Skew Generalized Quasi-Cyclic Codes over Finite Fields
aa r X i v : . [ c s . I T ] S e p Skew Generalized Quasi-Cyclic Codes OverFinite Fields
Jian Gao, Linzhi Shen, Fang-Wei Fu
Chern Institute of Mathematics and LPMC, Nankai UniversityTianjin, 300071, P. R. China
Abstract
In this work, we study a class of generalized quasi-cyclic (GQC) codes called skewGQC codes. By the factorization theory of ideals, we give the Chinese Remainder Theoremover the skew polynomial ring, which leads to a canonical decomposition of skew GQC codes.We also focus on some characteristics of skew GQC codes in details. For a 1-generatorskew GQC code, we define the parity-check polynomial, determine the dimension and give alower bound on the minimum Hamming distance. The skew quasi-cyclic (QC) codes are alsodiscussed briefly.
Keywords
Skew cyclic codes; Skew GQC codes; 1-generator skew GQC codes; Skew QCcodes
Mathematics Subject Classification (2000) · · Recently, it has been shown that codes over finite rings are a very important classof codes and many types of codes with good parameters could be constructed overrings [2, 3, 22]. Skew polynomial rings are an important class of non-commutativerings. More recently, applications in the construction of algebraic codes have beenfound [1, 5, 6, 7, 8], where codes are defined as ideals or modules in the quotientring of skew polynomial rings. The principle motivation for studying codes in thissetting is that polynomials in skew polynomials rings have more factorizations thanthat in the commutative case. This suggests that it may be possible to find goodor new codes in the skew polynomial ring with lager minimum Hamming distance.Some researchers have indeed shown that such codes in skew polynomial rings haveresulted in the discovery of many new linear codes with better minimum Hammingdistances than any previously known linear codes with same parameters [1, 6].Quasi-cyclic (QC) codes over commutative rings constitute a remarkable general-ization of cyclic codes [2, 4, 12, 18, 22]. More recently, many codes were constructedover finite fields which meet the best value of minimum distances with the samelength and dimension [2, 22]. In [1], Abualrub et al. have studied skew QC codesover finite fields as a generalization of classical QC codes. They have introduced the1otation of similar polynomials in skew polynomial rings and shown that parity-checkpolynomials for skew QC codes are unique up to similarity. They also constructedsome skew QC codes with minimum Hamming distances greater than previously bestknown linear codes with the given parameters. In [5], Bhaintwal studied skew QCcodes over Galois rings. He gave a necessary and sufficient condition for skew cycliccodes over Galois rings to be free, and presented a distance bound for free skew cycliccodes. Futhermore, he also discussed the sufficient condition for 1-generator skew QCcodes to be free over Galois rings. A canonical decomposition and the dual codes ofskew QC codes were also given.The notion of generalized quasi-cyclic (GQC) codes over finite fields were intro-duced by Siap and Kulhan [21] and some further structural properties of such codeswere studied by Esmaeili and Yari [13]. Based on the structural properties of GQCcodes, Esmaeili and Yari gave some construction methods of GQC codes and ob-tained some optimal linear codes over finite fields. In [9], Cao studied GQC codes ofarbitrary length over finite fields. He investigated the structural properties of GQCcodes and gave an explicit enumeration of all 1-generator GQC codes. As a naturalgeneralization, GQC codes over Galois rings were introduced by Cao and structuralproperties and explicit enumeration of GQC codes were also obtained in [10]. Butthe problem of researching skew GQC codes over finite fields has not been consideredto the best of our knowledge.Let F q be a finite field, where q = p m , p is a prime number and m is a positiveinteger. The Frobenius automorphism θ of F q over F p is defined by θ ( a ) = a p , a ∈ F q .The automorphism group of F q is called the Galois group of F q . It is a cyclic group oforder m and is generated by θ . Let σ be an automorphism of F q . The skew polynomialring R = F q [ x, σ ] is the set of polynomials over F q , where the addition is defined asthe usual addition of polynomials and the multiplication is defined by the followingbasic rule ( ax i )( bx j ) = aσ i ( b ) x i + j , a, b ∈ F q . From the definition one can see that R is a non-commutative ring unless σ is anidentity automorphism.Let | σ | denote the order of σ and assume | σ | = t . Then there exists a positiveinteger d such that σ = θ d and m = td . Clearly, σ fixes the subfield F p d of F q . Let Z ( F q [ x, σ ]) denote the center of R . For f, g ∈ R , g is called a right divisor (resp. leftdivisor ) of f if there exists r ∈ R such that f = rg (resp. f = gr ). In this case, f iscalled a left multiple (resp. right multiple ) of g . Let the division be defined similarly.Then • If g, f ∈ Z ( F q [ x, σ ]), then g · f = f · g . • Over finite fields, a skew polynomial ring is both a right Euclidean ring and aleft Euclidean ring. 2et f, g ∈ R . A polynomial h is called a greatest common left divisor (gcld) of f and g if h is a left divisor of f and g ; and if u is another left divisor of f and g , then u is a left divisor of h . A polynomial e is called a least common left multiple (lclm)of f and g if e is a right multiple of f and g ; and if v is another right multiple of f and g , then v is a right multiple of e . The greatest common right divisor (gcrd) and least common right multiple (lcrm) of polynomials f and g are defined similarly.The main aim of this paper is to study the structural properties of skew generalizedquasi-cyclic (GQC) codes over finite fields. The rest of this paper is organized asfollows. In Section 2, we survey some well known results of skew cyclic codes and givethe BCH-type bound for skew cyclic codes. By the factorization theory of ideals, wegive the Chinese Remainder Theorem in skew polynomial rings. In Section 3, using theChinese Remainder Theorem, we give a necessary and sufficient condition for a codeto be a skew GQC code. And this leads to a canonical decomposition of skew GQCcodes. In Section 4, we mainly describe some characteristics of 1-generator GQCcodes including parity-check polynomials, dimensions and the minimum Hammingdistance bounds. In Section 5, we discuss a special class of skew GQC codes calledskew QC codes. Let σ be an automorphism of the finite field F q and n be a positive integer suchthat the order of σ divides n . A linear code C of length n over F q is called skewcyclic code or σ -cyclic code if for any codeword ( c , c , . . . , c n − ) ∈ C , the vector( σ ( c n − ) , σ ( c ) , . . . , σ ( c n − )) is also a codeword in C . In polynomial representation,a linear code of length n over F q is a skew cyclic code if and only if it is a left ideal ofthe ring R/ ( x n − x n −
1) denotes the two-sided ideal generated by x n − f ( x ) ∈ R generates a two-sided ideal, then a left ideal of R/ ( f ( x )) is alinear code over F q . Such a linear code will be called a skew linear code or a σ -linearcode . Let C be a linear code of length n over F q . The Euclidean dual of C is definedas C ⊥ = { v ∈ F nq | u · v = 0 , ∀ u ∈ C } . In this paper, we suppose that the order of σ divides n and gcd( n, q ) = 1. In thefollowing, we list some well known results of skew cyclic codes in Theorem 2.1. Theorem 2.1 [6, 7]
Let C be a skew cyclic code ( σ -cyclic code) of length n over F q generated by a right divisor g ( x ) = P n − k − i =0 g i x i + x n − k of x n − . Then(i) The generator matrix of C is given by g · · · g n − k − · · · σ ( g ) · · · σ ( g n − k − ) 1 · · ·
00 . . . . . . . . . ...... . . . . . . · · · . . . 00 · · · σ k − ( g ) · · · σ k − ( g n − k − ) 1 (1) and | C | = q n − deg( g ( x )) .(ii) Let x n − h ( x ) g ( x ) and h ( x ) = P k − i =0 h i x i . Then C ⊥ is also a skew cycliccode of length n generated by e h ( x ) = x deg( h ( x )) ϕ ( h ( x )) = 1+ σ ( h k − ) x + · · · + σ k ( h ) x k ,where ϕ is an anti-automorphism of σ defined as ϕ ( P ti =0 a i x t ) = P ti =0 x − i a i , where P ti =0 a i x i ∈ R . The generator matrix of C ⊥ is given by σ ( h k − ) · · · σ k ( h ) 0 · · ·
00 1 σ ( h k − ) · · · σ k +1 ( h ) · · ·
00 0 . . . . . . 0... . . . . . . · · · . . . ...0 · · · σ n − k ( h k − ) · · · σ n − ( h ) (2) and | C ⊥ | = q k .(iii) For c ( x ) ∈ R , c ( x ) ∈ C if and only if c ( x ) h ( x ) = 0 in R .(iv) C is a cyclic code of length n over F q if and only if the generator polynomial g ( x ) ∈ F p d [ x ] / ( x n − . The monic polynomials g ( x ) and h ( x ) in Theorem 2.1 are called the generatorpolynomial and the parity-check polynomial of the skew cyclic code C , respectively. Theorem 2.2
Let C be a skew cyclic code with the generator polynomial g ( x ) andthe check polynomial h ( x ) . Then a polynomial f ( x ) ∈ R/ ( x n − generates C if andonly if there exists a polynomial p ( x ) ∈ R such that f ( x ) = p ( x ) g ( x ) where p ( x ) and h ( x ) are right coprime.Proof Let f ( x ) ∈ R/ ( x n −
1) generate C . Then there exist polynomials p ( x ) , q ( x ) ∈ R/ ( x n −
1) such that f ( x ) = p ( x ) g ( x ) and g ( x ) = q ( x ) f ( x ) in R/ ( x n − g ( x ) = q ( x ) p ( x ) g ( x ). In R , we have g ( x ) = q ( x ) p ( x ) g ( x ) + r ( x )( x n −
1) = q ( x ) p ( x ) g ( x ) + r ( x ) h ( x ) g ( x )for some r ( x ) ∈ R . It follows that(1 − q ( x ) p ( x ) − r ( x ) h ( x )) g ( x ) = 0 . R is a principal ideal domain, we have 1 − q ( x ) p ( x ) − r ( x ) h ( x ) = 0, whichimplies that p ( x ) and h ( x ) are right coprime.Conversely, suppose f ( x ) = p ( x ) g ( x ) where p ( x ) and h ( x ) are right coprime. Thenthere exist polynomials u ( x ) , v ( x ) ∈ R such that u ( x ) p ( x )+ v ( x ) h ( x ) = 1. Multiplyingon right by g ( x ) both sides, we have u ( x ) p ( x ) g ( x ) + v ( x ) h ( x ) g ( x ) = g ( x ), whichimplies that u ( x ) p ( x ) g ( x ) = u ( x ) f ( x ) g ( x ) in R/ ( x n − g ( x ) ∈ ( f ( x )) l ,where ( f ( x )) l denotes the left ideal generated by f ( x ) in R/ ( x n − g ( x )) l ⊆ ( f ( x )) l . Clearly, ( f ( x )) l ⊆ ( g ( x )) l , and hence ( f ( x )) l = ( g ( x )) l = C . ✷ Let F [ Y q , ◦ ] = { a Y + a Y q + · · · + a n Y q n | a , a , . . . , a n ∈ F q } , where q = p d .For f = a Y + a Y q + · · · + a n Y q n and g = b Y + b Y q + · · · + b t Y q t , define f + g to be ordinary addition of polynomials and define f ◦ g = f ( g ). Thus, f ◦ g = c Y + c Y q + · · · + c n + t Y q n + t , where c i = P j + s = i a j b q s s . It is easy to see that F [ Y q , ◦ ] under addition and composition ◦ forms a non-commutative ring called OrePolynomial ring (see [20]).Define φ : R → F q [ Y q , ◦ ] , X a i x i X a i Y q i . Lemma 2.3 [20, Theorem II.13]
The above mapping φ : R → F [ Y q , ◦ ] is a ringisomorphism between the skew polynomial ring R = F q [ x, σ ] and the Ore Polynomialring F q [ Y q , ◦ ] . ✷ For a skew cyclic code over F q , it can also be described in terms of the n -th rootof unity. By the above mapping φ , one can verify that φ ( x n −
1) = Y q n − Y . Since σ = θ d , the fixed subfield is F p d = F q . Let F q s be the smallest extension of F q containing F q n as a subfield. Then F q s is the splitting field of φ ( x n −
1) over F q . Anelement α ∈ F q s is called a right root of f ∈ R if x − α is a right divisor of f .Let the extension of σ to an automorphism of F q s be also denoted by σ . For any α ∈ F q s define N σ,i ( α ) = σ i − ( α ) σ i − ( α ) · · · σ ( α ) α, i >
0, with N σ, = 1. Lemma 2.4 [15, Proposition 1.3.11]
Let f ( x ) = P ki =0 a i x i ∈ R . Then(i) The remainder r on right division of f ( x ) by x − α is given by r = a N σ, ( α ) + a N σ, ( α ) + · · · + a k N σ,k ( α ) .(ii) Let β ∈ F q s . Then ( x − β ) | r f ( x ) if and only if P ki =0 a i N σ,i ( β ) = 0 . ✷ Note that σ ( α ) = θ d ( α ) = α p d = α q , and hence N σ,i ( α ) = α qi − q − . The followingresult can also be found in [11, Lemma 4], here we give another proof by Lemma 2.3. Lemma 2.5
Let f ( x ) ∈ R , and F q s be the smallest extension of F q in which φ ( f ( x )) splits. Then a non-zero element α ∈ F q s is a root of φ ( f ( x )) if and only if α q /α isa right root of f ( x ) . roof If α q /α is a right root of f ( x ), then x − α q /α is a right divisor of f ( x ). FromLemma 2.3, we have Y q − α q /αY is a factor of φ ( f ( x )). Therefore α is the root of φ ( f ( x )).Conversely, suppose α is the root of φ ( f ( x )). Let f ( x ) = k ( x )( x − α q /α ) + r ,where r ∈ F q . Then φ ( f ( x )) = φ ( k ( x )) ◦ φ ( x − α q /α ) + φ ( r ). From the discussionabove, α is the root of φ ( x − α q /α ). Therefore α is also the root of φ ( r ), i.e., rα = 0.Since α is a non-zero element in F q s , we have r = 0. This implies that α q /α is aright root of f ( x ). ✷ Since φ ( x n −
1) = Y q n − Y splits into linear factors in F q s , it follows from Lemma2.3 that x n − F q s [ x, σ ]. It is well known that thenon-zero roots of Y q n − Y are precisely the elements of { , γ, . . . , γ q n − } , where γ isa primitive element of F q n . Therefore, by Lemma 2.5, x − ( γ i ) q /γ i is the right factorof the skew polynomial x n −
1. It means that there are several different factorizationsof the skew polynomial x n − F q . Theorem 2.6
Let C be a skew cyclic code of length n generated by a monic rightfactor g ( x ) of the skew polynomial x n − in R . If x − γ j is a right divisor of g ( x ) forall j = b, b + 1 , . . . , b + δ − , where b ≥ and δ ≥ , then the minimum Hammingdistance of C is at least δ .Proof Let c ( x ) = P n − i =0 c i x i be a codeword of C . Then c ( x ) is a left multiple of g ( x ), and hence x − γ j is a right divisor of c ( x ), for all 0 ≤ j ≤ b + δ −
2. FromLemma 2.4, x − γ j is a right divisor of c ( x ) if and only if P n − i =0 c i N σ,i ( γ j ) = 0, j = b, b + 1 , . . . , b + δ −
2. Therefore the matrix N σ, ( γ b ) · · · N σ,n − ( γ b )1 N σ, ( γ b +1 ) · · · N σ,n − ( γ b +1 )... ... . . . ...1 N σ, ( γ b + δ − ) · · · N σ,n − ( γ b + δ − ) (3)is a parity-check matrix. Any δ − δ − × ( δ −
1) matrixand denote D as its determinant. Since D is a Vandermonde determinant, D = 0 ifand only if N σ,i ( γ ) = N σ,j ( γ ), for i = j . It is equivalent to γ qi − qj q − = γ qj qi − j − q − = 1 . In particular, γ q j ( q i − j − = 1 implies that ( q n − | q j ( q i − j − q n − , q j ) =1, it follows that ( q n − | ( q i − j − l suchthat i − j = nl . It means that q nl − q − = k ( q n −
1) for some positive integer k . Thus( q − | q nl − q n − = P m − i =0 q ni . It implies that γ q − | γ , which is impossible. This showsthat any δ − C is is at least δ . ✷ xample 2.7 Consider R = F [ x, σ ], where σ = θ is a Frobenius automorphism of F over F . The polynomial g ( x ) = x − α is a right factor of x −
1, where α is aprimitive element of F . Since φ ( x −
1) = Y − Y , it follows that φ ( x −
1) splitsin F . Let ξ be a primitive element of F . Then α = ξ and φ ( g ( x )) = Y − α Y has a root ξ . Therefore, by Lemma 2.5, ( ξ ) /ξ = ξ is a right root of g ( x ). Let C be a skew cyclic code of length 4 generated by g ( x ) over F . Then C is a codewith dim( C ) = 3 and d H ( C ) ≥
2. In fact, C is an optimal [4 , ,
2] skew cyclic codeover F . Also for each i = 1 , , . . . ,
40, ( ξ i ) /ξ i = ξ i is a right root of x −
1, whichimplies that there are 10 different factorizations of skew polynomial x − F .We now consider the factorization theory of (two-sided) ideals or two-sided ele-ments. An element a ∗ ∈ R is called two-sided element if Ra ∗ = a ∗ R . Theorem 2.8 [20, Theorem II.12]
If a polynomial f ∗ generates a two-sided ideal in R , then f ∗ has the form ( a + a x t + · · · + a n x nt ) x m , where a i ∈ F q and t = | σ | . Obvious F q [ x ] = { b + b x + · · · + b n x n | b i ∈ F q } forms a commutative subring of R . Then the center of R is F q [ x ] ∩ F q [ x t ] where F q [ x t ] = { a + a x t + · · · + a n x nt | a i ∈ F q } , i.e., the center of R is F q [ x t ] = F p d [ x t ] (see [20]).If Ra ∗ is a non-zero (two-sided) maximal ideal in R , or equivalently, a ∗ = 0 and R/Ra ∗ is a simple ring, then we call the two-sided element a ∗ a two-sided maximal (t.s.m) element. Let a, b be non-zero elements in R . Then a is said to be left similar to b ( a ∼ l b ) if and only if R/Ra ∼ = R/Rb . Two elements are left similar if and onlyif they are right similar.Let a be a non-zero element in R . If a is not a unit in R , then a can be writtenas a = p p · · · p s , where p , p , . . . , p s are irreducible. Moreover, if a = p p · · · p s = p ′ p ′ · · · p ′ t , where p i and p ′ j are irreducible, then s = t and there exists a permutation(1 ′ , ′ , . . . , s ′ ) of (1 , , . . . , s ) such that p i ∼ p ′ i ′ (see [15]). Lemma 2.9 [15, Theorem 1.2.17 ′ , Theorem 1.2.19] Let a ∗ ba a non-zero two-sidedelement in R and a ∗ not a unit. Then(i) a ∗ = p ∗ p ∗ · · · p ∗ m , where p ∗ i , ≤ i ≤ m , are t.s.m elements and such a factorizationis unique up to order and unit multipliers.(ii) Let p ∗ i = p i, p i, · · · p i,n , where p i,j , ≤ i ≤ m, ≤ j ≤ n are irreducible. Then p i, , p i, , . . . , p i,n are all similar. Example 2.10
Consider R = F [ x, σ ], where σ = θ is a Frobenius automorphismof F over F . The fixed field of σ is F . Let f ( x ) = x − x − ∈ R . Since f ( x ) ∈ F [ x ], f ( x ) is a two-sided element of R . A factorization of f ( x ) in R is f ( x ) = ( x + 1)( x − x + 1 and x − x t , t ≥
1, to be a two-sided element.
Remark 2.1
Note that there is an error in Example 4 in [5], where the authorclaimed that the fixed field of σ is F . But it is well known that F is not a subfieldof F at all. We have corrected it in Example 2.10.Suppose x n − x n − f f · · · f k , where f , f , . . . , f k are irreducible polynomials. Since x n − x n − f ∗ f ∗ · · · f ∗ t , where each f ∗ i is a t.s.melement and is a product of all polynomials similar to an irreducible factor f i of x n − q, n ) = 1, it follows that all factors f ∗ , f ∗ , . . . , f ∗ t are distinct. Also since( f ∗ i ) is maximal, we can see that f ∗ i and f ∗ j are coprime for all i = j . Denote b f ∗ i asthe product of all f ∗ j except f ∗ i , we have the following Chinese Remainder Theorem in the skew polynomial ring F q [ x, σ ]. Theorem 2.11
Let x n − f ∗ f ∗ · · · f ∗ t be the unique representation of x n − asa product of pairwise coprime t.s.m elements in R . Since gcd( b f ∗ i , f ∗ i ) = 1 , there existpolynomials b i , c i ∈ R such that b i b f ∗ i + c i f ∗ i = 1 . Let e i = b i b f ∗ i ∈ R . Then(i) e , e , . . . , e t are mutually orthogonal in R ;(ii) e + e + · · · + e t = 1 in R ;(iii) R i = ( e i ) is a two-sided ideal of R and e i is the identity in ( e i ) ;(iv) R = R L R L · · · L R t ;(v) For each i = 1 , , . . . , t , the map ψ : R/ ( f ∗ i ) → R i g + ( f ∗ i ) ( g + ( x n − e i is a well-defined isomorphism of rings;(vi) R ∼ = R/ ( f ∗ ) L R/ ( f ∗ ) L · · · L R/ ( f ∗ t ) .Proof (i) Suppose e i = 0 for some i = 1 , , . . . , t , i.e., b i b f ∗ i ∈ ( x n −
1) in R . Then b i b f ∗ i ∈ ( f ∗ i ). Thus 1 = b i b f ∗ i + c i f ∗ i ∈ ( f ∗ i ), which is a contradiction. Hence, for each i = 1 , , . . . , t , e i = 0. Thus we have b i b f ∗ i b j b f ∗ j ∈ ( x n −
1) for i = j . This implies that e i e j = 0 in R .(ii) We have b b f ∗ + · · · + b t b f ∗ t − ∈ ( f ∗ i ), for all i = 1 , , . . . , t . Therefore b b f ∗ + · · · + b t b f ∗ t − ∈ ( x n − e + · · · + e t = 1 in R .(iii) Let Re i = ( e i ) l . Then ( e i ) l ⊆ ( b f ∗ i ). On the other hand, b f ∗ i = b f ∗ i ( b i b f ∗ i + c i f ∗ i ) = b f ∗ i b i b f ∗ i in R , which implies ( b f ∗ i ) ⊆ ( e i ) l . Therefore ( e i ) l = ( b f ∗ i ). Similarly, one canprove that e i R = ( e i ) r = ( b f ∗ i ), which implies that ( e i ) is a two-sided ideal of R .Clearly, e i is the identity in ( e i ).(iv) For any a ∈ R , a can be represented as a = ae + ae + · · · + ae t . Since ae i ∈ ( e i ), R = ( e )+( e )+ · · · +( e t ). Assume that a + a + · · · + a t = 0, where a i ∈ ( e i ).8ultiplying on the left (or on the right) by e i , we obtain that a e i + a e i + · · · + a t e i = a i e i = a i , for i = 1 , , . . . , t . Therefore R = R L R L · · · L R t .(v) Let g +( f ∗ i ) = g ′ +( f ∗ i ), for g, g ′ ∈ R . Then g − g ′ ∈ ( f ∗ i ). But b i b f ∗ i ∈ ( f ∗ j ) for all i = j . Therefore ( g − g ′ ) b i b f ∗ i ∈ ( x n − g +( x n − e i = ( g ′ +( x n − e i in R ,which implies that the map ψ is well-defined. Clearly, ψ is a surjective homomorphismof rings. Let g + ( f ∗ i ) ∈ R/ ( f ∗ i ) statisfy ( g + ( x n − e i = ( x n − gb i b f ∗ i ∈ ( x n − ⊆ ( f ∗ i ). Thus g ∈ ( f ∗ i ), i.e., g + ( f ∗ i ) = ( f ∗ i ). This implies that the kernel of ϕ is zero.(vi) From (iv) and (v), one can deduce this result immediately. ✷ In this section, we investigate the structural properties of skew GQC codes. We givethe definition of skew GQC codes first.
Definition 3.1
Let R = F q [ x, σ ] be a skew polynomial ring and m , m , . . . , m l positive integers. Let t be a divisor of each m i , where t is the order of σ and i =1 , , . . . , l . Denote R i = R/ ( x m i − for i = 1 , , . . . , l . Any left R -submodule of the R -module R = R × R × · · · × R l is called a skew generalized quasi-cyclic (GQC)code over F q of block length ( m , m , . . . , m l ) and length P li =1 m i . Let m i ≥ i = 1 , , . . . , l , and gcd( q, m i ) = 1. Then, by Lemma 2.9, x m i − x m i − f ∗ i f ∗ i · · · f ∗ ir i , where f ∗ ij , j = 1 , , . . . , r i , are pairwisecoprime monic t.s.m elements in R . Let { g ∗ , g ∗ , . . . , g ∗ s } = { f ∗ ij | ≤ i ≤ l, ≤ j ≤ r i } .Then we have x m i − g ∗ d i g ∗ d i . . . g ∗ d is s , where d ik = 1 if g ∗ k = f ∗ i,j for some 1 ≤ j ≤ r i and d i,k = 0 if gcd( g ∗ k , x m i −
1) = 1, forall 1 ≤ i ≤ l and 1 ≤ k ≤ s . Suppose n j = | { i | f ∗ i,λ = g ∗ j , ≤ λ ≤ r i , ≤ i ≤ l, ≤ j ≤ s } | . Let M j = ( R/ ( g ∗ j )) n j . Then we have Theorem 3.2
Let R = R × R × · · · × R l , where R i = R/ ( x m i − for all i = 1 , , . . . , l . Then there exists an R -module isomorphism φ from R onto M ×M ×· · ·×M s such that a linear code C is a skew GQC code of block length ( m , m , . . . , m l ) and length P li =1 m i over F q if and only if for each ≤ k ≤ s there is a unique left R -module M k of M k such that φ ( C ) = M × M × · · · × M s .Proof Denote g ∗ = g ∗ g ∗ · · · g ∗ s , b g ∗ k = g ∗ g ∗ k , e g ∗ i,k = x m i − g ∗ d ik k , i = 1 , , . . . , l, k = 1 , , . . . , s. Then there exists a polynomial w ∗ i,k ∈ R such that b g ∗ k = w ∗ i,k e g ∗ i,k , i = 1 , , . . . , l, k = 1 , , . . . , s. g ∗ k and b g ∗ k are coprime, there exist polynomials b k , s k ∈ R such that b k b g ∗ k + s k g ∗ k =1, which implies that b k w ∗ i,k e g ∗ i,k + c k g ∗ k = 1 in R . Let ε ik = b k w ∗ i,k e g ∗ i,k + ( x m i −
1) = b k b g ∗ k + ( x m i − ∈ R i . Then, by Theorem 2.11, we have(i) ε ik = 0 if and only if gcd( g ∗ k , x m i −
1) = 1 , k = 1 , , . . . , s. (ii) ε i , ε i , . . . , ε is are mutually orthogonal in R i .(iii) ε i + ε i + · · · + ε is = 1 in R i .(iv) Let R ik = ( ε ik ) be the principle ideal of R i generated by ε ik . Then ε ik is theidentity of R ik and R ik = ( b k b g ∗ k ). Hence R ik = { } if and only if gcd( g ∗ k , x m i −
1) = 1.(v) R i = L sk =1 R ij .(vi) For each k = 1 , , . . . , s , the mapping φ ik : R ik → R/ ( g ∗ d ik k ), defined by φ ik : f b k b g ∗ k + ( x m i − f + ( g ∗ d ik k ) , where f ∈ R, is a well defined isomorphism of rings.(vii) R i = R/ ( x m i − ∼ = L sj =1 R/ ( g ∗ d ij j ).From (vi), we have a well defined R -module isomorphism Φ k from b k b g ∗ k R onto R/ ( g ∗ d ik k ) × · · · × R/ ( g ∗ d ik k ), which defined byΦ k : ( α , . . . , α l ) ( φ k ( α ) , . . . , φ lk ( α l )) , where α i ∈ R ik , i = 1 , , . . . , l. Φ k can introduce a natural R -module isomorphism µ k from b k b g ∗ k R onto M k .For any c = ( c , c , . . . , c l ) ∈ R , from (v) we deduce c = ( b b g ∗ c + · · · + b s b g ∗ s c , . . . ,b b g ∗ c l + · · · + b s b g ∗ s c l ) = b b g ∗ c + · · · + b s b g ∗ s c , where b k b g ∗ k c ∈ b k b g ∗ k R × · · · × b k b g ∗ k R l forall k = 1 , , . . . , s . Hence R = b b g ∗ R + · · · + b s b g ∗ s R . Let c , c , . . . , c s ∈ R satisfying b b g ∗ c + · · · + b s b g ∗ s c s = 0. Since ( x m i − | g ∗ for all i = 1 , , . . . , l , it follows that g ∗ R = { } . Then for each k = 1 , , . . . , s , from b k b g ∗ k + s k g ∗ k = 1, g ∗ = g ∗ k b g ∗ k and g ∗ | b g ∗ τ b g ∗ σ for all 1 ≤ τ = σ ≤ s , we deduce b k b g ∗ k c k = 0. Hence R = L sj =1 b j b g ∗ j R .Define φ : β + β + · · · + β s ( µ ( β ) , µ ( β ) , . . . , µ s ( β s )) where β k ∈ b k b g ∗ k R , k =1 , , . . . , s . Then φ is an R -module isomorphism from R onto M × · · · × M s . Forany left R -module M j , it is obvious that M × · · · × M s is a left R -submodule of M × · · · × M s . Therefore there is a unique left R -submodule C of R such that φ ( C ) = M × · · · × M s . ✷ Since M k = ( R/ ( g ∗ k )) n k = L li =1 R/ ( g ∗ d ik k ) is up to an R -module isomorphism,Theorem 3.2 can lead to a canonical decomposition of skew GQC codes as follows. Theorem 3.3
Let C be a skew GQC code of block length ( m , m , . . . , m l ) and length P li =1 m i over F q . Then C = s M i =1 C i where C i , ≤ i ≤ s , is a linear code of length l over R/ ( g ∗ d ik i ) and each j -th, ≤ j ≤ l ,component in C i is zero if d ji = 0 and an element of the ring R/ ( g ∗ i ) otherwise. ✷ m = m = · · · m l = m . Then a skew GQC code C is a skew quasi-cyclic (QC) code of length ml over F q . From Theorems 3.2 and 3.3, we have the followingresult. Corollary 3.4
Let R = F q [ x, σ ] , gcd( m, q ) = 1 and x m − g ∗ g ∗ · · · g ∗ s , where g ∗ , g ∗ , . . . , g ∗ s are pairwise coprime monic t.s.m elements in R . Then we have(i) There is an R -module isomorphism φ from R = ( R/ ( x m − l , l ≥ , onto ( R/ ( g ∗ )) l × ( R/ ( g ∗ )) l × · · · × ( R/ ( g ∗ s )) l .(ii) C is a skew QC code of length ml over F q if and only if there is a left R -submodule M i of ( R/ ( g ∗ i )) l , i = 1 , , . . . , s , such that φ ( C ) = M × M × · · · × M s .(iii) A skew QC code C of length ml can be decomposed as C = L si =1 C i , where each C i is a linear code of length l over R/ ( g ∗ i ) , i = 1 , , . . . , s . ✷ A skew GQC code C of block length ( m , m , . . . , m l ) and length P li =1 m i is calleda ρ -generator over F q if ρ is the smallest positive integer for which there are codewords c i ( x ) = ( c i, ( x ) , c i, ( x ) , . . . , c i,l ( x )), 1 ≤ i ≤ ρ , in C such that C = Rc ( x ) + Rc ( x ) + · · · + Rc ρ ( x ).Assume that the dimension of each C i , i = 1 , , . . . , s , is k i , and set K = max { k i | ≤ i ≤ s } . Now by generalizing Theorem 3 of [13], we get Theorem 3.5
Let C be a ρ -generator skew GQC code of block length ( m , m , . . . , m l ) and length P li =1 m i over F q . Let C = L si =1 C i , where each C i , i = 1 , , . . . , s , is withdimension k i and K = max { k i | ≤ i ≤ s } . Then ρ = K . In fact, any skew GQC code C with C = L si =1 C i , where each C i , i = 1 , , . . . , s , is with dimension k i satisfying ρ = max ≤ i ≤ s k i , is a ρ -generator skew GQC code.Proof Let C be a ρ -generator skew GQC code generated by the elements c ( j ) ( x ) =( c ( j )1 ( x ) , c ( j )2 , . . . , c ( j ) l ( x )) ∈ R , j = 1 , , . . . , ρ . Then for each i = 1 , , . . . , s , C i isspanned as a left R -module by e c ( j ) ( x ) = ( e c ( j )1 ( x ) , e c ( j )2 ( x ) , . . . , e c ( j ) l ( x )), where e c ( j ) ν ( x ) = c ( j ) ν ( x ) (mod g ∗ i ) if g ∗ i is a factor of x m i − e c ( j ) ν ( x ) = 0 otherwise, ν = 1 , , . . . , l .Hence k i ≤ ρ for each i , and so K ≤ ρ .On the other hand, since K = max ≤ i ≤ s k i , there exist q ( j ) i ( x ) ∈ R l , 1 ≤ j ≤ K ,such that q ( j ) i ( x ) span C i , 1 ≤ i ≤ s , as a left R -module. Then, by Theorem 3.3, foreach 1 ≤ j ≤ K , there exists q ( j ) ( x ) ∈ C such that q ( j ) i ( x ) = q ( j ) ( x ) (mod g ∗ i ) and C is generated by q ( j ) i ( x ), 1 ≤ j ≤ K . Hence ρ ≤ K , which implies that ρ = K . ✷ If C is a 1-generator skew GQC code of block length ( m , m , . . . , m l ) and length P li =1 m i over F q , then by Theorem 3.5, each C i , i = 1 , , . . . , s , is either trivial or an[ l,
1] linear code over R/ ( g ∗ i ). Conversely, any linear code C is a 1-generator GQCcode when each C i , i = 1 , , . . . , s , is with dimension at most 1. Example 3.6
Let R = F [ x, σ ], where σ is the Frobenius automorphism of F over F . Let R = R/ ( x − × R/ ( x −
1) and C be a 2-generator skew GQC code11f block length (4 ,
8) and length 4 + 8 = 12 generated by c ( x ) = ( x − x, x − αx )and c ( x ) = ( x , x − αx ), where α is a 4-th primitive element in F over F . Since x − x − x −
2) and x − x − x − x − α )( x − α ), by Theorem3.2, R ∼ = ( R/ ( x − × ( R/ ( x − × R/ ( x − α ) × R/ ( x − α ) . Then up to an R -module isomorphism R ∼ = ( R/ ( x − , R/ ( x − L ( R/ ( x − , R/ ( x − L (0 , R/ ( x − α )) L (0 , R/ ( x − α )) . (4)This implies that the skew GQC code C can be decomposed into C = L i =1 C i , where • C is the [2 ,
2] linear code with the basis (0 , (1 − α ) x ) and ( x, (1 − α ) x ) over R/ ( x − • C is the [2 ,
2] linear code with the basis ( x, (2 − α ) x ) and (2 x, (2 − α ) x ) over R/ ( x − • C is the [2 ,
1] linear code with the basis (0 , αx ) over R/ ( x − α ); • C is the [2 ,
1] linear code with the basis (0 , ( α − α ) x ) over R/ ( x − α ).Let k i be the dimension of C i , i = 1 , , ,
4. Thenmax k i = 2 = the number of generators of C. -generator skew GQC codes In this section, we discuss some structural properties of 1-generator skew GQC codesover F q . Let R = F q [ x, σ ] and R = R/ ( x m − × R/ ( x m − × · · · × R/ ( x m l − Definition 4.1
Let C be a -generator skew GQC code generated by c ( x ) = ( c ( x ) ,c ( x ) , . . . , c l ( x )) ∈ R . The the monic polynomial h ( x ) of minimum degree satisfying c ( x ) h ( x ) = 0 is called the parity-check polynomial of C . Let C be a 1-generator skew GQC code of block length ( m , m , . . . , m l ) and length P li =1 m i with the generator ( c ( x ) , c ( x ) , . . . , c l ( x )), c i ( x ) ∈ R i = R/ ( x m i − , i =1 , , . . . , l . Define a well defined R -homomorphism ϕ i from R onto R i such that ϕ i ( c ( x ) , c ( x ) , . . . , c l ( x )) = c i ( x ). Then ϕ i ( C ) is a skew cyclic code of length m i generated by c i ( x ) in R i . From Theorem 2.2, we have ϕ i ( C ) = ( p i ( x ) g i ( x )), where g i ( x ) is a right divisor of x m i − p i ( x ) and h i ( x ) = ( x m i − /g i ( x ) are rightcoprime. Therefore, h i ( x ) = ( x m i − /g i ( x ) = ( x m i − / gcld( c i ( x ) , x m i −
1) is theparity-check polynomial of ϕ i ( C ). It means that h ( x ) = lclm { h ( x ) , h ( x ) , . . . , h l ( x ) } is the parity-check polynomial of C . Define a map ψ from R to R such taht ψ ( a ( x )) = c ( x ) a ( x ) . This is an R -module homomorphism with the kernel ( h ( x )) r , which impliesthat C ∼ = R/ ( h ( x )) r . Thus dim( C ) = deg( h ( x )).12s stated above, we have the following result. Theorem 4.2
Let C be a -generator skew GQC code of block length ( m , m , . . . , m l ) and length P li =1 m i generated by c ( x ) = ( c ( x ) , c ( x ) , . . . , c l ( x )) ∈ R . Then theparity-check polynomial of C is h ( x ) = lclm { h ( x ) , h ( x ) , . . . , h l ( x ) } , where h i ( x ) =( x m i − / gcld( c i ( x ) , x m i − , i = 1 , , . . . , l , and the dimension of C is equal to thedegree of h ( x ) . ✷ Let h ( x ) and h ( x ) be the parity-check polynomials of 1-generator skew GQCcodes C and C , respectively. If C = C , then h ( x ) = h ( x ), which impliesthat deg( h ( x )) = deg( h ( x )). It means that R/ ( h ( x )) r = R/ ( h ( x )) r . Conversely,suppose h ( x ) and h ( x ) are similar. Then we have R/ ( h ( x )) r ∼ = R/ ( h ( x )) r , whichimplies that C = C . Then from the discussion above, we have C = C if andonly if h ( x ) ∼ h ( x ), i.e., any 1-generator skew GQC code has a unique parity-checkpolynomial up to similarity. Theorem 4.3
Let C be a -generator skew GQC code of block length ( m , m , . . . , m l ) and length P li =1 m i generated by c ( x ) = ( c ( x ) , c ( x ) , . . . , c l ( x )) ∈ R . Suppose h i ( x ) is given as in Theorem 4.2 and h ( x ) = lclm { h ( x ) , h ( x ) , . . . , h l ( x ) } . Let δ i denotethe number of consecutive powers of a primitive m i -th root of unity that among theright zeros of ( x m i − /h i ( x ) . Then(i) d H ( C ) ≥ P i K ( δ i + 1) , where K ⊆ { , , . . . , l } is a set of maximum size such that lclm i ∈ K h i ( x ) = h ( x ) .(ii) If h ( x ) = h ( x ) = · · · = h l ( x ) , then d H ( C ) ≥ P li =1 ( δ i + 1) .Proof Let a ( x ) ∈ C be a nonzero codeword. Then there exists a polynomial f ( x ) ∈ R such that a ( x ) = f ( x ) c ( x ). Since for each i = 1 , , . . . , l , the i -th component is zero ifand only if ( x m i − | f ( x ) c i ( x ), i.e., if and only if h i ( x ) | f ( x ). Therefore a ( x ) = 0if and only if h ( x ) | f ( x ). So a ( x ) = 0 if and only if h ( x ) ∤ f ( x ). This implies that c ( x ) = 0 has the most number of zero blocks whenever h ( x ) = lclm i ∈ K h i ( x ), wherelclm i ∈ K h i ( x ) | f ( x ), and K is a maximal subset of { , , . . . , l } having this property.Thus, d H ( C ) ≥ P i/ ∈ K d i , where d i = d H ( ϕ i ( C )) ≥ δ i + 1. Clearly, K = ∅ if and onlyif h ( x ) = h ( x ) = · · · = h l ( x ). Therefore, from the discussion above, we have if h ( x ) = h ( x ) = · · · = h l ( x ), then d H ( C ) = P li =1 d i ≥ P li =1 ( δ i + 1). ✷ From Theorems 4.2 and 4.3, we have the following corollary immediately.
Corollary 4.4
Let C be a -generator skew QC code of length ml generated by c ( x ) =( c ( x ) , c ( x ) , . . . , c l ( x )) ∈ ( R/ ( x m − l . Suppose h i ( x ) = ( x m − / gcld( c i ( x ) , x m − , i = 1 , , . . . , l , and h ( x ) = lclm { h ( x ) , h ( x ) , . . . , h l ( x ) } . Then(i) The dimension of C is the degree of h ( x ) .(ii) Let δ i denote the number of consecutive powers of a primitive m i -th root of unitythat among the right zeros of ( x m − /h i ( x ) . Then d H ( C ) ≥ P i K ( δ i + 1) , where K ⊆ { , , . . . , l } is a set of maximum size such that lclm i ∈ K h i ( x ) = h ( x ) .(iii) If h ( x ) = h ( x ) = · · · = h l ( x ) , then δ i = δ for each i = 1 , , . . . , l and d H ( C ) ≥ ( δ + 1) . ✷ Example 4.5
Let R = F [ x, σ ], where σ is the Frobenius automorphism of F over F . The polynomial g ( x ) = x − α is a right divisor of x −
1, where α is aprimitive element of F . Consider the 1-generator GQC code C of block length (4 , c ( x ) = ( g ( x ) , g ( x )). Then, by Theorem 4.3, h ( x ) = ( x − / ( x − α ) and d H ( C ) ≥
2. A generator matrix for C is given as follows G = − α − α − α − α − α − α − α − α − α − α − α − α − α − α . (5) C is an optimal [12 , ,
4] skew GQC code over F actually. Example 4.6
Let R = F [ x, σ ], where σ is the Frobenius automorphism of F over F . The polynomial g ( x ) = x − α is a right divisor of x −
1, where α is a primitiveelement of F . Consider the 1-generator skew QC code C of length ml = 12 andindex 3 generated by c ( x ) = ( g ( x ) , g ( x ) , g ( x )) over F . Then h ( x ) = h ( x ) = h ( x ) = h ( x ) = ( x − / ( x − α ). Thus, by Corollary 4.4, C is a skew QC code of length 12and index 3 with dimension 3 and the minimum Hamming distance at least 3 × C is given as follows G = − α − α − α − α − α − α − α − α − α . (6)From the generator matrix G , we see that C is an [12 , ,
6] skew QC code over F . Skew quasi-cyclic (QC) codes as a special class of skew generalized quasi-cyclic (GQC)codes, have the similar structural properties to skew GQC codes such as Corollary3.4 and Corollary 4.4. But in this section, we use another view presented in [16] toresearch skew QC codes over finite fields. The dual codes of skew QC codes are alsodiscussed briefly.For convenience, we write an element a ∈ F lmq as a m - tuple a = ( a , a , . . . , a m − ),where a i = ( a i, , a i, , . . . , a i, ( l − ) ∈ F lq . Let the map T σ,l on F lmq be defined as follows T σ,l ( a , a , . . . , a m − ) = ( σ ( a m − ) , σ ( a ) , . . . , σ ( a m − )) , where σ ( a i ) = ( σ ( a i, ) , σ ( a i, ) , . . . , σ ( a i,l − )). Define a one-to-one correspondence14 : F lmq → R l , ( a , , a , , . . . , a ,l − , a , , a , , . . . , a ,l − , . . . , a m − , , a m − , , . . . , a m − ,l − ) a ( x ) = ( a ( x ) , a ( x ) , . . . , a l − ( x )) , where a j ( x ) = P m − i =0 a i,j x i for j = 0 , , . . . , l −
1. Then a skew QC code C of length lm with index l defined as in Corollary 3.4 is equivalent to a linear code of length lm ,which is invariant under the map T σ,l .Let v = ( v , , v , , . . . , v ,l − , v , , v , , . . . , v ,l − , . . . , v m − , , v m − , , . . . , v m − ,l − ) ∈ F mlq . Let { , ξ, ξ , . . . , ξ l − } be a basis of F q l over F q . Define an isomorphism between F mlq and F mq l , for i = 0 , , . . . , m −
1, associating each l -tuple ( v i, , v i, , . . . , v i,l − ) withthe element v i ∈ F q l where v i = v i, + v i, ξ + · · · + v i,l − ξ l − . Then every elementin F mlq is a one-to-one correspondence with an element in F mq l . The operator T σ,l on( v , , v , , . . . , v ,l − , v , , v , , . . . , v ,l − , . . . , v m − , , v m − , , . . . , v m − ,l − ) ∈ F mlq cor-responds to the element ( σ ( v m − ) , σ ( v ) , . . . , σ ( v m − )) ∈ F mq l under the above iso-morphism. The vector v ∈ F mlq can be associated with the polynomial v ( x ) = v + v x + · · · + v m − x m − ∈ e R = F q l [ x, σ ]. Clearly, there is an R/ ( x m − F mlq and e R [ x ] / ( x m −
1) that is defined by φ ( v ) = v ( x ). It followsthat there is a one-to-one correspondence between the left R/ ( x m − e R/ ( x m −
1) and the skew QC code of length ml with index l over F q . In addi-tion, a skew QC code of length ml with index l over F q can also be regarded as an R -submodule of e R/ ( x m −
1) because of the equivalence of F mlq and e R/ ( x m − C be a skew QC code of length ml with index l over F q , and generated bythe elements v ( x ) , v ( x ) , . . . , v ρ ( x ) ∈ e R/ ( x m −
1) as a left R/ ( x m − e R [ x ] / ( x m − C = { a ( x ) v ( x ) + a ( x ) v ( x ) + · · · + a ρ ( x ) v ρ ( x ) | a i ( x ) ∈ R/ ( x m − , i = 1 , , . . . , ρ } . As discussed above, C is also an R -submodule of e R/ ( x m − R -submodule of e R/ ( x m − C is generated by the following set { v ( x ) , xv ( x ) , . . . , x m − v ( x ) , v ( x ) , xv ( x ) , . . . , x m − v ( x ) , . . . , v ρ ( x ) , xv ρ ( x ) , . . . ,x m − v ρ ( x ) } .Since R/ ( x m −
1) is a subring of e R [ x ] / ( x m −
1) and C is a left R/ ( x m − e R/ ( x m − C is in particular a left submodule of an e R/ ( x m − e R/ ( x m − e C of length m over e R . Therefore, d H ( C ) ≥ d H ( e C ), where d H ( C ) and d H ( e C ) are the minimum Hamming distance of C and e C , respectively. Lally [16, Theorem 5] has obtained another lower bound on theminimum Hamming distance of the QC code over finite fields. In the following, wegeneralized these results to skew QC codes. Theorem 5.1
Let C be a ρ -generator skew QC code of length ml with index l over F q and generated by the set { v i ( x ) = e v i, + e v i, x + · · · + e v i,m − x m − , i = 1 , , . . . , ρ } ⊆ R/ ( x m − . Then C has lower bound on minimum Hamming distance given by d H ( C ) ≥ d H ( e C ) d H ( B ) , where e C is a skew cyclic code of length m over e R with generator polynomial gcld( v ( x ) ,v ( x ) , . . . , v ρ ( x ) , x m − and B is a skew linear code of length l generated by {V i,j , i =1 , , . . . , ρ, j = 0 , , . . . , m − } ⊆ F lq where each V i,j is the vector corresponding to thecoefficients e v i,j ∈ F q l with respect to a F q -basis { , ξ, . . . , ξ l − } . ✷ Define the Euclidean inner product of u, v ∈ F lmq by u · v = m − X i =0 l − X j =0 u i,j v i,j . Let C be a skew QC code of length lm with index l , u ∈ C and v ∈ C ⊥ . Since σ m = 1, we have u · T σ,l ( v ) = P m − i =0 u i · σ ( v i + m − ) = P m − i =0 σ ( σ m − ( u i ) · v i + m − ) = σ ( T m − σ,l ( u ) · v ) = σ (0) = 0, where i + m − m . Hence T σ,l ( v ) ∈ C ⊥ ,which implies that the dual code of skew QC code C is also a skew QC code of thesame index.We define a conjugation map − on R such that ax i = σ − i x m − i , for ax i ∈ R . On R l , we define the Hermitian inner product of a ( x ) = ( a ( x ) , a ( x ) , . . . , a l − ( x )) and b ( x ) = ( b ( x ) , b ( x ) , . . . , b l − ( x )) ∈ R l by h a ( x ) , b ( x ) i = l − X i =0 a ( x ) · b i ( x ) . By generalizing Proposition 3.2 of [17], we get
Proposition 5.2
Let u, v ∈ F lmq and u ( x ) and v ( x ) be their polynomial representationsin R l , respectively. Then T kσ,l ( u ) · v = 0 for all ≤ k ≤ m − if and only if h u ( x ) , v ( x ) i = 0 . ✷ Let C be a skew QC code of length lm with index l over F q . Then, by Theorem5.1, C ⊥ = { v ( x ) ∈ R l | h c ( x ) , v ( x ) i = 0 , ∀ c ( x ) ∈ C } . Furthermore, by Corollary 3.4 (iii), we have C ⊥ = L si =1 C ⊥ i .In [19], some results for ρ -generator QC codes and their duals over finite fields aregiven. These results can also be generalized to skew ρ -generator QC codes over finitefields. By generalizing Corollary 6.3, Corollary 6.4 in [19] and Theorem 3.5 in thispaper, we get the following result. Theorem 5.3
Let C be a ρ -generator skew QC code of length lm with index l over F q . Let C = L si =1 C i , where each C i , i = 1 , , . . . , s , is with dimension k i . Then i) C is a K -generator skew QC code and C ⊥ is an ( l − K ′ ) -generator skew QC code,where K = max ≤ i ≤ s k i and K ′ = min ≤ i ≤ s k i .(ii) Let l ≥ . If C ⊥ is also an ρ -generator skew QC code, then min ≤ i ≤ s k i = l − ρ and l ≤ ρ .(iii) If C is a self-dual ρ -generator skew QC code, then l is even and l ≤ ρ . ✷ For a 1-generator skew QC code of length lm with index l and the canonicaldecomposition C = L si =1 C i , C ⊥ is also a 1-generator skew QC code if and only if l = 2 and dim( C i ) = 1 for each i = 1 , , . . . , s . The structural properties of skew cyclic codes and skew GQC codes over finite fieldsare studied. Using the factorization theory of ideals, we give the Chinese RemainderTheorem in the skew polynomial ring F q [ x, σ ], which leads to a canonical decomposi-tion of skew GQC codes. Moreover, we give some characteristics of ρ -generator skewGQC codes. For 1-generator skew GQC codes, we give their parity-check polynomialsand dimensions. A lower bound on the minimum Hamming distance of 1-generatorskew GQC codes is given. These special codes may lead to some good linear codesover finite fields. Finally, skew QC codes are also discussed in details.In this paper, we restrict on the condition that the order of σ divides each m i , i = 1 , , . . . , l . If we remove this condition, then the polynomial x m − R/ ( x m −
1) is not a ring anymore. In thiscase, the cyclic code in R/ ( x m −
1) will not be an ideal. It is just a left R -submodule,and we call it a module skew cyclic code . A GQC code in R is also a left R -submoduleof R , and we call it a module skew GQC code . Most of our results on skew cyclic codesand skew GQC codes in this paper depend on the fact that x m − R . Since in the module skew case this is not ture anymore, some results statedin this paper cannot be held. Therefore, the structural properties of module skewcyclic and module skew GQC codes are also interesting open problems for furtherconsideration. Another interesting open problem is to find some new or good linearcodes over finite fields from skew GQC codes. Acknowledgments
This research is supported by the National Key Basic ResearchProgram of China (973 Program Grant No. 2013CB834204), the National NaturalScience Foundation of China (Nos. 61171082, 60872025, 10990011).
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