Some Repeated-Root Constacyclic Codes over Galois Rings
aa r X i v : . [ c s . I T ] A ug Some Repeated-Root Constacyclic Codes over Galois Rings ∗ Hongwei Liu , Youcef Maouche , School of Mathematics and Statistics, Central China Normal University,Wuhan, Hubei, 430079, China Department of Mathematics, University of Sciences and Technology HOUARI BOUMEDIENE, Alger, Algeria
Abstract.
Codes over Galois rings have been studied extensively during thelast three decades. Negacyclic codes over GR(2 a , m ) of length 2 s have beencharacterized: the ring R ( a, m, −
1) =
GR(2 a ,m )[ x ] h x s +1 i is a chain ring. Further-more, these results have been generalized to λ -constacyclic codes for any unit λ of the form 4 z − z ∈ GR(2 a , m ). In this paper, we study more general casesand investigate all cases where R p ( a, m, γ ) = GR( p a ,m )[ x ] h x ps − γ i is a chain ring. Inparticular, necessary and sufficient conditions for the ring R p ( a, m, γ ) to be achain ring are obtained. In addition, by using this structure we investigate all γ -constacyclic codes over GR( p a , m ) when R p ( a, m, γ ) is a chain ring. Neces-sary and sufficient conditions for the existence of self-orthogonal and self-dual γ -constacyclic codes are also provided. Among others, for any prime p , thestructure of R p ( a, m, γ ) = GR( p a ,m )[ x ] h x ps − γ i is used to establish the Hamming andhomogeneous distances of γ -constacyclic codes. Keywords.
Constacyclic codes; Hamming distances; Repeated-root codes;Codes over rings; Galois rings; Chain rings.2010
Mathematics Subject Classification.
Primary 94B15, 94B05; Secondary11T71.
The importance of codes over finite rings has been recognized since the 1990s; the works of Nechaev [25]and Hammons et al. [18], [5] showed that some well-known nonlinear binary codes such as Kerdock andPreparata codes can be constructed from linear codes over Z , the ring of integers modulo 4. Since thencodes over Z in particular, and codes over finite rings in general, have received a great deal of attention.Constacyclic codes are a generalization of cyclic codes, and they play a very significant role in thetheory of error-correcting codes. Most works on constacyclic codes over finite rings concentrate on the ∗ E-Mail addresses: [email protected] (H. Liu), [email protected] (M. Youcef). s over Z were considered by Abualrub and Oehmke in [1], where such codes were characterized in termsof their sets of generators. The structure of negacyclic codes of length 2 s over Z m was obtained in [14].Moreover, [3] presented a transform approach to classify negacyclic codes of even length over Z .A Galois ring is a Galois extension of the ring Z p a , the ring of integers modulo a prime power p a . In particular, Z p a such as Z is a Galois ring. The class of Galois rings has been used widely asan alphabet for cyclic and negacyclic codes, for instance [6], [28]-[34], [14], [2]-[4]. In 2005, Dinh [9]investigated negacyclic codes of length 2 s over the Galois ring GR(2 a , m ), and showed that the ring R ( a, m, −
1) =
GR(2 a ,m )[ x ] h x s +1 i is in fact a chain ring. In 2007, Dinh [10] computed the Hamming, Lee,homogeneous, and Euclidean distances of all those negacyclic codes over Z a . In [12], the results of Dinhwere extended to λ -constacyclic codes over GR(2 a , m ), for any unit λ of the form 4 z −
1; it was shownthat the λ -constacyclic codes of length 2 s over GR(2 a , m ) are precisely the ideals generated by ( x + 1) i ofthe chain ring R ( a, m, λ ) = GR(2 a ,m )[ x ] h x s − λ i , for i = 0 , , · · · , a s . Using this structure, the Hamming, Lee,homogeneous, and Rosenbloom-Tsfasman (RT) distances of all those λ -constacyclic codes are obtainedin the same paper [12].The aim of this paper is to study the structures and distances of γ -constacyclic codes of length p s over the Galois ring GR( p a , m ) for any unit γ of the form ζ + pζ + p z , where z is an arbitrary elementof GR( p a , m ) and ζ , ζ are nonzero elements of the set T ( p, m ). Here T ( p, m ) denotes a complete set ofrepresentatives of the cosets GR( p a ,m ) p GR( p a ,m ) = F p m in GR( p a , m ). Each unit of this form is called a unit ofType (1). In Section 3, we prove that the cases − z − p = 2. Moreover, for any prime p , we show that the ring R p ( a, m, γ ) = GR( p a ,m )[ x ] h x ps − γ i is a chain ring if andonly if γ is of Type (1). We also derive the duals of all such γ -constacyclic codes as well as necessary andsufficient conditions for the existence of self-orthogonal and self-dual γ -constacyclic codes. In Sections4 and 5, by using the structure obtained in Section 3, the Hamming and homogeneous distances of all γ -constacyclic codes are established respectively. We conclude this paper with open problems in Section6. In this paper, all rings under consideration are associative and commutative rings with identity. An ideal I of a ring R is called principal if it is generated by one element. A ring R is a principal ideal ring if eachof its ideals is principal. We say R is a local ring if it has a unique maximal ideal. Furthermore, a ring2 is called a chain ring if the set of all ideals of R is linearly ordered under set-theoretic inclusion.The following proposition is known for the class of finite commutative chain rings (see [14, Prop.2.1]). Proposition 2.1.
Let R be a finite commutative ring. Then the following conditions are equivalent: ( i ) R is a local ring and the maximal ideal M of R is principal. ( ii ) R is a local principal ideal ring. ( iii ) R is a chain ring. We have the following well-known properties of chain rings.
Proposition 2.2.
Let R be a finite commutative chain ring with maximal ideal M = h r i . Denote thequotient ring ¯ R = RM , and let β be the nilpotency of r . Then ( a ) There is some prime p and positive integers k, l with k ≥ l such that | R | = p k , | ¯ R | = p l , thecharacteristic of R is powers of p and ¯ R is a field. ( b ) The ideals of R are h r i i , where i = 0 , , · · · , β , and they are strictly inclusive: R = h r i ) h r i ) · · · ) h r β − i ) h r β i = h i . ( c ) For i = 0 , · · · , β , |h r i i| = | ¯ R | β − i . In particular, | R | = | ¯ R | β , i.e., k = lβ . A polynomial in Z p a [ x ] is called a basic irreducible polynomial if its reduction modulo p is irreduciblein Z p [ x ]. The Galois ring of characteristic p a and dimension m , denoted by GR( p a , m ), is the Galoisextension of degree m of the ring Z p a . Equivalently,GR( p a , m ) = Z p a [ u ] h h ( u ) i , where h ( u ) is a monic basic irreducible polynomial of degree m in Z p a [ u ]. Note that if a = 1, thenGR( p, m ) = F p m , and if m = 1 then GR( p a ,
1) = Z p a . We list some well-known facts about Galois rings(cf. [24, 21, 27]), which will be used throughout this paper. Proposition 2.3.
Let
GR( p a , m ) = Z pa [ u ] h h ( u ) i be a Galois ring. Then the following hold: ( i ) Each ideal of
GR( p a , m ) is of the form h p k i = p k GR( p a , m ) , for k a . In particular, GR( p a , m ) is a chain ring with maximal ideal h p i = p GR( p a , m ) and residue field F p m . ( ii ) For i a , | p i GR( p a , m ) | = p m ( a − i ) . ( iii ) Each element of
GR( p a , m ) can be represented as vp k , where v is a unit and k a . In thisrepresentation k is unique and v is unique modulo h p a − k i . iv ) h ( u ) has a root ζ in GR( p a , m ) , which is also a primitive ( p m − th root of unity. The set T ( p, m ) = (cid:8) , , ζ, ζ , · · · , ζ p m − (cid:9) is a complete set of representatives of the cosets GR( p a ,m ) p GR( p a ,m ) = F p m in GR( p a , m ) . Each element r ∈ GR( p a , m ) can be written uniquely as r = ζ + pζ + · · · + p a − ζ a − with ζ i ∈ T ( p, m ) , i a − . ( v ) For ≤ i < j ≤ p m − , all ζ i − ζ j are units of GR( p a , m ) . For a finite ring R , consider the set R n of n -tuples of elements from R as a module over R . Anynonempty subset C ⊆ R n is called a code of length n over R , and the code C is linear if in addition, C isan R -submodule of R n . Let λ be a unit of the ring R , then the λ -constacyclic shift τ λ on R n is the shift τ λ (cid:0) x , x , · · · , x n − (cid:1) = (cid:0) λx n − , x , x , · · · , x n − (cid:1) , and a code C is said to be λ -constacyclic if τ λ ( C ) = C , i.e., if C is closed under the λ -constacyclic shift τ λ . In light of this definition, when λ = 1, λ -constacyclic codes are just cyclic codes, and when λ = − λ -constacyclic codes are called negacyclic codes.Each codeword c = ( c , c , · · · , c n − ) of a code C is customarily identified with its polynomial repre-sentation c ( x ) = c + c x + · · · + c n − x n − , and the code C is in turn identified with the set of all polynomialrepresentations of its codewords. Then in the ring R [ x ] h x n − λ i , xc ( x ) corresponds to a λ -constacyclic shift of c ( x ). From that, the following well-known fact is straightforward: Proposition 2.4.
A linear code C of length n over R is λ -constacyclic if and only if C is an ideal of R [ x ] h x n − λ i . Given n -tuples x = ( x , x , · · · , x n − ) , y = ( y , y , · · · , y n − ) ∈ R n , their inner product is defined asusual h x, y i = x y + x y + · · · + x n − y n − ∈ R. Two n -tuples x, y are called orthogonal if h x, y i = 0. For a linear code C over R , its dual code C ⊥ is theset of n -tuples over R that are orthogonal to all codewords of C , i.e., C ⊥ = (cid:8) x (cid:12)(cid:12) h x, y i = 0 , ∀ y ∈ C (cid:9) . A code C is called self-orthogonal if C ⊆ C ⊥ , and it is called self-dual if C = C ⊥ . The following result iswell-known (cf. [14]). Proposition 2.5.
Let R be a finite chain ring of size p α . The number of codewords in any linear code C of length n over R is p k , for some integer k , ≤ k ≤ αn . Moreover, the dual code C ⊥ has p αn − k codewords, so that | C | · | C ⊥ | = | R | n . Note that the dual of a cyclic code is a cyclic code, and the dual of a negacyclic code is a negacycliccode. In general, we have the following implication of the dual of a λ -constacyclic code. Proposition 2.6.
The dual of a λ -constacyclic code is a λ − -constacyclic code. Constacyclic codes of length p s over GR( p a , m ) As mentioned in Section 2, there exists a primitive ( p m − ζ such that the set T ( p, m ) = { , , ζ, ζ , · · · , ζ p m − } is a complete set of representatives of the cosets GR( p a ,m ) p GR( p a ,m ) = F p m in GR( p a , m ). Each element r ∈ GR( p a , m ) can be written uniquely as r = ζ + pζ + p ζ + · · · + p a − ζ a − with ζ i ∈ T ( p, m ), 0 ≤ i ≤ a −
1. To simplify notations, we will say that an element γ ∈ GR( p a , m ) is of Type (0) if it has the form γ = ζ + p ζ + · · · + p a − ζ a − = ζ + p z , with ζ = 0, and γ is said to be of Type (1) if it is of the form γ = ζ + pζ + p ζ + · · · + p a − ζ a − = ζ + pζ + p z , for ζ = 0 = ζ and z ∈ GR( p a , m ). Clearly, the elements of Type (0) and Type (1) are invertible in GR( p a , m ). Furthermore,the sets of Type (0) and Type (1) form a partition of the set of all units of GR( p a , m ) when a ≥
2. Wecall a γ -constacyclic code is of Type (0) (resp. Type (1)) if the unit γ is of Type (0) (resp. Type (1)). ByProposition 2.4, γ -constacyclic codes of length p s over GR( p a , m ) are exactly the ideals of the ambientring R p ( a, m, γ ) = GR( p a , m )[ x ] h x p s − γ i . Proposition 3.1.
Let b and λ be two units of GR( p a , m ). For any positive integer n , there existpolynomials α n ( x ) , β n ( x ) , θ n ( x ) ∈ Z [ x ], such that • If p = 2, then ( x + b ) n = x n + b n + 2 α n ( x ) = x n + b n + 2(( bx ) n − + 2 β n ( x )). Moreover, α n ( x )is invertible in R ( a, m, λ ). • If p is odd, then ( x + b ) p n = x p n + b p n + p ( x + b ) θ n ( x ). Proof.
We prove this proposition by induction on n . If p = 2 and n = 1, then ( x + b ) = x + b + 2 bx , α ( x ) = bx and β ( x ) = 0. Obviously, α ( x ) = bx is a unit in R ( a, m, λ ). Assume n > n . Then( x + b ) n = (( x + b ) n − ) = ( x n − + b n − + 2 α n − ( x )) = x n + b n + 4 α n − ( x ) + 4 b n − α n − ( x ) + 4 x n − α n − ( x ) + 2( bx ) n − = x n + b n + 2 α n ( x ) , where α n ( x ) = ( bx ) n − + 2 β n ( x ) and β n ( x ) = α n − ( x ) + b n − α n − ( x ) + x n − α n − ( x ). We know thatboth of x and b are invertible in R ( a, m, λ ), and so ( bx ) n − is also invertible in R ( a, m, λ ). As 2 isnilpotent in R ( a, m, λ ), the proof is completed for p = 2.5ow suppose p is odd. Let m be a positive integer, then we have( x p m − + b p m − ) p = x p m + b p m + p − X i =1 (cid:18) pi (cid:19) ( b p m − ) i ( x p m − ) p − i = x p m + b p m + p − X i =1 (cid:26)(cid:18) pi (cid:19) ( x p m − ) p − i ( b p m − ) i + (cid:18) pp − i (cid:19) ( x p m − ) i ( b p m − ) p − i (cid:27) = x p m + b p m + p − X i =1 (cid:18) pi (cid:19) b ip m − x ip m − n x p m − ( p − i ) + b p m − ( p − i ) o . Clearly, p m − ( p − i ) is odd, thus there exist polynomials β ′ i ( x ) ∈ Z [ x ], 0 ≤ i ≤ p − , such that x p m − ( p − i ) + b p m − ( p − i ) = ( x + b ) β ′ i ( x ). Then( x p m − + b p m − ) p = x p m + b p m + p − X i =1 (cid:18) pi (cid:19) b ip m − x ip m − ( x + b ) β ′ i ( x )= x p m + b p m + p ( x + b ) p − X i =1 (cid:0) pi (cid:1) p b ip m − x ip m − β ′ i ( x ) . Hence, we have ( x p m − + b p m − ) p = x p m + b p m + p ( x + b ) β ′ m ( x ) , (1)where β ′ m ( x ) = p − X i =1 (cid:0) pi (cid:1) p b ip m − x ip m − β ′ i ( x ) . Plugging in m = 1 yields that the conclusion is true for n = 1. Assume n > n . Then( x + b ) p n = (( x + b ) p n − ) p = ( x p n − + b p n − + p ( x + b ) α n − ( x )) p = ( x p n − + b p n − ) p + p X i =1 (cid:18) pi (cid:19) ( x p n − + b p n − ) p − i ( p ( x + b ) α n − ( x )) i = ( x p n − + b p n − ) p + p ( x + b ) t ( x ) , where t ( x ) = p X i =1 (cid:18) pi (cid:19) ( x p n − + b p n − ) p − i ( p ( x + b ) α n − ( x )) i p ( x + b ) . By using Equation (1) and inductive hypothesis, we get( x + b ) p n = x p n + b p n + p ( x + b ) β ′ n ( x ) + p ( x + b ) t ( x ) = x p n + b p n + p ( x + b ) θ n ( x ) , where θ n ( x ) = β ′ n ( x ) + t ( x ). The proof is complete. (cid:3) R p ( a, m, λ ) is a local ring, and hence in R p ( a, m, λ ) the sum of two noninvertibleelements is noninvertible, and the sum of a noninvertible element and an invertible element is invertible. Lemma 3.2.
Let λ be a unit of Type (1) of GR( p a , m ), i.e., λ = ζ + pζ + p z with some z ∈ GR( p a , m )and ζ , ζ nonzero elements of T ( p, m ). Then there exists an invertible element α in T ( p, m ), such that h ( x − α ) p s i = h p i in R p ( a, m, λ ), and the element x − α is nilpotent with nilpotency ap s . Proof.
We have that T ( p, m ) \{ } ≃ F ∗ p m , and T ( p, m ) \{ } is generated by ζ . Note that gcd( p s , | F ∗ p m | ) =gcd( p s , p m −
1) = 1. This implies that ζ p s is also a generator of T ( p, m ) \{ } . Therefore, there existsinteger i , 0 ≤ i ≤ p m − ζ ip s = ζ . Let α = ζ i then α p s = ζ . If p = 2, by Proposition 3.1 wehave ( x − α ) s = x s + ( − α ) s + 2 α s ( x )= λ + α s + 2[( − xα ) s − + 2 β s ( x )]= ζ + 2 ζ + 4 z + ζ + 2[( xα ) s − + 2 β s ( x )]= 2[( xα ) s − + ζ + ζ + 2( β s ( x ) + z )] . To complete our proof we first need to prove that ( xα ) s − + ζ is noninvertible. Now suppose to thecontrary that ( xα ) s − + ζ is invertible in R ( a, m, λ ), then( xα ) s − − ζ = [( xα ) s − + ζ ] − ζ , is invertible in R ( a, m, λ ), which implies that (( xα ) s − − ζ )(( xα ) s − + ζ ) = ( xα ) s − ζ is also invertiblein R ( a, m, λ ). This is a contradiction, since in R ( a, m, λ )( xα ) s − ζ = λα s − ζ = ( ζ + 2 ζ + 4 z ) ζ − ζ = 2( ζ ζ + 2 ζ z ) . Therefore, ( xα ) s − + ζ is noninvertible in R ( a, m, λ ). Clearly, 2( β n ( x )+ z ) is noninvertible in R ( a, m, λ ),which implies that ζ + (( xα ) s − + ζ ) + 2( β n ( x ) + z ) is invertible. Hence, h ( x − α ) s i = h i , and x − α has nilpotency a s .If p is odd, by using Proposition 3.1 again,( x − α ) p s = x p s + ( − α ) p s + p ( x − α ) α s ( x )= λ − α p s + p ( x − α ) α s ( x )= ζ + pζ + p z − ζ + p ( x − α ) α s ( x )= p ( ζ + pz + ( x − α ) α s ( x )) . Because p is nilpotent in GR( p a , m ), x − α is also nilpotent. It follows that pz + ( x − α ) α s ( x ) is anoninvertible element in R p ( a, m, λ ), which implies that ζ + pz + ( x − α ) α s ( x ) is invertible. Hence, h ( x − α ) p s i = h p i , and x − α has nilpotency ap s . (cid:3) Theorem 3.3.
Let λ = ζ + pζ + p z ∈ GR( p a , m ) be a unit of Type (1). Then the ring R p ( a, m, λ )is a chain ring with maximal ideal h x − α i , where α p s = ζ . The λ -constacyclic codes of length p s p a , m ) are precisely the ideals h ( x − α ) i i of the ring R p ( a, m, λ ), where 0 ≤ i ≤ ap s . Each λ -constacyclic code h ( x − α ) i i has exactly p m ( p s a − i ) codewords. Proof.
Let f ( x ) ∈ R p ( a, m, λ ), then f ( x ) can be expressed as f ( x ) = b + b ( x − α ) + b ( x − α ) + · · · + b p s − ( x − α ) p s − , where b i ∈ GR( p a , m ). Clearly, b ( x − α ) + b ( x − α ) + · · · + b p s − ( x − α ) p s − is noninvertible in R p ( a, m, λ ). Since R p ( a, m, λ ) is a local ring, f ( x ) is noninvertible if and only if b ∈ p GR( p a , m ). Moreover, by Lemma 3.2, p ∈ h ( x − α ) p s i ( h x − α i . Hence, h x − α i is the set of allnoninvertible elements of R p ( a, m, λ ), which implies that R p ( a, m, λ ) is a chain ring with maximal ideal h x − α i . By Lemma 3.2 again, the nilpotency of x − α is ap s , so the ideals of R p ( a, m, λ ) are h ( x − α ) i i ,0 ≤ i ≤ ap s . The rest of the theorem follows readily from the fact that λ -constacyclic codes of length p s over GR( p a , m ) are ideals of the chain ring R p ( a, m, λ ). (cid:3) Lemma 3.4.
Let γ = ζ + pζ + p z and γ = ζ + pζ + p z be two units of Type (1). Let γ = 1 + p z and γ = 1 + p z be two units of Type (0). Let a ≥ a ≥ a , i.e., p a = 0 in GR( p a , m ). Then • γ γ is of Type (1), i.e., the product of a unit of Type (1) and a unit of Type (0) is a unit of Type(1). • γ γ is of Type (0), i.e., the product of two units of Type (0) is a unit of Type (0). • γ − = ζ − (1 − p ( ζ − ζ + pζ − z )) Q a − j =1 [1 + p j ( ζ − ζ + pζ − z ) j ] is of Type (1), i.e., the inverseof a unit of Type (1) is a unit of Type (1). • γ − = (1 − p z ) Q a − j =1 [1 + ( p z ) j ] is of Type (0), i.e., the inverse of a unit of Type (0) is a unitof Type (0). Proof.
The first and the second statements follow readily. For the third statement, observe that(1 − p ( ζ − ζ + pζ − z ))(1 + p ( ζ − ζ + pζ − z )) a − Y j =1 [1 + p j ( ζ − ζ + pζ − z ) j ]= (1 − p ( ζ − ζ + pζ − z ) ) a − Y j =1 [1 + p j ( ζ − ζ + pζ − z ) j ] = 1 − p a ( ζ − ζ + pζ − z ) a = 1 . Therefore, γ − ζ = (1 − p ( ζ − ζ + pζ − z )) a − Y j =1 [1 + p j ( ζ − ζ + pζ − z ) j ] .
8o complete the proof, it suffices to show that ζ − − p ( ζ ζ − + pζ − z ) is of Type (1). Since − ζ ζ − is invertible in GR( p a , m ), − ζ ζ − = ζ ′ + pz, where 0 = ζ ′ ∈ T ( p, m ). It implies that ζ − − p ( ζ ζ − + pζ − z ) = ζ − + pζ ′ + p z ′ , where z ′ = − ζ − z ∈ GR( p a , m ). Hence ζ − − p ( ζ ζ − + pζ − z ) is of Type (1). Note that for1 ≤ j ≤ a −
1, 1 + p j ( ζ − ζ + pζ − z ) j is of Type (0). The rest follows from the first two statements.The proof of the fourth statement is similar to that of the third statement. (cid:3) Proposition 3.5.
Let γ = ζ + pζ + p z ∈ GR( p a , m ) be a unit of Type (1), and let C = h ( x − α ) i i ⊆R p ( a, m, γ ) be a γ -constacyclic code of length p s over GR( p a , m ), for some i ∈ { , , · · · , ap s } , where α p s = ζ . The dual of C is a γ − -constacyclic code of length p s over GR( p a , m ), and C ⊥ = h ( x − α − ) ap s − i i ⊆ R p ( a, m, γ − ) which contains precisely p mi codewords. Proof.
By Proposition 2.6, C ⊥ is a γ − -constacyclic code of length p s over GR( p a , m ). By Lemma 3.4, γ − = ζ − + pζ ′ + p z ′ is also a unit of Type (1). Thus, Theorem 3.3 is applicable for C ⊥ and R p ( a, m, γ − ).Observe that ( α − ) p s = ζ − . Hence, C ⊥ is an ideal of the form h ( x − α − ) j i ⊆ R p ( a, m, γ − ), where0 ≤ j ≤ ap s . On the other hand, by Proposition 2.5, | C | · | C ⊥ | = | GR( p a , m ) | p s = p p s am , which implies that | C ⊥ | = p p s am | C | = p p s am p m ( p s a − i ) = p mi . Therefore, C ⊥ must be the ideal h ( x − α − ) p s a − i i of R p ( a, m, γ − ). (cid:3) The following definition was introduced in [13].
Definition 3.6.
Let C be a linear code of length n over a finite ring R such that C is both α - and β -constacyclic, for distinct units α , β of R . Then C is called a multi-constacyclic code, or more specifically,an [ α, β ]-multi-constacyclic code.It is known that a code C of length n over a finite field F is a multi-constacyclic code if and only if C = { } or C = F n . There are non-trivial multi-constacyclic codes over a finite ring R . Proposition 3.7.
Let λ = ζ + pζ + p z , λ = ζ + pζ ′ + p z be two distinct units of Type (1), andlet C = h ( x − α ) i i ⊆ R p ( a, m, λ ) be a λ -constacyclic code of length p s over GR( p a , m ). Then C is alsoa λ -constacyclic code, i.e., C is a [ λ , λ ]-multi-constacyclic code. Proof.
By the division algorithm, there exist nonnegative integers j, t such that i = tp s + j , 0 ≤ j < p s .Using Lemma 3.2, then we have C = h ( x − α ) i i = h ( x − α ) tp s ( x − α ) j i = h p t ( x − α ) j i . Let c be an arbitrary codeword of C , then c has the form c = p t ( c , c , · · · , c p s − ). Note that C is a λ -constacyclic code, and we have p t ( λ c p s − , c , · · · , c p s − ) = p t (( ζ + pζ + p z ) c p s − , c , · · · , c p s − )= p t ( ζ c p s − , c , · · · , c p s − ) + p t +1 (( ζ + pz ) c p s − , , · · · , ∈ C.
9n the other hand, p t +1 ∈ h p t +1 i = h ( x − α ) ( t +1) p s i ⊆ h ( x − α ) tp s + j i = C. This implies that ( p t +1 , , · · · , ∈ C . Since C is a linear code and p t +1 ( ζ + pz ) c p s − , p t +1 ( ζ ′ + pz ′ ) c p s − ∈ GR( p a , m ), we have p t +1 (( ζ + pz ) c p s − , , · · · ,
0) and p t +1 (( ζ ′ + pz ) c p s − , , · · · , ∈ C, which yields that p t ( λ c p s − , c , · · · , c p s − ) = p t ( ζ c p s − , c , · · · , c p s − ) + p t +1 (( ζ ′ + pz ) c p s − , , · · · , ∈ C. Thus, C is also a λ -constacyclic code. (cid:3) Corollary 3.8.
Let λ = ζ + pζ + p z and λ = ζ + pζ ′ + p z be two units of Type (1). Let C = h ( x − α ) i i ⊆ R p ( a, m, λ ) be a λ -constacyclic code of length p s over GR( p a , m ). Then C is also theideal h ( x − α ) i i of the ring R p ( a, m, λ ), i.e., let c ( x ) ∈ GR( p a , m )[ x ] be a polynomial of degree less than p s , then there exists a polynomial g ( x ) ∈ GR( p a , m )[ x ] such that c ( x ) ≡ g ( x )( x − α ) i mod ( x p s − λ ) ifand only if there exists a polynomial g ′ ( x ) ∈ GR( p a , m )[ x ] such that c ( x ) ≡ g ′ ( x )( x − α ) i mod ( x p s − λ ). Proof.
By Proposition 3.7, C is also a λ -constacyclic code which contains p m ( ap s − i ) codewords. ByProposition 2.4, C is an ideal of the ring R p ( a, m, λ ), because λ is of Type (1) and α p s = ζ . Thus,Theorem 3.3 is applicable for C and R p ( a, m, λ ). Hence, C is the ideal h ( x − α ) i i of the ring R p ( a, m, λ ). (cid:3) Remark 3.9.
Corollary 3.8 gives us very important information about λ -constacyclic codes over GR( p a , m ),where λ is a unit of Type (1). This corollary shows that the λ -constacyclic codes depend on ζ only, whichmeans that there exist just p m − p s over GR( p a , m ) of Type (1). Moreover, inSection 4, we will show that those codes are similar to ¯ λ -constacyclic codes of length p s over F p m . Theorem 3.10.
Let γ = ζ + pζ + p z ∈ GR( p a , m ) be a unit of Type (1), let α p s = ζ , and let C = h ( x − α ) i i be a γ -constacyclic code of length p s over GR( p a , m ). Then the following are true. • If ζ = ζ − , then C is a γ -constacyclic self-orthogonal code of length p s over GR( p a , m ) if and onlyif l ap s m ≤ i ≤ ap s . • If ζ = ζ − , then C is a γ -constacyclic self-orthogonal code of length p s over GR( p a , m ) if and onlyif (cid:6) a (cid:7) p s ≤ i ≤ ap s . Proof.
It follows from Proposition 3.5 that the dual of C is C ⊥ = h ( x − α − ) ap s − i i ⊆ R p ( a, m, γ − ) . If C is self-orthogonal then we have | C | ≤ | C ⊥ | , which gives 2 i ≥ ap s .If ζ = ζ − , by Proposition 3.7, C ⊥ is also a γ -constacyclic code. Observing that α p s = ζ = ζ − = ( α − ) p s and by Corollary 3.8, it follows that C ⊥ = h ( x − α ) ap s − i i ⊆ R p ( a, m, γ ). Hence, C isself-orthogonal if and only if h ( x − α ) i i ⊆ h ( x − α ) ap s − i i if and only if l ap s m ≤ i ≤ ap s .10f ζ = ζ − , by Proposition 2.3 and Lemma 3.4, ζ − ζ − is invertible in GR( p a , m ) and γ − = ζ − + pζ ′ + p z ′ . Now we consider the polynomial x − α in R p ( a, m, γ − ). If p = 2, by Proposition 3.1,we have ( x − α ) s = x s + α s + 2 α s ( x )= γ − + ζ + 2 α s ( x )= ζ − + 2 ζ ′ + 4 z ′ + ζ + 2 α s ( x )= ζ + ζ − + 2( ζ ′ + 2 z ′ + α s ( x )) . If p is odd, then ( x − α ) p s = x p s + ( − α ) p s + p ( x − α ) β s ( x )= γ − − ζ + p ( x − α ) β s ( x )= ζ − − ζ + p ( ζ + pz ′ + ( x − α ) β s ( x )) . This gives that ( x − α ) p s is invertible in R p ( a, m, γ − ). Hence, x − α is also invertible in R p ( a, m, γ − ).By the division algorithm, there exist nonnegative integers t and j , such that i = tp s + j , and by Lemma3.2, we get C = h ( x − α ) i i = h p t ( x − α ) j i and C ⊥ = h ( x − α − ) ap s − i i = h p a − t − ( x − α − ) p s − j i . If j = 0, then C ⊆ C ⊥ if and only if t ≥ (cid:6) a (cid:7) if and only if i ≥ p s (cid:6) a (cid:7) .Now we assume that j = 0. If t < a − t −
1, then | C | > | C ⊥ | , and hence, in this case C is notself-orthogonal. If t = a − t − C ⊆ C ⊥ then p t ( x − α ) j ∈ C ⊥ , which implies that p t ∈ C ⊥ , since x − α is invertible in R p ( a, m, γ − ) by the discussion above. Then j = 0 and it followsthat C is not self-orthogonal in this case either.If t ≥ a − t , then p t ∈ h p a − t i = h ( x − α − ) p s ( a − t ) i ⊆ h p a − t − ( x − α − ) p s − j i = C ⊥ . Therefore, C is self-orthogonal if and only if t ≥ a − t if and only if i ≥ p s (cid:6) a (cid:7) + 1. (cid:3) Corollary 3.11.
Let γ = ζ + pζ + p z be a unit of Type (1) of GR( p a , m ). Then the following are true. • If ζ = ζ − , then there exists a self-dual γ -constacyclic code of length p s over GR( p a , m ) if andonly if ap is even. In this case, h ( x − α ) p s a/ i is the unique self-dual γ -constacyclic code of length p s over GR( p a , m ). • If ζ = ζ − , then there exists a self-dual γ -constacyclic code of length p s over GR( p a , m ) if andonly if a is even. In this case, h p a/ i is the unique self-dual γ -constacyclic code of length p s overGR( p a , m ). 11 roof. Let C be a γ -constacyclic code of length p s over GR( p a , m ), then C = h ( x − α ) i i and C ⊥ = h ( x − α − ) ap s − i i , where 0 ≤ i ≤ ap s . Note that C = C ⊥ if and only if | C | = | C ⊥ | and C ⊆ C ⊥ . If | C | = | C ⊥ | then i = ap s − i . The rest of the proof follows from Theorem 3.10.If ζ = ζ − and a is an odd number or ζ = ζ − and ap is odd, by Theorem 3.10, if C is self-orthogonalthen ap s − i < i . Hence, self-dual γ -constacyclic codes do not exist in this case. (cid:3) Remark 3.12.
The γ -constacyclic codes of Type (1) of GR(2 a , m ), namely γ = ζ + 2 ζ + · · · + 2 a − ζ a − and ζ = 0 = ζ , are a generalization of negacyclic codes which are investigated in [9], and λ -constacycliccodes, where λ is unit of the form 4 z − ζ = ζ = · · · = ζ a − = 1, then γ = 1 + 2 + 2 + · · · + 2 a − = −
1, and in this case a γ -constacyclic code is just anegacyclic code; if we take ζ = ζ = 1, in this case γ = 1 + 2 + 4 ζ + · · · + 2 a − ζ a − = − z . Hence, γ -constacyclic codes generalize both negacyclic and λ -constacyclic codes.In the following, we study the structure of R p ( a, m, γ ), where γ is of Type (0). Theorem 3.13. (Cf. [13]) Let λ be a unit of a finite chain ring R of characteristic p a such that there is anelement λ ∈ R such that λ p s = λ . In R [ x ] h x ps − λ i , x − λ is nilpotent with nilpotency index a p s − ( a − p s − . Proposition 3.14.
Let γ = ζ + p z be a unit of Type (0), and let α ∈ GR( p a , m ) such that α p s = ζ .Then the ring R p ( a, m, γ ) is a local ring with the unique maximal ideal h x − α, p i . If z = 0, then x − α has nilpotency a p s − ( a − p s − . Proof.
By the same method used in Lemma 3.2, we can prove that x − α is noninvertible in R p ( a, m, λ ).Let f ( x ) ∈ R p ( a, m, γ ), then f ( x ) can be expressed as f ( x ) = b + b ( x − α ) + b ( x − α ) + · · · + b p s − ( x − α ) p s − , where b i ∈ GR( p a , m ). Clearly, b ( x − α )+ b ( x − α ) + · · · + b p s − ( x − α ) p s − is noninvertible in R p ( a, m, γ ).Note that, R p ( a, m, γ ) is a local ring, then f ( x ) is noninvertible if and only if b ∈ p GR( p a , m ). Thusthe ideal h p, x − α i is the set of all noninvertible elements of R p ( a, m, γ ). Hence, R p ( a, m, γ ) is a localring with maximal ideal h p, x − α i . Now suppose p ∈ h x − α i . Then there are polynomials f ( x ) and f ( x ) ∈ GR( p a , m )[ x ] such that p = f ( x )( x − α ) + f ( x )( x p s − γ ). Putting x = α then p = f ( α )( α − α ) + f ( α )( α p s − ζ − p z )= f ( α )( ζ − ζ − p z ) = − p zf ( α ) , which is impossible because the nilpotency index of p is equal to a , and the nilpotency index of p zf ( α )is less than a . Obviously, x − α
6∈ h p i . Thus, h p, x − α i is not a principal ideal of R p ( a, m, γ ), whichimplies that R p ( a, m, γ ) is not a chain ring. The rest of the theorem follows from Theorem 3.13. (cid:3) As mentioned earlier, the sets of Type (0) and Type (1) form a partition of the set of all units ofGR( p a , m ) when a ≥
2, then from Proposition 3.14 and Theorem 3.3 we have the following theorem.
Theorem 3.15.
Let γ = ζ + pζ + p z be a unit in GR( p a , m ), then the ring R p ( a, m, γ ) is chain ringif and only if γ is of Type (1), i.e., R p ( a, m, γ ) is chain ring if and only if ζ and ζ are both nonzero.12 Hamming distance
As mentioned in Section 3, γ -constacyclic codes over the Galois ring GR( p a , m ) are exactly ideals of thering R p ( a, m, γ ). If γ is of Type (1), then γ -constacyclic codes are precisely the ideals h ( x − α ) i i of thechain ring R p ( a, m, γ ), where i = 0 , , · · · , p s a .In this section, we will use the structure of γ -constacyclic codes of length p s over GR( p a , m ) tocompute their Hamming distances. By Theorem 3.3 and Lemma 3.2, for 0 ≤ i ≤ ( a − p s , h ( x − α ) i i ⊇ h ( x − α ) ( a − p s i = h p a − i . That means the Hamming distances of the codes h ( x − α ) i i , for 0 ≤ i ≤ ( a − p s are 1. For the remainingvalues of i , i.e., ( a − p s ≤ i ≤ ap s −
1, the main tool is the Hamming distances of all p m -ary constacycliccodes of length p s over F p m , that were established in [11]. Theorem 4.1. (Cf. [[11], Theorem 4.11]) Let C be a λ -constacyclic code of length p s over F p m , then C = h ( λ x + 1) i i ⊆ F pm [ x ] h x ps − λ i , for i ∈ { , , · · · , p s } and λ p s = − λ − . Its Hamming distance d ( C ) iscompletely determined by d ( C ) = , if i = 0 ,l + 2, if lp s − + 1 ≤ i ≤ ( l + 1) p s − , where 0 ≤ l ≤ p − , ( t + 1) p k , if p s − p s − k + ( t − p s − k − + 1 ≤ i ≤ p s − p s − k + tp s − k − ,where 1 ≤ t ≤ p −
1, and 1 ≤ k ≤ s − , , if i = p s . Proposition 4.2.
Let γ = ζ + pζ + p z ∈ GR( p a , m ) be a unit of Type (1), and let C = h ( x − α ) i i ⊆R p ( a, m, γ ) be a γ -constacyclic code, where p s ( a − ≤ i ≤ ap s . Then the Hamming distance of C iscompletely determined by d ( C ) = , if i = p s ( a − ,l + 2, if p s ( a −
1) + lp s − + 1 ≤ i ≤ p s ( a −
1) + ( l + 1) p s − , where 0 ≤ l ≤ p − , ( t + 1) p k , if ap s − p s − k + ( t − p s − k − + 1 ≤ i ≤ ap s − p s − k + tp s − k − ,where 1 ≤ t ≤ p −
1, and 1 ≤ k ≤ s − , , if i = ap s . Proof.
By Lemma 3.2, h ( x − α ) p s i = h p i in R p ( a, m, γ ). By the division algorithm, there existsnonnegative integer 0 ≤ j < p s , such that i = ( a − p s + j . Therefore, h ( x − α ) i i = h ( x − α ) ( a − p s + j i = h p a − ( x − α ) j i . Now, the ideals h p a − ( x − α ) i i of R p ( a, m, γ ) are indeed the sets of elements from the ideals h ( x − α ) i i of F pm [ x ] h x ps − ζ i multiplied by p a − . Hence, the proof follows from Theorem 4.1. (cid:3) From Theorem 3.3 and Propositions 4.2, we now have the Hamming distances of all γ -constacycliccodes of length p s over GR( p a , m ). 13 heorem 4.3. Let γ ∈ GR( p a , m ) be a unit of Type (1), and C be a γ -constacyclic code of length p s over GR( p a , m ), i.e., C = h ( x − α ) i i ⊆ R p ( a, m, γ ), for some integer i ∈ { , , · · · , p s a } . Then theHamming distance of C can be completely determined as follows: d ( C ) = , if 0 ≤ i ≤ p s ( a − ,l + 2, if p s ( a −
1) + lp s − + 1 ≤ i ≤ p s ( a −
1) + ( l + 1) p s − where 0 ≤ l ≤ p − , ( t + 1) p k , if ap s − p s − k + ( t − p s − k − + 1 ≤ i ≤ ap s − p s − k + tp s − k − ,where 1 ≤ t ≤ p −
1, and 1 ≤ k ≤ s − , , if i = ap s . The homogeneous weight on finite rings is a generalization of the Hamming weight on finite fields andthe Lee weight on the residue ring of integers modulo 4. It was first introduced in [8] over integer residuerings, and later over finite Frobenius rings. This weight has numerous applications for codes over finiterings, such as constructing extensions of the Gray isometry to finite chain rings [20, 19, 16], or providinga combinatorial approach to MacWilliams equivalence theorems (cf. [22, 23, 35]) for codes over finiteFrobenius rings [17]. In this section, we shall compute the homogeneous distance of Type (1) constacycliccodes over Galois rings.Let a ≥
2, then the homogeneous weight on GR( p a , m ) is a weight function on GR( p a , m ) given asw h : GR( p a , m ) −→ N , r , if r = 0,( p m − p m ( a − , if r ∈ GR( p a , m ) (cid:15) p a − GR( p a , m ), p m ( a − , if r ∈ p a − GR( p a , m ) (cid:15) { } . The homogeneous weight of a word x = ( x , x , · · · , x n − ) of length n over GR( p a , m ) is the rational sumof the homogeneous weights of its components, i.e., w h ( x ) = n − P i =0 w h ( x i ) . The homogeneous distance (or minimum homogeneous weight) d h of a linear code C is the minimumhomogeneous weight of nonzero codewords of C : d h ( C ) = min (cid:8) w h ( x − y ) (cid:12)(cid:12) x, y ∈ C, x = y (cid:9) = min (cid:8) w h ( c ) (cid:12)(cid:12) c ∈ C, c = 0 (cid:9) . Theorem 5.1.
Let γ ∈ GR( p a , m ) be a unit of Type (1), and let C be a γ -constacyclic code of length p s over GR( p a , m ), i.e., C = h ( x − α ) i i ⊆ R p ( a, m, γ ), for some integer i ∈ { , , · · · , p s a } . Then thehomogeneous distance d h ( C ) of C can be completely determined:14 h ( C ) = , if i = ap s , ( p m − p m ( a − , if 0 ≤ i ≤ p s ( a − ,p m ( a − , if ≤ p s ( a −
2) + 1 ≤ i ≤ p s ( a − , ( l + 2) p m ( a − , if p s ( a −
1) + lp s − + 1 ≤ i ≤ p s ( a −
1) + ( l + 1) p s − , where 0 ≤ l ≤ p − , ( t + 1) p m ( a − k , if ap s − p s − k + ( t − p s − k − + 1 ≤ i ≤ ap s − p s − k + tp s − k − ,where 1 ≤ t ≤ p −
1, and 1 ≤ k ≤ s − . Proof.
By Lemma 3.2, h ( x − α ) p s i = h p i , and therefore h ( x − α ) p s j + t i = h p j ( x − α ) t i .If 0 ≤ i ≤ p s ( a − h i ⊇ C ⊇ h p a − i . Since d h ( h i ) = d h ( h p a − i ) = ( p m − p m ( a − , d h ( C ) = ( p m − p m ( a − .If p s ( a −
2) + 1 ≤ i ≤ p s ( a − h p a − ( x − α ) i ⊇ C ⊇ h p a − i . Clearly, d h ( h p a − i ) = p m ( a − and d h ( h p a − ( x − α ) i ) ≥ p m − p m ( a − ≥ p m ( a − . Thus, p m ( a − ≤ d h ( h p a − ( x − α ) i ) ≤ d h ( C ) ≤ d h ( h p a − i ) = p m ( a − . This implies that d h ( C ) = p m ( a − .If p s ( a −
1) + 1 ≤ i ≤ p s a −
1, then C = h p a − ( x + 1) j i , 1 ≤ j ≤ p s −
1. Let c ∈ C , then c can beexpressed as c = p s − X i =0 p a − c i x i , where c i ∈ GR( p a , m ). Hence, d h ( C ) = d ( C ) p m ( a − and the rest of the proof follows from Theorem 4.3. (cid:3) In this paper we investigated γ -constacyclic codes over GR( p a , m ) of length p s . We showed that theambient ring R p ( a, m, γ ) is a chain ring if and only if γ is of Type (1). Moreover, if γ is of Type (1),then the complete algebraic structure, the Hamming and homogeneous weight for such γ -constacycliccodes are provided. If γ is of Type (0), the ring R p ( a, m, γ ) is a local ring with maximal ideal h p, x − α i ,but not a chain ring. However, the complete algebraic structure of constacyclic codes of Type (0) is stillunknown in general. It is interesting to give the complete algebraic structure of such kind of constacycliccodes and investigate their distances. It is also a challenge to characterize all self-dual and self-orthogonalconstacyclic codes of Type (0). Moreover, it will be very interesting to generalize the results in this paperto constacylic codes over finite commutative chain rings. Acknowledgement.
The authors are very grateful to the two anonymous reviewers and the editor fortheir detailed comments and suggestions that improved the quality of this paper. We are also grateful toProfessor Yun Fan for helpful discussions. This work was supported by NSFC (Grant No. 11171370).15 eferences [1] T. Abualrub and R. Oehmke, “On the generators of Z cyclic codes of length 2 e ”, IEEE Trans. Inf.Theory, vol. 49, no 9, 2126-2133, 2003.[2] N. S. Babu and K. H. Zimmermann, “Decoding of linear codes over Galois rings”, IEEE Trans. Inf.Theory, vol. 47, no 4, 1599-1603, 2001.[3] T. Blackford, “Negacyclic codes over Z of even length”, IEEE Trans. Inf. Theory, vol. 49, no 6,1417-1424, 2003.[4] T. Blackford, “Cyclic codes over Z of oddly even length”, Discrete Appl. Math., vol. 128, no 1,27-46, 2003.[5] A. R. Calderbank, A. R. Hammons, P. V. Kumar, N. J. A. Sloane, and P. Sol´e, “A linear con-struction for certain Kerdock and Preparata codes”, Bull. Amer. Math. Soc., vol. 29, no 2, 218-222,1993.[6] A. R. Calderbank and N. J. A. Sloane, “Modular and p -adic cyclic codes”, Des. Codes Cryptogr.,vol. 6, no 1, 21-35, 1995.[7] G. Castagnoli, J. L. Massey, P. A. Schoeller, and N. von Seemann, “On repeated-root cyclic codes”,IEEE Trans. Inf. Theory, vol. 37, no 2, 337-342, 1991.[8] I. Constaninescu, “Lineare Codes ¨uber Restklassenringen ganzer Zahlen und ihre Automorphismenbez¨uglich einer verallgemeinerten Hamming-Metrik”, Ph.D. dissertation, Technische Universit¨at,M¨unchen, Germany, 1995.[9] H. Q. Dinh, “Negacyclic codes of length 2 s over Galois rings”, IEEE Trans. Inf. Theory, vol. 51, no12, 4252-4262, 2005.[10] H. Q. Dinh, “Complete distances of all negacyclic codes of length 2 s over Z a ”, IEEE Trans. Inf.Theory, vol. 53, no 1, 147-161, 2007.[11] H. Q. Dinh, “Constacyclic codes of length p s over F p m + u F p m ”, J. Algebra, vol. 324, no 5, 940-950,2010.[12] H. Q. Dinh, H. Liu, X. Liu, and S. Sriboonchitta, “On structure and distances of some classes ofrepeated-root constacyclic codes over Galois rings”, Finite Fields Appl., vol. 43, 86-105, 2017.[13] H. Q. Dinh, H. D. Nguyen, S. Sriboonchitta, and T. M. Vo, “Repeated-root constacyclic codes ofprime power lengths over finite chain rings”, Finite Fields Appl., vol. 43, 22-41, 2017.[14] H. Q. Dinh and S. R. L´opez-Permouth, “Cyclic and negacyclic codes over finite chain rings”, IEEETrans. Inf. Theory, vol. 50, no 8, 1728-1744, 2004.[15] S. T. Dougherty and S. Ling, “Cyclic codes over of even length”, Des. Codes Cryptogr., vol. 39, no2, 127-153, 2006. 1616] M. Greferath and S. E. Schmidt, “Gray isometries for finite chain rings and a nonlinear ternary(36 , ,
15) code”, IEEE Trans. Inf. Theory, vol. 45, no 7, 2522-2524, 1999.[17] M. Greferath and S. E. Schmidt, “Finite-ring combinatorics and MacWilliams’s equivalence theo-rem”, J. Combin. Theory, Series A, vol. 92, no 1, 17-28, 2000.[18] A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sol´e, “The Z -linearity ofKerdock, Preparata, Goethals, and related codes”, IEEE Trans. Inf. Theory, vol. 40, no 2, 301-319,1994.[19] W. Heise, T. Honold, and A. A. Nechaev, “Weighted modules and representations of codes”, In :Proceedings of the ACCT, 123-129, 1998.[20] T. Honold and I. Landjev, “Linearly representable codes over chain rings”, Abhandlungen aus demmathematischen Seminar der Universit Hamburg, vol. 69, no. 1, 187-203, 1999.[21] W. C. Huffman and V. Pless, Fundamentals of error-correcting codes, Cambridge university press,2010.[22] F. J. MacWilliams, “Error Correcting Codes for Multiple Level Transmission”, Bell System Tech.J., vol. 40, no 1, 281-308, 1961.[23] F. J. MacWilliams, Combinatorial problems of elementary abelian groups, PhD. Dissertaion, Har-vard University, Cambridge, MA, 1962.[24] B. R. McDonald, Finite rings with identity, Marcel Dekker Incorporated, 1974.[25] A. A. Nechaev, “Kerdock code in a cyclic form”, Discrete Math. Appl., vol. 1, no 4, 365-384, 1991.[26] C. S. Nedeloaia, “Weight distributions of cyclic self-dual codes”, IEEE Trans. Inf. Theory, vol. 49,no 6, 1582-1591, 2003.[27] V. Pless, R. A. Brualdi, and W. C. Huffman, Handbook of coding theory, Elsevier Science Inc.,1998.[28] M. Shi, P. Sol´e, and B. Wu, “Cyclic codes and the weight enumerator of linear codes over F + v F + v F ”, Appl. Comput. Math., vol. 12, no 2, 247-255, 2013.[29] M. Shi and Y. Zhang, “Quasi-twisted codes with constacyclic constituent codes”, Finite FieldsAppl., vol. 39, 159-178, 2016.[30] M. Shi, S. Zhu, and S. Yang, “A class of optimal p -ary codes from one-weight codes over F p [ u ] / h u m i ”,J. Franklin Inst., vol. 350, no 5, 729-737, 2013.[31] R. Sobhani and M. Esmaeili, “Cyclic and negacyclic codes over the Galois ring GR( p , m )”, DiscreteAppl. Math., vol. 157, no 13, 2892-2903, 2009.[32] L. Tang, C. B. Soh, and E. Gunawan, “A note on the q -ary image of a q mm