorem 2.9 and its proof, we have that C = L rj =1 h u l j e j ( x ) i where 0 ≤ l j ≤ j = 1 , . . . , r . By Corollary 4.1 and the proof of Theorem 2.9, we seethat C ⊥ E = L rj =1 h u − l j e τ ( j ) ( x ) i . Since τ is a bijection on the set { , . . . , r } ,we have C = L rj =1 h u l τ ( j ) e τ ( j ) ( x ) i . From this and by Corollary 2.8(i), wededuce that C is self-dual if and only if l τ ( j ) = 4 − l j for all j = 1 , . . . , r . Thenwe have one of the following two cases.(i) Let 1 ≤ j ≤ ρ . Then τ ( j ) = j , and l τ ( j ) = 4 − l j if and only if l j = 4 − l j , which is equivalent to that l j = 2.(ii) Let j = ρ + i , where 1 ≤ i ≤ ǫ . In this case, τ ( j ) = j + ǫ and τ ( j + ǫ ) = j . Then l τ ( j ) = 4 − l j and l τ ( j + ǫ ) = 4 − l j + ǫ , i.e., l j + ǫ = 4 − l j and l j = 4 − l j + ǫ , if and only if l j + ǫ = 4 − l j . (cid:3)
5. An example
Let F = { , } and R = F [ u ] / h u i = F + u F + u F + u F ( u = 0).Then R is a finite chain ring of 2 = 16 elements. It is known that x − x + 1 = f ( x ) f ( x ) f ( x ) where f ( x ) = x + 1, f ( x ) = x + x + 1 and f ( x ) = x + x + 1are irreducible polynomials in F [ x ]. Obviously, d j = deg( f j ( x )) where d = 1and d = d = 3.For 1 ≤ j ≤
3, let F j ( x ) = x +1 f j ( x ) and find g j ( x ) , h j ( x ) ∈ F [ x ] such that g j ( x ) F j ( x ) + h j ( x ) f j ( x ) = 1 . Then we calculate ε j ( x ) = g j ( x ) F j ( x ) (mod x +1) . Dividing ε j ( x ) by x + 1, we obtain a unique pair ( e j, ( x ) , e j, ( x )) of poly-nomials in F [ x ] such that ε j ( x ) = e j, ( x )+( x +1) e j, ( x ) and deg( e j,i ( x )) ≤ i = 0 ,
1. Then we have e j ( x ) = e j, ( x ) + u e j, ( x ) ∈ R [ x ] / h x − (1 + u ) i by Corollary 2.8(ii). Precisely, we have e ( x ) = x + ( u + 1) x + x + ( u + 1) x + x + ( u + 1) x + 1; e ( x ) = x + x + ( u + 1) x + 1; e ( x ) = x + ( u + 1) x + ( u + 1) x + 1,where τ ( e ( x )) = e ( x − ) = e ( x ) and τ ( e ( x )) = e ( x − ) = e ( x ) (mod x − (1 + u )). Hence ρ = ǫ = 1. 14 By Theorem 2.9, there are 5 = 125 distinct (1 + u )-constacyclic codesover R of length 7: C ( l ,l ,l ) = (cid:10) g ( l ,l ,l ) ( x ) (cid:11) (mod x − (1 + u )) , where g ( l ,l ,l ) ( x ) = u l e ( x ) + u l e ( x ) + u l e ( x ), 0 ≤ l , l , l ≤ . Moreover,the number of codewords contained in C ( l ,l ,l ) is equal to |C ( l ,l ,l ) | = 2 (4 − l )+3(4 − l )+3(4 − l ) = 2 − ( l +3( l + l )) . • By Theorems 4.2, there are 5 distinct self-dual (1 + u )-constacycliccodes over R of length 7: C (2 ,l, − l ) = (cid:10) g (2 ,l, − l ) ( x ) (cid:11) (mod x − (1 + u )) , ≤ l ≤ , where g (2 , , ( x ) = u x + u x + ( u + 1) x + u x + ( u + 1) x + x + u + 1; g (2 , , ( x ) = ( u + u ) x + ( u + u ) x + ( u + u ) x + ( u + u ) x + ( u + u ) x + ( u + u + u ) x + u + u + u ; g (2 , , ( x ) = u ; g (2 , , ( x ) = ( u + u ) x + ( u + u + u ) x + ( u + u ) x + ( u + u ) x +( u + u + u ) x + ( u + u ) x + u + u + u ; g (2 , , ( x ) = ( u + 1) x + x + x u + x + x u + xu + u + 1. Acknowledgments
Part of this work was done when Yonglin Cao was vis-iting Chern Institute of Mathematics, Nankai University, Tianjin, China.Yonglin Cao would like to thank the institution for the kind hospitality. Thisresearch is supported in part by the National Natural Science Foundation ofChina (Grant Nos. 11471255).
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