Quasi-Cyclic Codes Over Finite Chain Rings
aa r X i v : . [ c s . I T ] S e p Quasi-Cyclic Codes Over Finite Chain Rings
Jian Gao, LinZhi Shen, Fang-Wei Fu
Chern Institute of Mathematics and LPMC, Nankai UniversityTianjin, 300071, P. R. ChinaEmail: [email protected]
Abstract
In this paper, we mainly consider quasi-cyclic (QC) codes over finite chain rings. Westudy module structures and trace representations of QC codes, which lead to some lower bounds onthe minimum Hamming distance of QC codes. Moreover, we investigate the structural properties of1-generator QC codes. Under some conditions, we discuss the enumeration of 1-generator QC codesand describe how to obtain the one and only one generator for each 1-generator QC code.
Keywords
Quasi-cyclic codes; Module structure; Trace representation; 1-Generator quasi-cycliccodes
Mathematics Subject Classification (2000) · · A finite commutative ring with identity is called a finite chain ring if its ideals are linearlyordered by inclusion. It is well known that every ideal of finite chain ring is principal andits maximal ideal is unique. Let R denote the finite chain ring, γ a generator of its maximalideal and F the residue field R/ h γ i . The ideals of R form a chain as follows h i = h γ s i ⊆ h γ s − i ⊆ · · · ⊆ h γ i ⊆ h i = R. The integer s is called the nilpotency index of R . If F ∼ = F q , then | R | = q s . In this paper, weassume that F is equivalent to F q .The classical examples of finite chain rings that are not finite fields are the integer residuering Z p s , the Galois ring GR( p m , s ) and the ring F p m + u F p m + · · · + u s − F p m , where p is a prime number and m , s are positive integers such that s ≥
2. Note that the ring F p m + u F p m + · · · + u s − F p m is isomorphic to F p m [ u ] / h u s i , the only finite chain ring withcharacter p and nilpotency index s . Define the ring epimorphism − : R → R = F q by r r , where r denotes r + h γ i . Extending the ring epimorphism − : R [ x ] → F q [ x ] by r + r x + · · · + r n x n r + r x + · · · + r n x n , and the image of f ( x ) ∈ R [ x ] under the map − is denoted by f ( x ) ∈ F q [ x ].Let f ( x ) and g ( x ) be polynomials of R [ x ]. A monic polynomial d ( x ) is called the greatestcommon divisor of f ( x ) and g ( x ) if d ( x ) is a divisor of f ( x ) and g ( x ), and if e ( x ) is a divisorof f ( x ) and g ( x ), then e ( x ) is a divisor of d ( x ). We denote d ( x ) = gcd( f ( x ) , g ( x )). Twopolynomials f ( x ) and g ( x ) are said to be coprime over R if there are two polynomials a ( x )1nd b ( x ) in R [ x ] such that a ( x ) f ( x ) + b ( x ) g ( x ) = 1. It is well known that f ( x ) and g ( x ) arecoprime over R if and only if f ( x ) and g ( x ) are coprime over F q . An interesting thing is thatin R [ x ] two coprime polynomials may have a common divisor with degree ≥
1. However, it isclear that the common divisor must be a unit in R [ x ]. Therefore if f ( x ) and g ( x ) are monicpolynomials over R , then their common divisor is only 1 for the case coprime of them.A polynomial f ( x ) ∈ R [ x ] is said to be basic irreducible if f ( x ) is irreducible in F q [ x ], and basic primitive if f ( x ) is primitive in F q [ x ]. If f ( x ) is a monic basic irreducible polynomialwith degree m over R , then the residue class ring R [ x ] / h f ( x ) i is called the m -th Galoisextension ring of R , and denoted as R . R is also a finite chain ring, with maximal ideal h γ i and nilpotency index s . If ξ is a root of f ( x ), then R = R [ ξ ], i.e., R is a free module of rank m over R with { , ξ, . . . , ξ m − } as a basis. If f ( x ) is a basic primitive polynomialover R , and ξ is the root of f ( x ), then the order of ξ is q m −
1. Let the
Teichm¨uller set be T = { , , ξ, . . . , ξ q m − } . Then each element r of R can be expressed uniquely as r = r + r γ + · · · + r s − γ s − , where r , r , . . . , r s − ∈ T . The units of R form a multiplicative group denoted by R ∗ , whichis a direct product h ξ i × G , where G = { π | π ∈ h γ i} is a group of order q ( s − m .Let R be a finite chain ring and R n be the set of n -tuples over R . C is a linear code oflength n over R if and only if C is an R -submodule of R n .Quasi-cyclic (QC) codes are an important class of linear codes and have some good algebrastructures [2-7, 16-18]. Recently, there are some research papers about QC codes over finitechain rings [2,3,5,7,10,17,22]. In [2], Aydin et al. studied QC codes over Z and obtainedsome new binary codes using the usual Gray map. Moreover, they characterized cyclic codescorresponding to free modules in terms of their generator polynomials. Bhaintwal et al.discussed QC codes over the prime integer residue ring Z q [3]. They viewed a QC codeof length mℓ with index ℓ as an Z q [ x ] / h x m − i -submodule of GR( q, ℓ )[ x ] / h x m − i , whereGR( q, ℓ ) was the ℓ -th Galois extension ring of Z q . A sufficient condition for 1-generator QCcode to be Z q -free was given and some distance bounds for 1-generator QC codes were alsodiscussed. In [7], Cui et al. considered 1-generator QC codes over Z . Under some conditions,they gave the enumeration of quaternary 1-generator QC codes of length mℓ with index ℓ ,and described an algorithm to obtain one and only one generator for each 1-generator QCcode. Based on the idea in [7], we studied 1-generator QC codes over another special finitechain ring F q + u F q [10]. More recently, Cao used polynomial theory presented in [20] tostudy 1-generator QC codes over arbitrary finite chain rings [5]. He discussed the paritycheck polynomials of 1-generator QC codes and gave an explicit enumerator and algorithmfor finding the generator of 1-generator QC codes with some fixed parity check polynomial.However, it should be noted that all research papers discussed above base on one fact thatthe block length of the QC code is coprime with the characteristic of finite chain ring. In [22],Siap, et al. studied 1-generator QC codes of arbitrary lengths over finite chain ring F + u F .They gave the generating set and the free condition of the 1-generator QC code. Using Garymap, they also got some optimal binary linear codes over finite field F .2n this paper, we mainly consider QC codes over general finite chain rings. The paper isorganized as follows. In section 2, we sketch some well known structural properties of cycliccodes over finite chain rings. Moreover, we discuss trace representations of cyclic codes. Somefurther results are obtained in this section (Proposition 2.7 and Theorem 2.8). In section 3,we discuss module structures of QC codes over finite chain rings, which are generalizationsof QC codes over finite fields. This point of view for studying QC codes could give a lowerbound on the minimum Hamming distance (Theorem 3.1) and a construct method of linearcodes over finite fields (Theorem 3.2). In section 4, we discuss the trace representation of QCcodes over finite chain rings, which lead to another lower bound on the minimum Hammingdistance (Theorem 4.3). In section 5, we investigate 1-generator QC codes. We give thestructure of annihilator of the 1-generator QC code (Theorem 5.1) and the free condition for1-generator QC code to be free (Theorem 5.2). In section 6, we give a sufficient and necessarycondition for the 1-generator QC code to be unique (Theorem 6.1). Under some conditionsand using another method different from the point of view presented in [5], we also give anexplicit enumeration formula for 1-generator QC codes with a fixed annihilator (Theorem6.4). In section 7, we describe how to obtain one and only one generator for each 1-generatorQC code (Theorem 7.2). Finally, we give an example to illustrate the main work in section 6and section 7. Let T be a cyclic shift operator T : R n → R n , which transforms v = ( v , v , . . . , v n − ) into vT = ( v n − , v , . . . , v n − ). A linear code C is called the cyclic code of length n if it is invariantunder T . In this paper, we assume n to be a positive integer not divisible by the characteristicof the residue field F = F q , so that x n − F q [ x ]. Therefore x n − R [ x ].Let f ( x ) be a factor of x n − R . Denote b f ( x ) = ( x n − /f ( x ). It is well known that thecyclic code of length n over R can be regarded as an ideal of R [ x ] / h x n − i . In the following,we list the Chinese Reminder Theorem (CRT) first, which will be used in other sections.
Theorem 2.1 (cf. [23] Theorem 2.9)
Let f , f , . . . , f r be pairwise coprime monic polynomialsof degree > over R , f = f f . . . f r and R f = R [ x ] / h f i . Let b f i = f /f i . Then there exist a i , b i ∈ R [ x ] such that a i f i + b i b f i = 1 . Let e i = b i b f i + h f i . Then (1) e , e , . . . , e r are mutually orthogonal non-zero idempotents of R f ; (2) 1 = e + e + · · · + e r in R f ; (3) Let R f e i = h e i i be the principal ideal of R f generated by e i . Then e i is the identity of R f e i and R f e i = h b f i + h f ii ; (4) R f = L ri =1 R f e i ; (5) The map R [ x ] / h f i i → R f e i defined by g + h f i i 7→ h g + h f ii e i is a well-defined isomorphismof rings; (6) R f = R [ x ] / h f i ≃ L ri =1 R [ x ] / h f i i . ✷ R , there have been some wellknown results as follows. Theorem 2.2 (cf. [9] Theorem 3.2, 3.4, 3.6)
Let x n − f f · · · f r be a representation of x n − as a product of monic basic irreducible pairwise coprime polynomials over R . Then (1) Any ideal of R [ x ] / h x n − i is a sum of ideals of the form γ j b f i + h x n − i , where ≤ j ≤ s , ≤ i ≤ r . (2) Let C be a cyclic code over R . Then there exist a unique family of pairwise coprimemonic polynomials F , F , . . . , F s in R [ x ] (possibly, some of them are equal to ) such that F F · · · F s = x n − and C = h b F , γ b F , . . . , γ s − b F s i . (3) R [ x ] / h x n − i is a principal ideal ring. Let C be a cyclic code with notation in (2) . Then C = h b F + γ b F + · · · + γ s − b F s − i . ✷ Let C be a cyclic code of length n generated by g ( x ) over R . Unlike the case over finitefields, g ( x ) may be not a divisor of x n −
1. It is related to whether C is a free R -module ornot. Theorem 2.3 (cf. [20] Proposition 4.11)
Let C be a linear code over finite chain ring R .Then the following properties are equivalent (1) C is the Hensel lift of a cyclic code over R ; (2) C is a cyclic code and free; (3) There exists a polynomial g ( x ) ∈ R [ x ] such that C = h g ( x ) i and g ( x ) | x n − ; (4) There exists a monic polynomial g ( x ) ∈ R [ x ] such that { g ( x ) } is the generating set instandard form for C ; (5) The dual code of C is the Hensel lift of a cyclic code over R . ✷ By Theorem 2.3, we can get the following proposition immediately.
Proposition 2.4
A nonzero cyclic code C of length n over R is a free R -module if and onlyif it is generated by a polynomial g ( x ) dividing x n − over R . Moreover, if C is a free cycliccode, then the rank of C is n − deg g ( x ) and the set { g ( x ) , xg ( x ) , . . . , x n − deg g ( x ) − g ( x ) } formsan R -basis for C . ✷ Cyclic code C of length n over R can also be described in terms of primitive roots of unity.Let ξ i , ξ i , . . . , ξ i k be n -th primitive roots of unity in some Galois extension ring R of R .Then the corresponding cyclic code C can be defined as C = { c ( x ) ∈ R [ x ] / h x n − i| c ( ξ i j ) =0 , ≤ j ≤ k } . The generator polynomial g ( x ) of C is the least common multiple of theminimal polynomials of ξ i j , 1 ≤ j ≤ k . Obviously, g ( x ) is a monic divisor of x n − R .From Proposition 2.4, C is free.In the following, we consider the BCH-type bound of the cyclic code over finite chain ring R . The process of proof is analogous to that of the cyclic code over finite field. Here we omitit. 4 roposition 2.5 Let g ( x ) be the generator polynomial of the free cyclic code C of length n and a monic divisor of x n − over R . Suppose g ( x ) has roots ξ b , ξ b +1 , . . . , ξ b + δ − , where ξ is an n -th primitive root of unity in some Galois extension ring of R . Then the minimumHamming distance of C is d ( C ) ≥ δ . ✷ Example 2.6
Let R = F + u F . It is well known that the polynomial f ( x ) = x + x + 1is an irreducible and primitive factor of x − F . Since F [ x ] is a subring of R [ x ], wecan regard f ( x ) as a polynomial over R , i.e., the trivial Hensel lift of f ( x ) ∈ F [ x ] to R [ x ].Denote R = R [ x ] / h f ( x ) i . Let ξ be a root of f ( x ). Then ξ is a basic primitive element in R , i.e., an element of order 2 − R . Suppose the generator polynomial of a cycliccode C of length 7 over R be defined by g ( x ) = lcm( M ( x ) , M ( x ) , M ( x )), where M ( x ), M ( x ), M ( x ) are minimal polynomials of 1, ξ and ξ , respectively. Then M ( x ) = x + 1, M ( x ) = M ( x ) = x + x + 1 implying g ( x ) = ( x + 1)( x + x + 1) = x + x + x + 1. Clearly, g ( x ) is a monic divisor of x − R . Therefore, from Proposition 2.4, C is free over R and the set { g ( x ) , xg ( x ) , x g ( x ) } forms an R -basis of C . By Proposition 2.5, we have thatthe minimum Hamming distance of C is d ( C ) ≥
4. The generator matrix for C is given asfollows . (1)Since g ( x ) has the Hamming weight 4, the minimum Hamming distance of C , d ( C ) = 4actually. Thus C is a free cyclic code over R with parameters [7 , , x n − f f · · · f r , where each f i is a monic basic irreducible polynomial over finitechain ring R , i = 1 , , . . . , r . Suppose that ξ is an n -th primitive root of unity and R isthe smallest Galois extension ring of R , which contains the n -th primitive root of unity ξ .Therefore x n − x − x − ξ ) · · · ( x − ξ n − ) over R . Define the map π as follows π : R [ x ] / h x n − i → n − M i =0 R [ x ] / h x − ξ i i c ( x ) = c + c x + · · · + c n − x n − ( c (1) , c ( ξ ) , . . . , c ( ξ n − )) . If c ( x ) ∈ R [ x ] / h x n − i , then from Theorem 2.1, we can deduce π is an R [ x ]-module homomor-phism. Denote c ( ξ i ) = A i and A ( z ) = P n − i =0 A i z n − i . The A ( z ) is called Mattsom-Solomonpolynomial associated with c ( x ). Clearly,( A , A , . . . , A n − ) = ( c , c , . . . , c n − ) . . . ξ . . . ξ n − ... ... ... ...1 ξ n − . . . ξ ( n − . (2)For this reason A ( z ) is sometimes called a discrete Fourier transform of c ( x ). The inversetransform is given by c j = 1 n n − X k =0 A k ξ − jk , j = 0 , , . . . , n − . R is an m -th Galois extension ring of finite chain ring R . It is well knownthat R is also with maximal ideal h γ i and the residue field F = R / h γ i is F q m . Every element r of R can also be expressed uniquely in the form r = r + r γ + · · · + r s − γ s − , where r , r , . . . , r s − belong to the Teichm¨uller set T = { , , ζ, . . . , ζ q m − } , where ζ is a( q m − R . Define the Frobenius map φ on R to be the mapinduced by the map r + r γ + · · · + r s − γ s − r q + r q γ + · · · + r qs − γ s − , acting as theidentity on R . Since the degree of the extension R over R is m , φ m is the identity map. Theset of automorphism of R over R forms a group with respect to the composition of maps,which is called the Galois group of R over R and is denoted by Gal( R /R ). It is well knownthat Gal( R /R ) = h φ i . For any r ∈ R , we define the trace of r to be T r R /R ( r ) = r + φ ( r ) + · · · + φ m − ( r ) . Since φ i ( r ) = r q i + r q i γ + · · · + r q i s − γ s − , we have T r R /R ( r ) = T r R /R ( r ) + T r R /R ( r ) γ + · · · + T r R /R ( r s − ) γ s − . By Hensel lift, there is a one-to-one correspondence between factors of x n − q -cyclotomic cosets of Z n . Denote by U i (1 ≤ i ≤ r ) the cyclotomic coset correspondence to f i .Let R i be the Galois extension ring of R corresponding to the basic irreducible polynomial f i , i.e., R i = R [ x ] / h f i i . Then for a fixed u i ∈ U i , we have nc j = r X i =1 T r R i /R ( A i ξ − ju i ) . Sometimes this is called the trace representation of the cyclic code over finite chain ring R .In the following, we give a slight different trace representation of the cyclic code over finitechain ring R . Proposition 2.7
Let C be a free cyclic code of length n over finite chain ring R . Suppose thatnon-negative integers i , i , . . . , i k are in different q -cyclotomic cosets in Z n . Let ξ be an n -thprimitive root of unity and ξ i , ξ i , . . . , ξ i k be roots of the polynomial m ( x ) = Q kj =1 M j ( x ),where m ( x ) is the generator polynomial of C ⊥ and M j ( x ) is the minimal polynomial of ξ i j over R . Then for any codeword c ( x ) = c + c x + · · · + c n − x n − of C , we have c v = k X j =1 T r R /R a j ξ vi j , where a j ∈ R , v = 0 , , . . . , n −
1, and R is the smallest Galois extension ring of R containingthe n -th primitive root of unity ξ . Proof
Let k = 1. Consider the following set C = { ( c , c , . . . , c n − ) ∈ R n | c v = T r R /R a j ξ vi , v = 0 , , . . . , n − } . C is a nonzero linear code of length n over R . If c a j ( x ) = P n − v =0 T r R /R a j ξ vi x v ,then c a j ξ − i ( x ) = c a j ( x ) x in R [ x ] / h x n − i implying that C is cyclic. On the other hand the freecyclic code h M ( x ) i is contained in the dual code C ⊥ of C , which implies that h M ( x ) i ⊥ ⊇ C .It should be noted that h M ( x ) i ⊥ is a minimal free cyclic code with rank equality to thedegree of M ( x ), i.e., the minimal polynomial of ξ i over R . Since R is a principal ideal ring,the cyclic code C is also free over R implying that C = h M ( x ) i ⊥ .For k ≥
2, using the facts that any free cyclic code is the direct sum of some minimal freecyclic codes, we can get the result in this proposition. ✷ It is easy to see that c v = 0 if each a j = 0, j = 1 , , . . . , k . But sometimes c v may beidentical zero even if there is { l , l , . . . , l d } ⊆ { , , . . . , k } such that a l z = 0, z = 1 , , . . . , d .Therefore we could ask a question that when c v is zero except the case a j = 0 for each j = 1 , , . . . , k ? In the following we give an answer about this question. Theorem 2.8
Let U vi j be a q -cyclomotic coset containing vi j mod n for each j = 1 , , . . . , k .Let a , a , . . . , a k ∈ R\{ } . Then c v = 0 if and only if | U vi j | = τ vi j = m and T r R / e R j ( a j ) = 0 ,where e R j is the τ vi j -th Galois extension ring of R for all j = 1 , , . . . , k .Proof Firstly, we will prove c v = 0 if and only if T r R /R a j ξ vi j = 0 for all j = 1 , , . . . , k .Let a j = a j + a j γ + · · · + a j,s − γ s − , where a jg ∈ T = { , , ζ, . . . , ζ q m − } , ζ is abasic primitive element in R , j = 1 , , . . . , k and g = 0 , , . . . , s −
1. Then c v = 0 ifand only if P kj =1 T r R /R a j ξ vi j = P kj =1 T r R /R ( a j ξ vi j ) + γ P kj =1 T r R /R ( a j ξ vi j ) + · · · + γ s − P kj =1 T r R /R ( a j,s − ξ vi j ) = 0 if and only if P kj =1 T r R /R ( a jg ξ vi j ) = 0 for all g = 0 , , . . . , s − T r R /R ( a jg ξ vi j ) = 0 for all j = 1 , , . . . , k and g = 0 , , . . . , s − T r R /R a j ξ vi j = 0 for all j = 1 , , . . . , k .Secondly, we will prove T r R /R a j ξ vi j = 0 if and only if | U vi j | = τ vi j = m and T r R / e R j ( a j ) =0. Since τ vi j necessarily divides m , e R is a subring of R . Therefore T r R / e R makes sense. From a j ∈ R \ { } , we have | U vi j | 6 = m . By Theorem 2.2 in [12], we deduce that there are q ( m − τ vij ) s a j ’s in R such that T r R /R a j ξ vi j = 0. The number of elements in the kernel of T r R / e R is also q ms /q τ vij s = q ( m − τ vij ) s . For any b j in this kernel, we have T r R /R ( b j ξ vi j ) = T r e R /R ( T r R / e R ( b j ξ vi j )) = T r e R /R ( ξ vi j T r R / e R ( b j )) = 0 . Thus we have a j must be in the kernel of T r R / e R . Conversely, reading the above equality fromleft to right, replacing b j by a j , proves the claim. ✷ Let T be a cyclic shift operator T : R N → R N , which transforms v = ( v , v , . . . , v N − )into vT = ( v N − , v , . . . , v N − ). A linear code C is called quasi-cyclic (QC) code if it isinvariant under T ℓ for some positive integer ℓ . The smallest ℓ such that T ℓ ( C ) = C is calledthe index of C . Clearly, ℓ is a divisor of N . Let N = nℓ . Define an R -module isomorphismas follows ρ : R nℓ → ( R [ x ] / h x n − i ) ℓ v , v , . . . , v ,ℓ − , v , v , . . . , v ,ℓ − , . . . , v n − , , v n − , , . . . , v n − ,ℓ − ) ( v ( x ) , v ( x ) , . . . , v ℓ − ( x )) , where v i ( x ) = P n − j =0 v ji x j , i = 0 , , . . . , ℓ −
1. Then the QC code C is equivalent to saying that,for any ( v ( x ) , v ( x ) , . . . , v ℓ − ( x )) ∈ ρ ( C ), ( xv ( x ) , xv ( x ) , . . . , xv ℓ − ( x )) ∈ ρ ( C ). Therefore, C is a QC code of length nℓ with index ℓ if and only if ρ ( C ) is an R [ x ] / h x n − i -submodule of( R [ x ] / h x n − i ) ℓ . This definition of the QC code is known as conventional row circulant. Butin this section, we generalize another definition of the QC code in [16] to finite chain ring R .Let v = ( v , v , . . . , v ,ℓ − , v , v , . . . , v ,ℓ − , . . . , v n − , , v n − , , . . . , v n − ,ℓ − ) ∈ R nℓ .Define an isomorphism between R nℓ and R n by associating with each ℓ -tuple ( v i , v i , . . . , v i,ℓ − ), i = 0 , , . . . , n −
1, and the element v i ∈ R represented as v i = v i + v i ξ + · · · + v ℓ − ξ ℓ − , wherethe set { , ξ, ξ , . . . , ξ ℓ − } forms an R -basis of R . Then every element in R nℓ is one-to-one cor-respondence with an element in R n . The operator T ℓ for some element ( v , v , . . . , v ,ℓ − , . . . ,v n − , , v n − , , . . . , v n − ,ℓ − ) ∈ R nℓ corresponds to the element ( v n − , v , . . . , v n − ) ∈ R n . In-dicating the block positions with increasing powers of x , the vector v ∈ R nℓ can be associatedwith the polynomial v + v x + · · · + v n − x n − ∈ R [ x ]. An R [ x ] / h x n − i -module isomorphismbetween R nℓ and R [ x ] / h x n − i , which is defined as ψ ( v ) = v + v x + · · · + v n − x n − . Inthis setting, multiplication by x of any element of R [ x ] / h x n − i is equal to applying T ℓ tooperate the element of R nℓ . It follows that there is a one-to-one correspondence between R [ x ] / h x n − i -submodule of R [ x ] / h x n − i and the QC code of length nℓ with index ℓ over R .In addition, let C be a QC code of length nℓ with index ℓ over R . It is also can be regardedas an R -submodule of R [ x ] / h x n − i because of the equivalence of R nℓ and R [ x ] / h x n − i .Let C be a QC code of length nℓ with index ℓ over R , and generated by elements v ( x ) , v ( x ) , . . . , v r ( x ) ∈ R [ x ] / h x n − i as an R [ x ] / h x n − i -submodule of R [ x ] / h x n − i . Then C = { a ( x ) v ( x )+ a ( x ) v ( x )+ · · · + a r ( x ) v r ( x ) | a i ( x ) ∈ R [ x ] / h x n − i , i = 1 , , . . . , r } . As dis-cussed above, C is also an R -submodule of R [ x ] / h x n − i . For an R -submodule of R [ x ] / h x n − i , C is generated by the following set { v ( x ) , xv ( x ) , . . . , x n − v ( x ) , . . . , v r ( x ) , xv r ( x ) , . . . , x n − v r ( x ) } .If C is generated by a single element v ( x ) as an R [ x ] / h x n − i -submodule of R [ x ] / h x n − i ,then C is called the 1 -generator QC code. Let the preimage of v ( x ) in R nℓ be v . Then for the1-generator QC code C , we have C is generated by the set { v, T ℓ v, . . . , T ℓ ( n − v } . It is theconventional of row circulant definition of 1-generator QC code. In fact, let v ( x ) = v + v x + · · · + v n − x n − be a polynomial in R [ x ] / h x n − i , where v i = v i + v i ξ + · · · + v i,ℓ − ξ ℓ − , i = 0 , , . . . , n −
1. Then v ( x ) becomes an ℓ -tuple of polynomials over R each of degree atmost n − R -basis { , ξ, ξ , . . . , ξ ℓ − } . Therefore, v ( x ) becomes an element of( R [ x ] / h x n − i ) ℓ . So C is an R [ x ] / h x n − i -submodule of ( R [ x ] / h x n − i ) ℓ , i.e. the conventionalrow circulant definition of QC code.Since R [ x ] / h x n − i is a subring of R [ x ] / h x n − i and C is an R [ x ] / h x n − i -submodule of R [ x ] / h x n − i , it is in particular a submodule of an R [ x ] / h x n − i -submodule of R [ x ] / h x n − i ,i.e. the cyclic code e C of length n over R . Therefore, d ( C ) ≥ d ( e C ) , d ( C ) and d ( e C ) are minimum Hamming distances of C and e C , respectively. Theorem 3.1
Let C be an r -generator QC code of length nℓ with index ℓ over R and generatedby the set { v ( x ) , v ( x ) , . . . , v r ( x ) } , where v i ( x ) ∈ R [ x ] / h x n − i , i = 1 , , . . . , v . Then C hasa lower bound on the minimum Hamming distance given by d ( C ) ≥ d ( e C ) d ( B ) , where e C is a cyclic code of length n over R with generator polynomials v ( x ) , v ( x ) , . . . , v r ( x ) ,and B is a linear code of length ℓ generated by the set {V ij , i = 1 , , . . . , r, j = 0 , , . . . , n − } ⊆ R ℓ where each V ij is the vector equivalent of the j -th coefficient of v i ( x ) with respect to an R -basis { , ξ, . . . , ξ ℓ − } .Proof The result follows from the Theorem 5 in [16]. ✷ In the reset of this section, we give an application of the above discussion. This applicationleads to construct QC codes over finite fields.Let R = F q + u F q + · · · + u ℓ − F q , where u ℓ = 0 and ℓ is a positive integer. Consider acyclic code e C of length n generated by a polynomial v ( x ) over R . Let C be a linear code oflength nℓ spanned by the set { v ( x ) , xv ( x ) , . . . , x n − v ( x ) } over F q . Then C is a 1-generatorQC code of length nℓ with index ℓ . If v ( x ) = v + v x + · · · + v n − x n − ∈ R [ x ] / h x n − i , theneach v i is an ℓ -tuple with respect to the fixed F q -basis { , u, . . . , u ℓ − } of R . Now let the set { v , v , . . . , v n − } generate a linear code B of length ℓ over F q . By Theorem 3.1, we have Theorem 3.2
Let C be a quasi-cyclic code of length nℓ with index ℓ over finite field F q generated by the set { v ( x ) , xv ( x ) , . . . , x n − v ( x ) } , where v ( x ) = v + v x + · · · + v n − x n − ∈ R [ x ] / h x n − i . Then (1) C has a lower bound on the minimum Hamming distance given by d ( C ) ≥ d ( e C ) d ( B ) ,where e C is a cyclic code of length n over R generated by the polynomial g ( x ) ∈ R [ x ] / h x n − i ,and B is a linear code of length ℓ generated by { v , v , . . . , v n − } where each v i is an ℓ -tuplewith respect to a fixed F q -basis { , u, . . . , u ℓ − } of R . (2) If the cyclic code e C in (1) is free and the generator polynomial g ( x ) has δ − consecutiveroots in some Galois extension ring of R , and if the set { v , v , . . . , v n − } generates a cycliccode B over finite field F q of length ℓ such that the generator polynomial of B has ε − consecutive roots in some Galois extension field of F q , then d ( C ) ≥ δε . ✷ Example 3.3
Let R = F + u F + u F . Suppose R = { , , u, v, uv, u , v , v } , where u = 0, v = 1 + u , v = 1 + u , v = 1 + u + u , uv = u + u . It is well known that x − x + v )( x + uvx + v x + v )( x + vx + ux + v ), where x + v , x + uvx + v x + v and x + vx + ux + v are basic irreducible polynomials over R . Let R = R [ x ] / ( x + uvx + v x + v ). Since x + uvx + v x + v is a basic primitive polynomial over R , the root ξ of x + uvx + v x + v is a primitive element in R . Taking v ( x ) = ( x + v )( x + uvx + v x + v ) = x + x + (1 + u + u ) x + u x + (1 + u ), then the cyclic code e C of length 7 generated by v ( x ) is free, and by Proposition 2.5, we have the minimum Hamming distance of e C at least9. The non-zero coefficients of v ( x ) correspond to the elements (1 , , , , , , , , , ,
0) with respect to the F -basis { , u, u } of R and they generate a cyclic code B of length 3 with the minimum Hamming distance 1 over F . Therefore, C is a 1-generatorQC code of length 21 with dimension 3 and minimum Hamming distance at least 4 × F . A generator matrix for C is given as follows
101 001 111 100 100 000 000000 101 001 111 100 100 000000 000 101 001 111 100 100 . (3)In fact the minimum Hamming distance of C is 8. Therefore, C is a QC code with parameters[21 , ,
8] over F . Let x n − f f . . . f r , where each f i , i = 1 , , . . . , r , is the monic basic irreduciblepolynomial with degree ℓ i over R . Then from Theorem 2.1, we have ( R [ x ] / h x n − i ) ℓ ≃ L ℓi =1 ( R [ x ] / h f i ) ℓ . Therefore if C is a QC code of length nℓ with index ℓ over R , then C = r M i =1 C i , where C i , i = 1 , , . . . , r , is a linear code of length ℓ over the ℓ i -th Galois extension ring R i of R . This is called the canonical decomposition of the QC code C . The trace representation ofcyclic code over finite chain R can give the following characterization result on QC code asfollows. Theorem 4.1 (cf. Theorem 5.1 [17])
Let x n − f f . . . f r , where each f i , i = 1 , , . . . , r ,is the basic irreducible polynomial with degree ℓ i over R . Denote R i = R [ x ] / h f i i . Let U i denote the q -cyclotomic coset mod n corresponding to f i . Fix a representatives u i ∈ U i fromeach cyclotomic coset. Let C i be a linear code of length ℓ over R i for all i = 1 , , . . . , r . For e c i ∈ C i and each j = 0 , , . . . , n − , let the vector c j = P ri =1 T r R i /R ( e c i ξ − ju i ) . Then the code C = { ( c , c , . . . , c n − ) | e c i ∈ C i } is a QC code of length nℓ with index ℓ over R . Conversely, every QC code of length nℓ withindex ℓ over R is obtained through this construction. ✷ Let R ⊂ e R ⊂ R be Galois extension. If ω ∈ R such that T r R / e R ( ω ) = 1, then for any ϑ ∈ e R we have T r e R/R ( ϑ ) = T r R /R ( ωϑ ) . Theorem 4.2
Let C be a QC code defined as above. Let ω , ω , . . . , ω r ∈ R be elements with T r R / R i ( ω i ) = 1 for all i = 1 , , . . . , r . Then (1) Any codeword ( c , c , . . . , c n − ) ∈ C is of the form c j = P ri =1 T r R /R ( e c i ω i ξ − ju i ) , for all j = 0 , , . . . , n − . The columns of any codeword c ∈ C lie in a free cyclic code B of length n over R , whichdual code B ⊥ has roots ξ − u , ξ − u , . . . , ξ − u r , where ξ is an n -th primitive root of unity in R ; (3) For any column b c ν = (Σ ri =1 T r R /R ( e c i,ν ω i ) , Σ ri =1 T r R /R ( e c i,ν ω i ξ − u i ) , . . . , Σ ri =1 T r R /R ( e c i,ν ω i ξ − ( n − u i )) , where e c i = ( e c i , e c i , . . . , e c iℓ ) ∈ R i and ν = 1 , , . . . , ℓ , we have b c ν = if and only if e c ,ν = e c ,ν = · · · = e c r,ν = 0 .Proof (1) Clearly, P ri =1 T r R /R ( e c i ω i ξ − ju i ) = P ri =1 T r R i /R ( e c i ξ − ju i ) = c j ;(2) For any column b c ν = (Σ ri =1 T r R /R ( e c i,ν ω i ) , Σ ri =1 T r R /R ( e c i,ν ω i ξ − u i ) , . . . , Σ ri =1 T r R /R ( e c i,ν ω i ξ − ( n − u i )) , the v -th component b c v = r X i =1 T r R /R ( e c i,ν ω i ξ − vu i ) , where e c i = ( e c i , e c i , . . . , e c iℓ ) ∈ R i and ν = 1 , , . . . , ℓ . Since e c i,ν ω i ∈ R , from Proposition 2.7,we have b c ν lies in a free cyclic code B of length n over R , which dual code B ⊥ has roots ξ − u , ξ − u , . . . , ξ − u r ;(3) b c ν = if and only if each v -th component is zero for all v = 0 , , . . . , n − ri =1 T r R /R ( e c i,ν ω i ξ − vu i ) = 0 if and only if T r R /R ( e c i,ν ω i ξ − vu i ) = 0 for all i = 1 , , . . . , r .( i ) If ℓ = ℓ = · · · = ℓ r = m , then T r R /R ( e c i,ν ω i ξ − vu i ) = 0 for all i = 1 , , . . . , r if and onlyif e c ,ν = e c ,ν = · · · = e c r,ν = 0;( ii ) If there exists a set { j , j , . . . , j d } ⊆ { , , . . . , r } such that ℓ j k < m for all k ∈{ j , j , . . . , j d } and ℓ p = m for all p ∈ { , , . . . , r }\{ j , j , . . . , j d } , then T r R /R ( e c i,ν ω i ξ − vu i ) =0 for all i = 1 , , . . . , r if and only if e c ℓ p ,ν = 0 and T r R / R i ( e c ℓ jk ω i ) = e c ℓ jk T r R / R i ( ω i ) = e c ℓ jk =0. Therefore, we have proved b c ν = if and only if e c ,ν = e c ,ν = · · · = e c r,ν = 0. ✷ As a consequence of Theorem 4.2, we can exhibit a minimum Hamming distance boundfor the QC code. We assume that d ( C ) ≥ d ( C ) ≥ · · · ≥ d ( C r ) . For any nonempty sunset I = { i , i , . . . , i t } ⊆ { , , . . . , r } with 1 ≤ i < i < · · · < i t ≤ r ,let B I = B i ,i ,...,i t be a free cyclic code of length n over R , which dual code B ⊥I hasroots ξ − u i , ξ − u i , . . . , ξ − u it . If ∅ 6 = I ⊂ I ⊆ { , , . . . , r } , then B I ⊂ B I and hence d ( B I ) ≥ d ( B I ).For I defined above, we define d I = d i ,i ,...,i t = d ( C i ) d ( B i ) if t = 1( d ( C i ) − d ( C i )) d ( B i ) + ( d ( C i ) − d ( C i )) d ( B i ,i )... if t ≥ d ( C i t − ) − d ( C i t )) d ( B i ,i ,...,i t − ) + d ( C i t ) d ( B i ,i ,...,i t ) . (4)11et J = I\{ i µ } for some µ ∈ { , , . . . , t } . Then d J ≥ d I . (See Lemma 4.7 in [13].)In the following, we give the minimum Hamming distance bound for QC codes over finitechain ring R , which is an interesting generalization of Theorem 4.8 in [13]. Theorem 4.3
Let C be a QC code as discussed above. Then the minimum Hamming distanceof C satisfies d ( C ) ≥ min { d r , d r − ,r , . . . , d , ,...,r } . Proof
Let c be a nonzero codeword of C . Suppose that e c i k ∈ C i for all k = 1 , , . . . , t , where { i , i , . . . , i t } ⊆ , , . . . , r and 1 ≤ i < i < · · · < i t ≤ r . If t = 1, then by Theorem 4.2 (2)there exists at least d ( C ) nonzero columns in C implying the minimum possible weight forsuch code is d ( C ) ≥ d ( C i ) d ( B i ). If t ≥
2, then the weight for such code C is minimized ifSupp( e c i t ) ⊆ Supp( e c i t − ) ⊆ · · · ⊆ Supp( e c i ) , where Supp( e c i k ) denotes the nonzero coordinates of e c i k for all k = 1 , , . . . , t . By the proofprocess of Theorem 4.5 in [13], the lowest possible weight for such code in this case is d ( C ) ≥ d I = ( d ( C i ) − d ( C i )) d ( B i ) + · · · + d ( C i t ) d B i ,i ,...,i t , which implies that d ( C ) ≥ min { d I | I = { i , i , . . . , i t } ⊆ { , , . . . , r } with i < i < · · · < i t } . Let
N ⊆ { , , . . . , r } and let i be the minimal element in N . Adjoining one element at atime, we have N ⊆ N ⊆ · · · ⊆ { i, i + 1 , . . . , r } . Then d N ≥ d N ≥ · · · ≥ d i,i +1 ,...,r . Hence, the minimum Hamming distance of C is equal to d ( C ) ≥ min { d r , d r − ,r , . . . , d , ,...,r } . ✷ Example 4.4
Consider a QC code C of length 14 with index 2 generated by ( a ( x ) , a ( x ))over R = F + u F , where a ( x ) = x + x + x and a ( x ) = x + x + x + x + 1. Since( R [ x ] / h x − i ) ∼ = ( R [ x ] / h x − i ) L ( R [ x ] / h x + x + 1 i ) L ( R [ x ] / h x + x + 1 i ) , we have C = M i =1 C i , where C is a linear code of length 2 generated by (1 ,
0) over R , C is a linear code of length 2generated by (1 , x + 1) over R [ x ] / h x + x + 1 i and C is a zero code over R [ x ] / h x + x + 1 i .Clearly, d ( C ) = d ( C ) = 1, d ( C ) = 0. Hence Theorem 4.3 yields d ( C ) ≥ min { , } = 3 . -generator quasi-cyclic codes Let R be finite chain ring with maximal ideal h γ i and the nilpotency index s . Denote R n = R [ x ] / h x n − i . Review the conventional row circulant definition of the QC code. Let T bea cyclic shift operator T : R N → R N , which transforms v = ( v , v , . . . , v N − ) into vT =( v N − , v , . . . , v N − ). A linear code C is called quasi-cyclic (QC) code if it is invariant under T ℓ for some positive integer ℓ . The smallest ℓ such that T ℓ ( C ) = C is called the index of C .Clearly, ℓ is a divisor of N . Let N = nℓ . Define a one-to-one correspondence ρ : R N → R ln ( a , , a , , . . . , a ,ℓ − , a , , a , , . . . , a ,ℓ − , . . . , a n − , , a n − , , . . . , a n − ,ℓ − ) a ( x ) = ( a ( x ) , a ( x ) , . . . , a ℓ − ( x ))where a j ( x ) = P n − i =0 a ij x i for j = 0 , , . . . , ℓ −
1. Then C is equivalent to for any a ( x ) =( a ( x ) , a ( x ) , . . . , a l − ) ∈ ρ ( C ), xa ( x ) ∈ ρ ( C ). Therefore, C is a QC code if and only if ρ ( C )is an R n -submodule of R ℓn .Let C = R n a ( x ), where a ( x ) is defined as above. Then C is called a 1-generator QC code.Define the annihilator of C as followsAnn( C ) = { c ( x ) ∈ R n | c ( x ) a i ( x ) = 0 , ∀ i = 0 , , . . . , ℓ − } . It is easy to check that Ann( C ) is an ideal of R n . Consider the map ϕ from R n to C , whichsends c ( x ) ∈ R n to c ( x ) a ( x ) ∈ C . Clearly, ϕ is a surjective R [ x ]-module homomorphism andthe kernel of ϕ is Ann( C ). Therefore R n / Ann( C ) is isomorphic to C and hence | C | = | R n | / | Ann( C ) | . Let I = h a ( x ) , a ( x ) , . . . , a ℓ − ( x ) i be the ideal generated by elements a ( x ), a ( x ), . . . , a ℓ − ( x ) ∈ R n . Then, by Theorem 2.2, there exists a unique set of pairwise coprimemonic polynomials F , F , . . . , F s over R (possibly, some of them are equal to 1) such that F F · · · F s = x n − I = h b F , γ b F , . . . , γ s − b F s i , where b F i = ( x n − /F i , i = 0 , , . . . , s . Theorem 5.1
Let a ( x ) = ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) ∈ R ln , and C = R n a ( x ) be a -generator QC code. Let F , F , . . . , F s be given above. Then the annihilator of C Ann( C ) = h b F , γ b F s , . . . , γ s − b F i , where b F i = ( x n − /F i , i = 0 , , . . . , s .Proof Let I = h a ( x ) , a ( x ) , a ℓ − ( x ) i and Ann( I ) be the annihilator of I . It is easy to verifythat Ann( I ) = Ann( C ). Clearly, b F , γ b F s , . . . , γ s − b F ∈ Ann( I ). On the other hand, since I ∼ = R n / Ann( I ), | Ann( I ) | = | R n | / | I | = q P si =1 i deg F i +1 = |h b F , γ b F s , . . . , γ s − b F i| , where F s +1 = F . Therefore Ann( C ) = Ann( I ) = h b F , γ b F s , . . . , γ s − b F i . ✷ F , F , . . . , F s be given above. If F = F = · · · = F s = 1, then x n − F F . Let C bea 1-generator QC code generated by ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )). Then I = h a ( x ) , a ( x ) , . . . , a ℓ − ( x )) i = h F i and Ann( C ) = h F i . By Proposition 2.4, we have I and Ann( C ) are all free cyclic codesover R . In the following, we give a sufficient condition for the 1-generator QC code to be freeand a BCH-type bound for the free 1-generator QC code. Theorem 5.2
Let C be a -generator QC code with annihilator h F i generated by a ( x ) =( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) ∈ R ℓn . (1) C is a free -generator QC code. Furthermore if deg F = k , then C is generated by theset { a ( x ) , xa ( x ) , . . . , x k − a ( x ) } with rank k . (2) Let b i ( x ) = a i ( x ) /F and gcd( b i ( x ) , F ) = 1 for all i = 0 , , . . . , ℓ − . Let ξ b , ξ b +1 , . . . , ξ b + δ − be the consecutive roots of F , where ξ is an n -th primitive root of unity in some Galois ex-tension ring R of R . Then the minimum Hamming distance of C is d ( C ) ≥ ℓδ .Proof (1) Define a map as follows φ : R [ x ] → R ℓn f ( x ) f ( x ) a ( x ) .φ is an R [ x ]-module homomorphism onto C . Clearly, the kernel of φ is the annihilator of C . It follows that C is isomorphic to R [ x ] / h F i . Since F F = x n −
1, by Theorem 2.1(5), we have C is isomorphic to the cyclic code h F i , which is free over R . Therefore C is a free 1-generator QC code over R . If deg F = k , then C can be generated by the set { a ( x ) , xa ( x ) , . . . , x k − a ( x ) } . From the isomorphism between C and the free cyclic code h F i ,one can deduce the elements in the set are linearly independent over R , which implies thatthe rank of C is k .(2) For a fixed i , i = 0 , , . . . , ℓ −
1, define a map φ i from ( R [ x ] / h x n − i ) ℓ to R [ x ] / h x n − i such that φ i ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) = a i ( x ). Then φ i ( C ) is a free cyclic code of length n generated by a i ( x ) = b i ( x ) F for all i = 0 , , . . . , ℓ −
1. Since x n − F F , we have one of thecomponents becomes zero if and only if all the others become zero because b i ( x ) F p ( x ) = 0if and only if ( x n − | b i ( x ) F p ( x ), i.e., F F | b i ( x ) F p ( x ) if and only if F | b i ( x ) p ( x ). Sincegcd( b i ( x ) , F ) = 1, we have F | p ( x ). Therefore b i ( x ) F p ( x ) = 0 for all i = 0 , , . . . , ℓ −
1. Andhence if c is a non-zero codeword in C , then φ i ( c ) is also non-zero. Since gcd( b i ( x ) , F ) = 1,it follows that φ i ( C ) is generated by F . By Proposition 2.5, we have d ( φ i ( C )) ≥ δ implying d ( C ) ≥ ℓδ . ✷ Example 5.3
Let R = F + u F and n = 24. Let C be a 1-generator QC code withgenerator ( a ( x ) , a ( x )) over R . Assume that a ( x ) = b ( x ) F = 1 × F = x + 2 x +4 x + 4 x + 3 x + 2 x + 3 x + 3 x + 3 x + x + 2 x + 2 x + 2 is a monic divisor of x − a ( x ) = b ( x ) F , where b ( x ) = 3 x + 2 x + x + 2 x . Then the annihilator of C is h F i = h x + 3 x + x + 3 x + 2 x + 3 x + 2 i over R . Since F having ξ i , 7 ≤ i ≤
18, among itszeros, where ξ is a 24-th primitive root of unity in R = F + u F , and gcd( b j ( x ) , F ) = 1 for j = 0 ,
1, the cyclic code of length 24 generated by a ( x ) and a ( x ) is free and the minimumHamming distance at least 13. Therefore C is free with the minimum Hamming distance at14east 2 ×
13 = 26. In fact, the minimum Hamming distance of C is 30. Therefore C is a free1-generator QC code of length 48, with the rank 8 and the minimum Hamming distance 30,i.e. an [48 , ,
30] linear code over R . -generator quasi-cyclic codes In this section, we assume that gcd( n, q ) = 1 and gcd( | q | n , ℓ ) = 1, where | q | n denotes theorder of q modulo n , i.e., the smallest integer t such that n | ( q t − R n = R [ x ] / h x n − i and h r i R be the ideal of R generated by r ∈ R .Let I = h a ( x ) , a ( x ) , . . . , a ℓ − ( x ) i be an ideal generated by a ( x ) , a ( x ) , . . . , a ℓ − ( x ) of R n . Then there exist pairwise coprime monic polynomials F , F , . . . , F s ∈ R n such that F F · · · F s = x n − I = h b F , γ b F , . . . , γ s − b F s i , where b F i = ( x n − /F i , i = 0 , , . . . , s .By reduction modulo γ , we have I = h b F i R n andgcd( a ( x ) , a ( x ) , . . . , a ℓ − ( x ) , x n −
1) = gcd( b F , x n −
1) = b F (5)It means that F a i ( x ) = 0 in R n [ x ], i = 0 , , . . . , ℓ −
1. Assume that there exist polynomials b i ( x ) ∈ R n such that γb i ( x ) = F a i ( x ) , i = 0 , , . . . , ℓ − . Then h F a ( x ) , . . . , F a ℓ − ( x ) i = h γb ( x ) , . . . , γb ℓ − , ( x ) i = h γF b F , . . . , γ s − F b F s i , whichimplies that h b ( x ) , b ( x ) , b ℓ − , ( x ) i R n = h F b F i R n andgcd( b ( x ) , b ( x ) , . . . , b ℓ − , ( x ) , x n −
1) = gcd( F b F , x n −
1) = b F (6)Repeating the above process, we get the following s equations gcd( a ( x ) , . . . , a l − ( x ) , x n −
1) = gcd( b F , x n −
1) = b F gcd( b ( x ) , . . . , b ℓ − , ( x ) , x n −
1) = gcd( F b F , x n −
1) = b F ...gcd( b ,s − ( x ) , . . . , b ℓ − ,s − ( x ) , x n −
1) = gcd( F · · · F s − b F s , x n −
1) = b F s . (7)Note that b F , b F , . . . , b F s are the Hensel lift of b F , b F , . . . , b F s , respectively. Therefore wecan determine the generators of I , i.e., b F , γ b F , . . . , γ s − b F s . Moreover the 1-generator QCcode C has the annihilator Ann( C ) = h b F , γ b F s , . . . , γ s − b F i if and only if the equations (7)hold. Theorem 6.1
Let a ( x ) = ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) ∈ R ℓn and C = R n a ( x ) be a -generatorQC code over R . Let c ( x ) = ( c ( x ) , c ( x ) , . . . , c ℓ − ( x )) ∈ R ℓn . Then R n a ( x ) = R n c ( x ) if andonly if there exists a polynomial h ( x ) ∈ R [ x ] such that c ( x ) = h ( x ) a ( x ) and h ( x ) are coprimewith the generator of Ann( C ) ,i.e. F = b F + γ b F s + · · · + γ s − b F .Proof Since Ann( C ) = h b F , γ b F s , . . . , γ s − b F i is an ideal of R n , from the proof process ofTheorem 3.6 in [9], we have Ann( C ) = h b F + γ b F s + · · · + γ s − b F i . Let c ( x ) = h ( x ) a ( x )and h ( x ) and F are coprime. Then R n c ( x ) ⊆ R n a ( x ). Since h ( x ) and F coprime, there exist15 ( x ) , v ( x ) ∈ R [ x ] such that u ( x ) h ( x )+ v ( x ) F = 1. It means that R n a ( x ) = R n a ( x )( u ( x ) h ( x )+ v ( x ) F ) = R n a ( x ) u ( x ) h ( x ) = u ( x ) R n b ( x ) in R n implying R n a ( x ) = R n c ( x )Conversely, if R n a ( x ) = R n c ( x ), then there are polynomials u ( x ) , h ( x ) ∈ R [ x ] such that c ( x ) = h ( x ) a ( x ) and a ( x ) = u ( x ) c ( x ). Therefore R n a ( x ) = u ( x ) h ( x ) R n c ( x ), which impliesthat (1 − u ( x ) h ( x )) R n a ( x ) = 0 in R n . Thus 1 − u ( x ) h ( x ) ∈ Ann( C ), i.e. there exists apolynomial v ( x ) ∈ R [ x ] such that 1 − u ( x ) h ( x ) = v ( x ) F . Hence h ( x ) and F are coprime. ✷ Let f ( x ) be a basic irreducible polynomial with degree ℓ over R . Then R = R [ x ] / h f ( x ) i be the ℓ -th Galois extension ring of R . We can map R ℓn into R n via the natural mapping σ : R ℓn → R n ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) ℓ − X i =0 a i ( x ) α i = A ( x ) , where the set { α , α , . . . , α ℓ − } is a basis for R over R .Recall that I = h a ( x ) , a ( x ) , . . . , a ℓ − ( x ) i = h b F , γ b F , . . . , γ s − b F s i . Let a i ( x ) = a i ( x ) b F + γa i ( x ) b F + · · · + γ s − a is b F s , where a i ( x ) , a i ( x ) , . . . , a is ( x ) ∈ R [ x ] for each i = 0 , , . . . , ℓ −
1. Then A ( x ) = b F P ℓ − i =0 a i ( x ) α i + γ b F P l − i =0 a i ( x ) α i + · · · + γ s − b F s P ℓ − i =0 a is ( x ) α i ∈h b F , γ b F , . . . , γ s − b F s i R n .Since gcd( n, q ) = 1 and gcd( | q | n , ℓ ) = 1, the polynomial x n − F q [ x ] and F q ℓ [ x ]. Therefore x n − R and R , respectively. In thiscase, if we assume a ( x ) = ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) ∈ R ℓn , A ( x ) = P ℓ − i =0 a i ( x ) α i , thengcd( a ( x ) , a ( x ) , . . . , a ℓ − ( x ) , x n −
1) = gcd( A ( x ) , x n −
1) ( cf. Lemma 3 in [21]).For all i = 1 , , . . . , s , b F i and F i are coprime in R [ x ] because of gcd( n, q ) = 1. Thereforethere exist u i ( x ) , v i ( x ) ∈ R [ x ] such that u i ( x ) b F + v i ( x ) F = 1. Let e i = 1 − v i ( x ) F i = u i ( x ) b F i .Then, by Theorem 2.1 (3), each e i is the identity of the ring h b F i i R n , i = 1 , , . . . , s . If weassume that e is the identity of h b F , γ b F , . . . , γ s − b F s i R n , then we have e = e + e + · · · + e s ,which implies the following direct sum decomposition h b F , γ b F , . . . , γ s − b F s i R n = h b F i R n ⊕ h γ b F i R n ⊕ · · · ⊕ h γ s − b F s i R n A ( x ) = A ( x ) + γA ( x ) + · · · + γ s − A s ( x ) , where A ( x ) = P ℓ − i =0 a i ( x ) α i , γ j − A j = A ( x ) e j for all j = 1 , , . . . , s . Lemma 6.2
Let A ( x ) = P ℓ − i =0 a i ( x ) α i = A ( x )+ γA ( x )+ · · · + γ s − A s ( x ) , where a i ( x ) ∈ R n and γ j − A j ( x ) ∈ h γ j − b F j i R n , i = 0 , , . . . , ℓ − , j = 1 , , . . . , s . Let a ( x ) = ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) .Then C = R n a ( x ) is a -generator QC code with annihilator h b F , γ b F s , . . . , γ s − b F s i if and onlyif A j ( x ) and F j are coprime for all j = 1 , , . . . , s .Proof Firstly, we will prove that gcd( A j ( x ) , x n −
1) = b F j if and only if A j ( x ) and F j are coprime for all j = 1 , , . . . , s . Let A j ( x ) and F j be coprime over R , j = 1 , , . . . , s .16hen gcd( A j ( x ) , F j ) = 1. On the other hand, since γ j − A j ( x ) ∈ h γ j − b F j i R n , we have b F j | gcd( A j ( x ) , x n − A j ( x ) , x n −
1) = b F j . Conversely, for any γ j − A j ( x ) ∈ h γ j − b F j i R n , if gcd( A j ( x ) , x n −
1) = b F j , then gcd( A j ( x ) , F j ) = 1. Therefore, A j ( x ) and F j are coprime in R [ x ].Next we will prove that C is a 1-generator QC code generated by a ( x ) with annihilator h b F , γ b F , . . . , γ s − b F s i if and only if gcd( A j ( x ) , x n −
1) = b F j for all j = 1 , , . . . , s . Note that A ( x ) = ℓ − X i =0 a i ( x ) α i F A ( x ) = ℓ − X i =0 b i ( x ) α i ... F · · · F s − A s ( x ) = ℓ − X i =0 b s − ,i ( x ) α i . Then by gcd( a ( x ) , a ( x ) , . . . , a ℓ − ( x ) , x n −
1) = gcd( A ( x ) , x n − b k ( x ) , b k ( x ) , . . . , b k,ℓ − ( x ) , x n −
1) = gcd( F · · · F k A k +1 ( x ) , x n − k = 1 , , . . . , s −
1, we have C is a 1-generator QC code with annihilator Ann( C ) = h b F , γ b F s , . . . , γ s − b F i if and only if b F = gcd( a ( x ) , a ( x ) , . . . , a ℓ − ( x ) , x n −
1) = gcd( A ( x ) , x n − b F k +1 = gcd( b k ( x ) , b k ( x ) , . . . , b k,ℓ − ( x ) , x n −
1) = gcd( F · · · F k A k +1 ( x ) , x n − k = 1 , , . . . , s − ✷ For each j = 1 , , . . . , s , define a map µ j from finite chain ring R with maximal ideal h γ i and nilpotency index s to the finite chain ring R (mod γ j ) = R ( j ) with maximal ideal h γ i and nilpotency index j , where j is a positive integer and 1 ≤ j ≤ s . Clearly, µ ( R ) isthe residue field F q and µ s ( R ) is R . Extend the map µ j to the polynomial ring R [ x ], suchthat for any polynomial f ( x ) ∈ R [ x ], µ j ( f ( x )) = f ( x )(mod γ j ). Then µ j is a surjective ringhomomorphism. Therefore we have the following ring isomorphism. φ j : h γ s − j b F s − j +1 i R n → γ s − j h µ j ( b F s − j +1 ) i R ( j ) n γ s − j f ( x ) γ s − j µ j ( f ( x )) . Denote G j = ( h µ j ( b F s − j +1 ) i R ( j ) n ) ∗ , the group of units of the ring h µ j ( b F s − j +1 ) i R ( j ) n . Theorem 6.3
Let A ( x ) = P ℓ − i =0 a i ( x ) α i = A ( x )+ γA ( x )+ · · · + γ s − A s ( x ) , where a i ( x ) ∈ R n and γ j − A j ( x ) ∈ h γ j − b F j i R n for all j = 1 , , . . . , s . Let a ( x ) = ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) . hen C = R n a ( x ) is a -generator QC code with annihilator h b F , γ b F s , . . . , γ s − b F i if andonly if µ j ( A s − j +1 ( x )) ∈ G j for all j = 1 , , . . . , s .Proof For j = 1 , , . . . , s , if µ j ( A s − j +1 ( x )) ∈ G j , then there exists µ j ( d ( x )) ∈ G j such that µ j ( A s − j +1 ( x )) µ j ( d ( x )) = µ j ( e s − j +1 ) = 1 − µ j ( v s − j +1 ( x )) µ j ( F s − j +1 ) , which implies that µ j ( A s − j +1 ( x )) µ j ( d ( x ))+ µ j ( v s − j +1 ( x )) µ j ( F s − j +1 ) = 1. Therefore A s − j +1 ( x )and F s − j +1 are coprime for all j = 1 , , . . . , s . From Lemma 6.2, C has the annihilator h b F , γ b F s , . . . , γ s − b F i .Conversely, for each j = 1 , , . . . , s , if A s − j +1 ( x ) and F s − j +1 are coprime, then there existpolynomials w s − j +1 ( x ), z s − j +1 ( x ) ∈ R [ x ] such that w s − j +1 ( x ) A s − j +1 ( x ) + z s − j +1 ( x ) F s − j +1 = 1 . It follows that µ j ( w s − j +1 ) µ j ( A s − j +1 ( x )) + µ j ( z s − j +1 ) µ j ( F s − j +1 ) = 1 in ring R ( j ) n . Multiply-ing by µ j ( e s − j +1 ) = µ j ( u s + j − ( x ) µ j ( b F s − j +1 ), we have µ j ( e s − j +1 ) = µ j ( w s − j +1 ) µ j ( A s − j +1 ( x )) µ j ( u s + j − ( x ) µ j ( b F s − j +1 ) . Therefore µ j ( A s − j +1 ( x )) ∈ G j for all j = 1 , , . . . , s . ✷ Theorem 6.4
Let M = { C | C is a − generator QC code over R with Ann( C ) = h b F , γ b F s , . . . , γ s − b F i} . For each j = 1 , , . . . , s , let F s − j +1 = f s − j +1 , . . . f s − j +1 ,r s − j +1 ,where each f s − j +1 ,p is the monic basic irreducible polynomial with degree e s − j +1 ,p , p =1 , , . . . , r s − j +1 . Then | M | = s Y j =1 r s − j +1 Y p =1 q jℓe s − j +1 ,p − q ( j − ℓe s − j +1 ,p q je s − j +1 ,p − q ( j − e s − j +1 ,p . Proof
Denote G j = ( h µ j ( b F s − j +1 ) i R ( j ) n ) ∗ , the group of units of the ring h µ j ( b F s − j +1 ) i R ( j ) n . Forsimplicity, we write µ j ( b F s − j +1 ) G j ∈ G j / G j as µ j ( b F s − j +1 ) in this proof process. Define a mapas follows η : G s / G s × G s − / G s − × · · · × G / G → M ( µ s ( A ) , µ s − ( A ) , . . . , µ ( A s )) R n a ( x ) , where for each j = 1 , , . . . , s , A j ( x ) is any inverse image of µ s − j +1 ( A j ) under the map µ j , A ( x ) = A ( x )+ γA ( x )+ · · · + γ s − A s ( x ) = P ℓ − i =0 a i ( x ) α i and a ( x ) = ( a ( x ) , a ( x ) , . . . , a l − ( x )).For any µ s − j +1 ( A ,j ) , µ s − j +1 ( A ,j ) ∈ G j / G j , j = 1 , , . . . , s , let A (1) ( x ) = ℓ − X i =0 a i ( x ) α i = A ( x ) + γA ( x ) + · · · + γ s − A s ( x ) ,A (2) ( x ) = ℓ − X i =0 a i ( x ) α i = A ( x ) + γA ( x ) + · · · + γ s − A s ( x ) , (1) ( x ) = ( a ( x ) , a ( x ) , . . . , a ,ℓ − ( x ))and a (2) ( x ) = ( a ( x ) , a ( x ) , . . . , a ,ℓ − ( x )) . If R n a (1) = R n a (2) , then by Theorem 6.1, there exists a polynomial h ( x ) ∈ R [ x ] such thatgcd( h ( x ) , F ) = 1. Therefore A (2) ( x ) = ℓ − X i =0 h ( x ) a i ( x ) α i = h ( x ) A ( x ) + γh ( x ) A ( x ) + · · · + γ s − A s ( x ) . Since for each j = 1 , , . . . , s , e j is the identity of h b F j i R n , we have γ j − h ( x ) A j ( x ) = γ j − h ( x ) e j A j ( x ). Moreover, µ s − j +1 ( b F j ) | µ s − j +1 ( h ( x )) µ s − j +1 ( e j ) since γ j − h ( x ) e j ∈ h γ j − b F j i .Then gcd( µ s − j +1 ( h ( x )) µ s − j +1 ( e j ) , x n − µ s − j +1 ( b F j )gcd( µ s − j +1 ( h ( x )) µ s − j +1 ( e j ) , µ s − j +1 ( F j ))= µ s − j +1 ( b F j ) , (8)which deduces that gcd( µ s − j +1 ( h ( x )) µ s − j +1 ( e j ) , µ s − j +1 ( F j )) = 1 (the second equality in (8)holds because of µ s − j +1 ( h ( x )) µ s − j +1 ( e j ) is a unit in G j ). Using the technique in the proof pro-cess of Theorem 6.3, we have µ s − j +1 ( h ( x )) µ s − j +1 ( e j ) ∈ G j . It follows that µ s − j +1 ( A j ( x )) = µ s − j +1 ( A j ( x )) for all j = 1 , , . . . , s . Hence η is injective.For any 1-generator QC code R n a ( x ) ∈ M , assume that A ( x ) = A ( x ) + γA ( x ) + · · · + γ s − A s ( x ) = P ℓ − i =0 a i ( x ) α i , where γ j − A j ( x ) ∈ h γ j − b F j i R n for all j = 1 , , . . . , s . ByTheorem 6.3, each µ j ( A s − j +1 ( x )) ∈ G j . Moreover η (( A ( x )) , A ( x ) , . . . , A s ( x )) = R n a ( x ).Therefore η is surjective. It means that η is a bijective map.From Theorem 2.1, for each j = 1 , , . . . , s , we have h µ j ( b F s − j +1 ) i R n ∼ = R ( j ) [ x ] / h F s − j +1 i and h µ j ( b F s − j +1 ) i ∼ = R ( j ) [ x ] / h F s − j +1 i implying | M | = | s M j =1 ( R ( j ) [ x ] / h F s − j +1 i ) ∗ / ( R ( j ) [ x ] / h F s − j +1 i ) ∗ | . Since F s − j +1 = f s − j +1 , . . . f s − j +1 ,r s − j +1 , where each f s − j +1 ,p is the monic basic irreduciblepolynomial with degree e s − j +1 ,p over R and R , p = 1 , , . . . , r s − j +1 , from Theorem 2.1, wehave | M | = s Y j =1 r s − j +1 Y p =1 q jℓe s − j +1 ,p − q ( j − ℓe s − j +1 ,p q je s − j +1 ,p − q ( j − e s − j +1 ,p . ✷ -generator quasi-cyclic codes In section 6, we have given the enumeration formula of 1-generator QC codes with the anni-hilator h b F , γ b F s , . . . , γ s − b F i . Another problem is that can we find out the one and only onegenerator of each 1-generator QC code with the annihilator h b F , γ b F s , . . . , γ s − b F i over finitechain ring R . 19et a ( x ) = ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )) and A ( x ) = P ℓ − i =0 a i ( x ) α i = A ( x ) + γA ( x ) + · · · + γ s − A s ( x ), where the set { α , α , . . . , α ℓ − } is a basis for R over R . From the proof ofTheorem 6.4, if we find µ j ( A s − j +1 ( x )) for all j = 1 , , . . . , s , we can determine the generatorof R n a ( x ), i.e., ( a ( x ) , a ( x ) , . . . , a ℓ − ( x )). By Theorem 2.1 (5), we have( R ( j ) [ x ] / h F s − j +1 i ) ∗ / ( R ( j ) [ x ] / h F s − j +1 i ) ∗ ∼ = r s − j +1 M p =1 ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ / ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ , where for each j = 1 , , . . . , s , F s − j +1 = f s − j +1 , f s − j +1 , . . . f s − j +1 ,r s − j +1 and each f s − j +1 ,p isa monic basic irreducible polynomial with degree e s − j +1 ,p over R and R , p = 1 , , . . . , r s − j +1 .Therefore µ j ( A s − j +1 ( x )) = µ j ( e s − j +1 ) r s − j +1 X p =1 ε jp , where e s − j +1 is the identity of h b F s − j +1 i R n and ε jp is the representation of the quotient group( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ / ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ for each j = 1 , , . . . , s and p = 1 , , . . . r s − j +1 .For simplicity, we choose the set { , α , . . . , α ℓ − } as the basis of R over R . Denote theset T s − j +1 ,p = { β p α + · · · + β p,ℓ − α ℓ − | β p , . . . , β p,ℓ − ∈ F q [ x ] / h f s − j +1 ,p i} . For each p = 1 , , . . . , r s − j +1 , let ξ p be an element in R ( j ) [ x ] / h f s − j +1 ,p i with order q ℓe s − j +1 ,p − X p = { , , . . . , q ℓe s − j +1 ,p − , ∞} , where we adopt ξ ∞ p = 0. Theorem 7.1
For p = 1 , , . . . , r s − j +1 , let the set Q jp = { ξ ap + γξ a + b p + · · · + γ j − ξ a + b j − p | a = 0 , , . . . , q ℓes − j +1 ,p − q es − j +1 ,p − ; b , . . . , b s − ∈ X p and ξ b , . . . , ξ b j − ∈ T s − j +1 ,p } . (9) Then Q jp is the complete set of representation of the cosets of ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ / ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ . Proof
Clearly, |Q jp | = q ℓe s − j +1 ,p − q e s − j +1 − q ℓe s − j +1 ,p q e s − j +1 ,p · · · q ℓe s − j +1 ,p q e s − j +1 ,p | {z } j − = q jℓe s − j +1 ,p − q ( j − ℓe s − j +1 ,p q je s − j +1 ,p − q ( j − e s − j +1 ,p , which implies that |Q jp | = ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ / ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ . Therefore we only need to prove that different elements in Q jp belong to different cosets. Forany A = ξ a p + γξ a + b p + · · · + γ j − ξ a + b j − , p and B = ξ a p + γξ a + b p + · · · + γ j − ξ a + b j − , p in ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ / ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ , if A = B then AB − ∈ ( R ( j ) [ x ] / h f s − j +1 ,p i ) ∗ .20t follows that A B − ∈ ( F q [ x ] / h f s − j +1 ,p i ) ∗ . Therefore q ℓes − j +1 ,p − q es − j +1 ,p − | ( a − a ), which deduces a = a . Repeating the above process, we have b λ = b λ , where λ = 1 , , . . . , j − ✷ Theorem 7.2
For each j = 1 , , . . . , s , let F s − j +1 = f s − j +1 , . . . f s − j +1 ,r s − j +1 , where each f s − j +1 ,p is the monic basic irreducible polynomial with degree e s − j +1 ,p , p = 1 , , . . . , r s − j +1 .Let e s + j − be the identity of h b F s − j +1 i R n . Then the elements s X j =1 r s − j +1 X p =1 γ s − j µ j ( e s − j +1 ) ε jp determine all the distinct -generator QC codes with annihilator h b F , γ b F s , . . . , γ s − b F i .Proof Since A ( x ) = P ℓ − i =0 a i ( x ) α i , it follows that if we get A ( x ) we can determine the gener-ator of the QC code. For A ( x ), we have A ( x ) = P sj =1 γ j − A j ( x ) = P sj =1 γ s − j A s − j +1 ( x ) ∼ = P sj =1 γ s − j µ j ( A s − j +1 ( x )) = P sj =1 γ s − j µ j ( e s − j +1 ) P r s − j +1 p =1 ε jp = P sj =1 P r s − j +1 p =1 γ s − j µ j ( e s − j +1 ) ε jp .It has proved the result. ✷ For all j = 1 , , . . . , s , if we let A s − j +1 ( x ) be any inverse image of µ j ( A s − j +1 ( x )) under themap µ j in R , then Theorem 7.2 gives an algorithm of finding the generator ( a ( x ) , a ( x ) , . . . , a ℓ − ( x ))over R actually.In the rest of this section, we give an example to illustrate the application of Theorem 6.4and Theorem 7.2. Example 7.3
Let ℓ = 2 and n = 7. Then x − x − x + 1)( x + x + 1)( x + x + 1) over R = F + u F . Let F = x + x + 1, F = x + 1 and F = x + x + 1. Consider the 1-generator QC code C withannihilator h b F , u b F i . Therefore in language of Theorem 6.4, we have s = 2, ℓ = 2, e = 3, e = 1. By Theorem 6.4, we have |M| = Q j =1 2 j × × e − j − ( j − × × e − j j × e − j − ( j − × e − j = × × − × − × × × − × × × − × = 9 ×
6= 54 . (10)Let R (2) = R (2) [ x ] / h x + x +1 i and ξ = x + h x + x +1 i . Then ξ is an element of order 2 − R (2) . Choose a basis of R over R as { , ξ } . Since F = x −
1, we have R (2) [ x ] / h F i = R (2) .Then µ ( e ) = b F = x + x + x + x + x + x + 1 is the identity of h b F i R (2) n , and ξ = ξF = ξx + ξx + ξx + ξx + ξx + ξx + ξ is an element of order 2 − R (2) [ x ] / h F i . In thelanguage of Theorem 7.1, we have T = { ξ β | β ∈ F } = { , ξ } and Q = { ξ a + uξ a + b | a =0 , , ξ b ∈ T } = { F , ξF , (1 + ξ ) F , (1 + uξ ) F , (1 + u + ξ ) F , ((1 + u ) ξ + u ) F } , which isthe complete set of representatives of cosets of ( R (2) [ x ] / h F i ) ∗ / ( R (2) [ x ] / h F i ) ∗ .In the language of Theorem 7.1, we have R (1) = F and R (1) [ x ] / h x + x + 1 i = F with the primitive element ξ = x + ξx + x + ξ x + ξ x + ξ and the identity µ ( e ) = x + x + x + 1, respectively. Then Q = { ξ a | a = 0 , , . . . , } . By Theorem 7.2, we get A ( x ) = A ( x ) + uA ( x ) ∼ = µ ( e ) ε + uµ ( e ) ε , where ε ∈ Q , ε ∈ Q and A ( x ), A ( x ) are any inverse images of µ ( e ) ε , µ ( e ) ε under the map µ , µ , respectively.21learly, the length of C is 14 and | C | = × = 4 × = 32. Define the Lee weight ofthe elements 0 , , u, u of F + u F as 0 , , ,
1, respectively. Moreover, the Lee weight ofan n -tuple in R n is the sum of the Lee weights of its components. The Gray map ϕ sendsthe elements 0 , , u, u of R to (0 , , (0 , , (1 , , (1 ,
0) over F , respectively. It is easyto verify that ϕ is a linear isometry from R n (Lee distance) to F n (Hamming distance).Therefore we have ϕ ( C ) is a linear code of length 28 with dimension 5 over F . Let g =1+ x + x + x + x + x + x , g = 1+ x + x + x , g = 1+ x + x + x and g = 1+ x + x . In thefollowing table, we list the generators and the minimum distances of all distinct 1-generatorQC codes with annihilator h b F , u b F i over R . In this paper, we mainly consider QC codes over finite chain rings. Module structures andtrace representation of QC codes are studied, which lead to some distance bounds on theminimum Hamming distance of QC codes. Particularly, we investigate the 1-generator QCcode giving the explicit structure of its annihilator that be used to calculate the number ofcodewords in QC code. Moreover, under some conditions, we discuss the enumerator and thegenerator of the 1-generator QC code with some fixed annihilator.More recently, there are some research papers on quasi-twisted (QT) codes, which arenatural generalizations of QC codes (see [1] [8] [11] [15]). They also have good algebraproperties and can produce some good linear codes over finite fields [1,8,11]. In [15], the authorused the generalized discrete Fourier transform (GDFT) to study structural properties of QTcodes of arbitrary lengths over finite fields. At the end of [15], the author gave a constructionalgorithm to construct QT codes. If we assume the block length of the QT code is coprimewith the characteristic of the finite chain ring, then all the results in this paper are valid forthe QT code. But the structural properties for QT codes of arbitrary lengths over finite chainrings are also interesting open problems for further consideration.
Acknowledgments
This research is supported by the National Key Basic Research Programof China (Grant No. 2013CB834204), and the National Natural Science Foundation of China(Grant Nos. 61171082, 10990011 and 60872025).
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