Note on quasi-twisted codes and an application

J. Appl. Math. Comput. DOI 10.1007/s12190-014-0787-0 ORIGINAL RESEARCH

Note on quasi-twisted codes and an application Jian Gao · Fang-Wei Fu

Received: 29 March 2014 © Korean Society for Computational and Applied Mathematics 2014

Abstract Recently, Jia proposed the decompositions and trace representations of quasi-twisted (QT) codes over finite fields (Finite Fields Appl 18:237–257, 2012). The present paper can be viewed as a complementary part of Jia’s work. We investigate some other useful properties of λ-QT codes over finite fields, including the lower Hamming distance bounds, enumerations and searching algorithm for generators. As an interesting application of λ-QT codes over finite fields, we study λ-QT codes over the finite non-chain ring Fq + vFq briefly. Keywords Constacyclic codes · Quasi-twisted codes · Minimum Hamming distance · Finite non-chain ring Mathematics Subject Classification

94B05 · 94B15

1 Introduction Quasi-twisted (QT) codes form an important class of linear codes. They can be viewed as the generalization of quasi-cyclic (QC) codes and constacyclic codes. The motivations of researching QT codes includes that they have good algebra structures and they can produce many record breakers in short lengths [1,4,7,8]. The paper [10], which is an interesting generalization of the work in [11], investigates some structural properties of QT codes over finite fields. The author uses the Chinese reminder theorem to decompose a QT code to a direct sum of component codes. Furthermore, the

J. Gao (B) · F.-W. Fu Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, China e-mail: [email protected] F.-W. Fu e-mail: [email protected]

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author also uses the generalized discrete Fourier transform to give an inverse formula to construct QT codes at the end of [10]. Codes over finite rings have been studied since the early 1970s. There are a lot of works on codes over finite rings after the discovery that certain good nonlinear binary codes can be constructed from cyclic codes over Z4 via the Gray map [9]. Since then, many researchers paid more and more attentions to study the codes over finite rings. In these studies, the group rings associated with codes are finite chain rings, and as some special cases of QT codes, QC codes and constacyclic codes have been characterized in several papers [2,3,5,6]. Recently, Zhu et al. [13,14] considered linear codes over the finite non-chain ring Fq +vFq . In [13], they study the cyclic codes over F2 +vF2 . It has shown that cyclic codes over this ring are principally generated. In the subsequent paper [14], they investigate a class of constacyclic codes over F p + vF p . In that paper, the authors prove that the image of a (1 − 2v)-constacyclic code of length n over F p + vF p under the Gray map is a cyclic code of length 2n over F p . Furthermore, they also assert that (1 − 2v)-constacyclic codes over F p + vF p are also principally generated. In this paper, we mainly consider some structural properties of λ-QT codes. The paper is organized as follows. In Sect. 2, we review some results on constacyclic codes over finite fields, and give a slightly different trace representation of the λ-constacyclic code in Theorem 1. In Sect. 3, we investigate some structural properties of λ-QT codes over finite fields, which will be used in Sect. 4. In this section, we give a slightly different trace representation and a different lower bound on the minimum Hamming distance of the λ-QT code in Theorem 2. The enumeration and a searching method of the generator of all different 1-generator λ-QT codes with a fixed p.c.p. are also given in Theorem 3 and Lemma 2, respectively. In Sect. 4, as an interesting application of Sect. 3, we study the λ-QT codes over the ring Fq + vFq . The relationship between λ-QT codes over Fq + vFq and QT codes over finite fields is given in Lemma 3 and 4. The enumeration and a searching algorithm of the generator of all different 1-generator λ-QT codes with a fixed annihilator are also given in Theorem 4 and Algorithm 1, respectively. 2 Constacyclic codes over finite fields Let Fq be a finite field of q = p m elements and let Fq∗ = Fq \{0} denote the multiplicative group of units in Fq , where p is a prime number and m is a positive integer. The Fq [X ] is a ring formed by the polynomials over Fq . A linear code C of length n with dimension k over Fq is a k-dimensional subspace of the vector space Fqn . The element of C is called the codeword. Let λ ∈ Fq∗ . A linear code C of length n is said to be λ-constacyclic if and only if, for any codeword (c0 , c1 , . . . , cn−1 ) ∈ C , (λcn−1 , c0 , . . . , cn−2 ) is also a codeword of C . Denote Rn as the quotient ring Fq [X ]/X n − λ. Define a map as follows, ϕ : Fqn → Fq [X ]/X n − λ = Rn (c0 , c1 , . . . , cn−1 ) → c0 + c1 X + . . . + cn−1 X n−1 .

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One can check that ϕ is an Fq -module isomorphism from Fqn to Rn . Moreover, C is a λ-constacyclic code of length n if and only if ϕ(C ) is an ideal of Rn . In this paper, we identify the λ-constacyclic code C with the ideal of Rn . Similar to the cyclic code over finite fields, let C be a λ-constacyclic code of length n over Fq . Then there exists a monic divisor g(X ) of X n − λ with least degree to n generate C . The g(X ) is called the generator polynomial of C . The polynomial Xg(X−λ) is called the annihilator polynomial of C . Let n be not divisible by the prime number p. Then the polynomial X n − λ has no multiple roots in Fq . Assume that λ does not have the nth root of unity. Then the polynomial X n − λ has roots β, βξ, . . . , βξ n−1 in some Galois extension field of Fq , where β is an nth root of λ and ξ is a primitive nth root of unity. Let s be the order of q modulo n, i.e s = min{k|q k ≡ 1(modn)}. Then ξ lies in Fq s . Let μ be the order of λ in the multiplicative group Fq∗ . Since β n = λ, it follows that β nμ = 1. Therefore β is a primitive nμth root of unity. Hence β lies in Fq l , where l is the order of q modulo nμ. It is well known that q l − 1 ≡ 0(modnμ) so q l − 1 ≡ 0(modn), which implies that s|l. Consequently, Fq s ⊆ Fq l implying that Fq l contains both β and ξ . Let ζ be a primitive element of Fq l . Then β and ξ can be expressed as β = ζ w , ξ = ζ wμ for some positive integer w. Therefore we have Xn − λ =

n−1  t=0



   n−1 X − β 1+tμ . X − βξ t =

(1)

t=0

Each irreducible factor of X n − λ corresponds to a q-cyclotomic coset modulo nμ. Example 1 Consider the polynomial X 6 − 3 over F5 . The order of the element 3 in F∗5 is 4 and there is no 6th root of 3 in F5 . According to the discussion above, we have X6 − 3 =

5  

     X − β 1+4t = X 2 + 3X + 3 X 2 + 2X + 3 X 2 + 3 ,

(2)

t=0

where β is a primitive 24th root of unity. The powers of β that appear in this factorization are 1, 5, 9, 13, 17, 21, and these are precisely union of three 5-cyclotomic cosets modulo 24, {1, 5}, {9, 21}, {13, 17}. In this paper, we assume n to be a positive integer not divisible by the prime number p. Then X n − λ has a unique decomposition as a product of some monic irreducible pairwise coprime polynomials over Fq . Let f (X ) be a monic factor of the polynomial n −λ f (X ) = Xf (X X n − λ. Denote  ). Proposition 1 [12] Let X n − λ = f 1 (X ) f 2 (X ) . . . fr (X ), where each f i (X ), i = 1, 2, . . . , r , is a irreducible polynomial over Fq . Denote Ri = Fq [X ]/ f i (X ). n −λ Let  f i (X ) = Xfi (X ) . Then there exist polynomials ai (X ), bi (X ) ∈ Rn such that ai (X ) f i (X ) + bi (X )  f i (X ) = 1. Assume that θi = bi (X )  f i (X ), then (i) θ1 , θ2 , . . . , θr are mutually orthogonal non-zero idempotents of Ri ;

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(ii) 1 = θ1 + θ2 + . . . + θr in Rn ; (iii) Let Rn θi = θi  be the principal ideal of Rn generated by θi . Then θi is the f i (X ) + X n − λ; identity of Rn θi and Rn θi =   (iv)

Rn = ⊕ri=1 Ri θi ;

(v) The map Fq [X ]/ f i (X ) → Rn θi defined by g(X ) +  f i (X ) → (g(X ) +  f i (X ))θi is a well-defined isomorphism of finite fields; (vi)

Rn  ⊕ri=1 Fq [X ]/ f i (X ).

Let X n − λ = f 1 (X ) f 2 (X ) . . . fr (X ), where each f i (X ), i = 1, 2, . . . , r , is a monic irreducible polynomial over Fq . Suppose that β is the nth root of λ and ξ is the primitive nth root of unity. Let Fq l be the smallest Galois extension field of Fq containing β and ξ . Then X n − λ = (X − β)(X − βξ ) . . . (X − βξ n−1 ). Define a map f as follows t f : Fq l [X ]/X n − λ → ⊕n−1 t=0 Fq l [X ]/X − βξ 

c(X ) = c0 + c1 X + . . . + cn−1 X n−1 → (c(β), c(βξ ), . . . , c(βξ n−1 )). Then f is a well-defined Fq l [X ]-module isomorphism. n−1 n−t . The polynomial A(Z ) is called Let At = c(βξ t ) and A(Z ) = t=0 At Z Mattsom-Solomon polynomial associated with c(X ). Clearly, ⎛ 

  ⎜ ⎜ A0 , A1 , . . . , An−1 = c0 , c1 , . . . , cn−1 ⎜ ⎝

1 β .. .

1 βξ .. .

... ... .. .

1 βξ n−1 .. .

⎞ ⎟ ⎟ ⎟. ⎠

β n−1 (βξ )n−1 . . . (βξ (n−1) )n−1 (3) For this reason, A(Z ) is sometimes called the discrete Fourier transform of c(X ). The inverse transform is given by ct =

n−1 1 Ak β −t ξ −tk , n

t = 0, 1, . . . , n − 1.

(4)

k=0

Since (β q−1 )n = λq−1 = 1 for λ ∈ Fq∗ , β q−1 is an nth root of unity. Then β q−1 can be expressed as a power of the primitive nth root of unity ξ , say β q−1 = ξ v , 0 ≤ v ≤ n − 1.

(5)

 k tk Let c(X ) = c0 + c1 X + . . . + cn−1 X n−1 ∈ Rn and At = c(βξ t ) = n−1 k=0 ck β ξ . Then n−1 n−1   q ck β kq ξ tkq = ck β k ξ k(tq+v) = Atq+v . (6) At = cq (βξ t ) = k=0

123

k=0

Note on quasi-twisted codes

Let m(X ) be a monic irreducible polynomial with degree  over Fq . Then the ring Fq [X ]/m(X ) is a finite field, which is the -th Galois extension of Fq . We denote Fq [X ]/m(X ) by Fq  . Assume that φ is a map from Fq  to Fq  defined by φ(a) = a q for any a ∈ Fq  . One can check that φ is a well-defined automorphism of Fq  . Moreover, the order of φ is  and φ generates the Galois group of Fq  over Fq . Define a trace function on Fq  down to Fq as follows T rFq  /Fq (a) = a + φ(a) + . . . + φ −1 (a), a ∈ Fq  .

(7)

Let X n − λ = f 1 (X ) f 2 (X ) . . . fr (X ), where f i (X ), i = 1, 2, . . . , r , is a monic irreducible polynomial with degree i over Fq . By Eq. (1), we have that there is a one-to-one correspondence between factors of X n − λ and the q-cyclotomic cosets Znμ . Denote by Ui (1 ≤ i ≤ r ) the q-cyclomotic coset correspondence to f i (X ). Let Fq i be the i -th Galois extension of Fq corresponding to the monic irreducible polynomial f i (X ), i.e. Fq i = Fq [X ]/ f i (X ). Then for a fixed 1 + u i μ ∈ Ui , u i ∈ {0, 1, . . . , n − 1}, by Eq. (4) we have ct =

r   1 T rFq i /Fq Ai β −t ξ −tu i , t = 0, 1, . . . , n − 1. n

(8)

i=1

Sometimes the Eq. (8) is called the trace representation of the λ-constacyclic codes over Fq . Let C be a λ-constacyclic code of length n over Fq . Then the Euclidean dual code C ⊥ of C is a λ−1 -constacyclic code. Let the roots of the polynomial X n − λ form the set N , and let the roots of the generator polynomial of C form the set O. Then the roots of the dual code C ⊥ form the set (N \O)−1 . In the following, we give a slightly different trace representation of the λ-constacyclic code. Theorem 1 Let C be a λ-constacyclic code over Fq . Suppose that there are nonnegative integers i 1 , i 2 , . . . , i k such that μi 1 − 1, μi 2 − 1, . . . , μi k − 1 are in different q-cyclotomic cosets of Znμ . Let β −1 ξ i1 , β −1 ξ i2 , . . . , β −1 ξ ik be roots of the polynomial  m(X ) = kj=1 M j (X ), where m(X ) is the generator polynomial of C ⊥ and each M j (X ), j = 1, 2, . . . , k, is the minimal polynomial of the element β −1 ξ i j over Fq . Then for any codeword c(X ) = c0 + c1 X + . . . + cn−1 X n−1 ∈ C , we have ct =

k 

T rFq l /Fq a j (β −1 ξ i j )t , t = 0, 1, . . . , n − 1,

(9)

j=1

where a j ∈ Fq l and Fq l is the smallest Galois extension field of Fq containing the λ’s nth root β and the primitive nth root of unity ξ . Proof Let k = 1. Consider the following set C1 =

    t c0 , c1 , . . . , cn−1 |ct = T rFq l /Fq a j β −1 ξ i1 , t = 0, 1, . . . , n − 1 . (10)

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Obviously, C1 is a linear code of length n over Fq . If ca j (X ) =

n−1 

T rFq l /Fq a j (β −1 ξ i1 )t X t ,

(11)

t=0

then ca j βξ −i1 (X ) = ca j (X )X in Rn , which implies that C1 is a λ-constacyclic code of length n over Fq . On the other hand, the λ−1 -constacyclic code M1 (X ) is contained in the dual code ⊥ C1 implying C1 contained in the λ-constacyclic code M1 (X )⊥ . Since M1 (X )⊥ is irreducible, it follows that C1 = M1 (X )⊥ = (C ⊥ )⊥ = C . For k ≥ 2, using the Proposition 1 that any λ-constacyclic code is the direct sum of some irreducible λ-constacyclic codes, it follows the result.   3 Quasi-twisted codes over finite fields Let C be a linear code of length N over Fq . Let λ ∈ Fq∗ and let be a positive integer. For each codeword (c0 , c1 , . . . , c N −1 ) in C , if there exists a smallest positive integer

such that the vector (λc N − , λc N − +1 , . . . , λc N −1 , c0 , . . . , c N − −1 ) also belongs to C , then the linear code C is called a λ-QT code of length N with index over Fq . Clearly, is a divisor of N . Let N = n. Define an Fq -module isomorphism from Fq n to Rn as follows ρ : Fq n → Rn

(c0,0 , . . . , c0, −1 , c1,0 , . . . , c1, −1 , . . . , cn−1,0 , . . . , cn−1, −1 ) → (c0 (X ), c1 (X ), . . . , c −1 (X )),

(12)

 t where ci (X ) = n−1 t=0 ct,i X , i = 0, 1, . . . , − 1. Then the λ-QT code C is equivalent to saying that, for any (c0 (X ), c1 (X ), . . . , c −1 (X )) ∈ ρ(C ), (X c0 (X ), X c1 (X ), . . . , X c −1 ) ∈ ρ(C ). Therefore C is a λ-QT code of length N with index over Fq if and only if ρ(C ) is an Rn -submodule of Rn . 3.1 Trace representations Let X n − λ = f 1 (X ) f 2 (X ) . . . fr (X ), where each f i (X ), i = 1, 2, . . . , r , is the monic irreducible polynomial with degree i over Fq . Then from Proposition 1, we have Rn  ⊕ri=1 Fq [X ]/ f i (X ). Denote Ri = Fq [X ]/ f i (X ) = Fq (βξ u i ), where β is an nth root of λ and ξ is a primitive nth root of unity. Then we have Rn  ⊕ri=1 Ri .

(13)

This implies that the λ-QT code C can be viewed as an (⊕ri=1 Ri )-submodule of ⊕ri=1 Ri and C = ⊕ri=1 Ci , where Ci is a linear code of length over Ri . These length linear codes over some Galois extension field of Fq are

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called the constituents of C . Let C be a K-generator λ-QT code generated as an Fq [X ]-submodule by the set {G1 (X ), G2 (X ), . . . , GK (X )}, where G j (X ) = (g j,0 (X ), g j,1 (X ), . . . , g j, −1 (X )) ∈ Rn for all j = 1, 2, . . . , K. Then Ci is spanned by the set {g j,0 (βξ u i ), g j,1 (βξ u i ), . . . , g j, −1 (βξ u i )}, i = 1, 2, . . . , r and j = 1, 2, . . . , K. In the following, we give the trace representations of λ-QT codes which is an interesting generalization of the trace representations of QC codes in [11]. Here we omit the proof. Proposition 2 Let X n − λ = f 1 (X ) f 2 (X ) . . . fr (X ), where each f i (X ), i = 1, 2, . . . , r , is the monic irreducible polynomial with degree i over Fq . Denote Ri = Fq [X ]/ f i (X ). Let Ui be the q-cyclotomic coset modulo nμ corresponding to f i (X ). Fix a representatives 1 + u i μ ∈ Ui from each q-cyclotomic coset. Let for all i = 1, 2, . . . , r . For  ci ∈ Ci and each Ci be a linear code of length over Ri  ci β −t ξ −tu i ). Then the code t = 0, 1, . . . , n − 1, let the vector ct = ri=1 T rFq i /Fq ( C =

   ci ∈ Ci c0 , c1 , . . . , cn−1 | 

(14)

is a λ-QT code of length n with index over Fq . Conversely, every λ-QT code of length n with index over Fq can be obtained through this construction. Let Fq ⊂  F ⊂ Fq l be a Galois extension. If ω ∈ Fq l such that T rF l / (ω) = 1, q F then for any α ∈  F we have T rFq l /Fq (αω) = T r F/Fq (T rFq l / F (αω)) = T r F/Fq (α).

(15)

Theorem 2 Let C be a λ-QT code as above. Let ω1 , ω2 , . . . , ωr ∈ Fq l be elements with T rFq l /Fq i (ωi ) = 1 for all i = 1, 2, . . . , r . Then (i) Any codeword (c0 , c1 , . . . , cn−1 ) ∈ C is of the form ct =

r 

T rFq l /Fq ( ci ωi β −t ξ −tu i ),

(16)

i=1

for all t = 0, 1, . . . , n − 1. (ii) The columns of any codeword c ∈ C lie in a λ-constacyclic code, which dual code has the roots β −1 ξ −u 1 , β −1 ξ −u 2 , . . . , β −1 ξ −ur , where β is the nth root of λ and ξ is the primitive nth root of unity. (iii) For any column  cν =

 r  i=1 r 

r      T rFq l /Fq  T rFq l /Fq  ci,ν ωi , ci,ν ωi β −1 ξ −u i , . . . , i=1

   −(n−1) −(n−1)u i ci,ν ωi β , T rFq l /Fq  ξ

(17)

i=1

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where  ci = ( ci,1 , ci,2 , . . . , ci, ) ∈ Fq i and ν = 1, 2, . . . , , we have  cν = 0 if c2,ν = . . . =  cr,ν = 0. and only if  c1,ν =  (iv) Assume that the constituents of C satisfy d(C1 ) ≥ d(C2 ) ≥ . . . ≥ d(Cr ).

(18)

For any nonempty subset {i 1 , i 2 , . . . , i d } ⊆ {1, 2, . . . , r } with 1 ≤ i 1 < i 2 < . . . < i d ≤ r , let Di1 ,i2 ,...,id be the λ-constacyclic code of length n over Fq , which dual code Di⊥1 ,i2 ,...,id has roots β −1 ξ −u 1 , β −1 ξ −u 2 , . . . , β −1 ξ −u d . Denote di1 ,i2 ,...,id = d(Ci1 )d(Di1 ) if d = 1 or di1 ,i2 ,...,id = (d(Ci1 ) − d(Ci2 ))d(Di1 ) + (d(Ci2 ) − d(Ci3 ))d(Di1 ,i2 ) + . . . + d(Cid )d(Di1 ,i2 ,...,id )

if d ≥ 2.

Then the minimum Hamming distance of the λ-QT code C satisfies d(C ) ≥ min{dr , dr −1,r , . . . , d1,2,...,r }.

(19)

Proof (i) By Eq. (16), r 

r     −t −tu  ci ωi β −t ξ −tu i = ci β ξ i T rFq l /Fq  T rFq i /Fq 

i=1

i=1

= ct

for t = 0, 1, . . . , n − 1. The second equality in the above equation holds following from the Proposition 2. (ii) For any column  cν =

 r 

r      ci,ν ωi , ci,ν ωi β −1 ξ −u i , . . . , T rFq l /Fq  T rFq l /Fq 

i=1 r 

i=1

   −(n−1) −(n−1)u i ci,ν ωi β , T rFq l /Fq  ξ

i=1

the t-th component  ct =

r 

  ci,v ωi β −t ξ −tu i , T rFq l /Fq 

i=1

where t = 0, 1, . . . , n−1 and ν = 1, 2, . . . , . Since ci,ν ωi ∈ Fq l , from Theorem 1, we have cν lies in a λ-constacyclic code of length n over Fq , which dual code has roots β −1 ξ −u 1 , β −1 ξ −u 2 , . . . , β −1 ξ −ur .

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(iii)  cv = 0 if and only if each t-th component is zero for all t = 0, 1, . . . , n − 1 if and only if r 

  ci,ν ωi β −t ξ −tu i = 0 T rFq l /Fq 

i=1

if and only if   ci,ν ωi β −t ξ −tu i = 0 T rFq l /Fq  for all i = 1, 2, . . . , r . (1) If 1 = 2 = . . . = r = l, then T rFq l /Fq ( ci,ν ωi β −t ξ −tu i ) = 0 for all i = 1, 2, . . . , r if and only if  c1,ν =  c2,ν = . . . =  cr,ν = 0. (2) If there exists a set { j1 , j2 , . . . , jk } ⊆ {1, 2, . . . , r } such that  jz < m for all jz ∈ { j1 , j2 , . . . , jk } and h = m for all h ∈ {1, 2, . . . , r }\{ j1 , j2 , . . . , jk }, ci,ν ωi β −t ξ −tu i ) = 0 for all i = 1, 2, . . . , r if and only if then T rFq l /Fq ( c jz ,ν ω jz ) =  c jz ,ν T rFq l /F  jz (ω jz ) =  c jz ,ν = 0.  c p,ν = 0 and T rFq l /F  jz ( q q Therefore, we have proved cν = 0 if and only if  c1,ν =  c2,ν = . . . =  cr,ν = 0. (iv) By (i), (ii) and (iii) in this theorem, we can prove (iv). The proof process is similar to that of Theorem 4.8 in [8], and here we omit it.   Example 2 Consider a 3-QT code C of length 12 with index 2 generated by G(X ) = (X 3 + 3X, X 3 + X 2 + 3X + 3) over F5 . Since (F5 [X ]/X 6 − 3)2  (F5 [X ]/X 2 + 3X + 3)2 ⊕(F5 [X ]/X 2 + 2X + 3)2 ⊕(F5 [X ]/X 2 + 3)2 , we have 3 Ci , C = ⊕i=1

where C1 is a linear code of length 2 generated by (4X + 4, X + 4) over F5 [X ]/X 2 + 3X +3, C2 is a linear code of length 2 generated by (4X +1, 2X +1) over F5 [X ]/X 2 + 2X + 3 and C3 is a zero-code. From (iv) in Theorem 2, we have d(C ) ≥ min{10, 4} = 4. By the help of Magma system, we have C is a [12, 4, 6] 3-QT code over F5 actually. 3.2 1-Generator quasi-twisted codes Let C be a λ-QT code over Fq . If C is generated by a set {G(X )}, where G(X ) = (g0 (X ), g1 (X ), . . . , g −1 (X )) ∈ Rn ,

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then C is called a 1-generator λ-QT code. Definition 1 Let C be a 1-generator λ-QT code generated by G(X ) = (g0 (X ), g1 (X ), . . . , g −1 (X )) ∈ Rn . Then the monic polynomial g(X ) = gcd(G(X ), X n − λ) = gcd(g0 (X ), g1 (X ), . . . , g −1 (X ), X n − λ) is called the generator polynomial (g.p.) of C . Moreover, the monic polynomial h(X ) with least degree satisfying h(X )G(X ) = 0 is called the parity-check polynomial (p.c.p.) of C . Both these polynomials are unique. If g(X ) and h(X ) are, respectively, the g.p. and p.c.p. of the λ-QT code C , then X n − λ = g(X )h(X ) and the dimension of C is dimC = deg(h(X )). Let gcd(n, q) = 1 and gcd(l, ) = 1. Then X n − λ has the same factorization in Fq [X ] and Fq [X ]. Denote Sn be Fq [X ]/X n − λ. Map Rn into Sn via the natural mapping σ : Rn → Sn (g0 (X ), g1 (X ), . . . , g −1 (X )) →

−1 

gε (X )αε ,

(20)

ε=0

where the set {α0 , α1 , . . . , α −1 } forms an Fq -basis of Fq . It is well known that the ideal g0 (X ), g1 (X ), . . . , g −1 (X ) in Rn is generated by a monic polynomial gε (X )g(X ) for each ε = 0, 1, . . . , − 1, then G(X ) = g(X ). Assume that gε (X ) =   g (X )α ∈ g(X ) g(X ) −1 ε Sn , the ideal generated by g(X ) in Sn . ε=0 Lemma 1 Let G(X ) = (g0 (X ), g1 (X ), . . . , g −1 (X )) ∈ Rn be a generator of the  −1 n 1-generator λ-QT code C . Let G(X ) = ε=0 gε (X )αε and h(X ) = Xg(X−λ) . Then (i) For any G  (X ) ∈ Rn , G  (X ) is a generator of C if and only if there exists a polynomial u(X ) ∈ Rn such that gcd(u(X ), h(X )) = 1 and G  (X ) = u(X )G(X ). (ii) C has h(X ) as p.c.p. if and only if gcd(G(X ), h(X )) = 1. (iii) Let G = (g(X ) Sn )∗ , the group of units of the ideal g(X ) Sn . Then C has h(X ) as p.c.p. if and only if G(X ) ∈ G. Proof (i) Let G  (X ) = u(X )G(X ) and gcd(u(X ), h(X )) = 1. Then Rn G  (X ) ⊆ Rn G(X ). Since u(X ) and h(X ) are coprime, there exist polynomials a(X ), b(X ) ∈ Fq [X ] such that a(X )u(X ) + b(X )h(X ) = 1. It means that Rn G(X ) = Rn G(X )(a(X )u(X ) + b(X )h(X )) = u(X )Rn G  (X )

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Note on quasi-twisted codes

in Rn implying Rn G(X ) = Rn G  (X ). Conversely, if Rn G(X ) = Rn G  (X ), then there exist polynomials a(X ), u(X ) ∈ Fq [X ] such that G  (X ) = u(X )G(X ) and G(X ) = a(X )G  (X ). Therefore Rn G(X ) = a(X )u(X )G  (X ), which implies (1 − a(X )u(X ))Rn G(X ) = 0 in Rn . Thus h(X ) is a divisor of the polynomial 1 − a(X )u(X ). Therefore there exists a polynomial b(X ) ∈ Fq [X ] such that 1 − a(X )u(X ) = h(X )b(X ), which implies that u(X ) and h(X ) are coprime. (ii) Since gcd(g0 (X ), g1 (X ), . . . , g −1 (X ), X n − λ) = gcd(G(X ), X n − λ), by (i) in the above, we have C is a 1-generator λ-QT code generated by G(X ) if and only if gcd(G(X ), X n −λ) = g(X ). Note that G(X ) ∈ g(X ) Sn , which implies that g(X ) divides gcd(G(X ), X n −λ). If G(X ) and h(X ) are coprime, then gcd(G(X ), X n − λ) = 1. Conversely, for any G(X ) ∈ g(X ) Sn , if gcd(G(X ), X n − λ) = g(X ), then G(X ) and h(X ) are coprime. (iii) Since gcd(l, ) = 1, it follows that X n −λ = g(X )h(X ) in Sn . Therefore there exist p(X ), q(X ) ∈ Sn such that p(X )g(X ) + q(X )h(X ) = 1. Let θ = p(X )g(X ) = 1 − q(X )h(X ). Then, by Proposition 1, θ is the identity of the subring g(X ) Sn . If G(X ) ∈ G, then there exists D(X ) ∈ G such that G(X )D(X ) = θ = 1 − q(X )h(X ), which implies that G(X ) and h(X ) are coprime. From (ii) in the above, we have C has h(X ) as p.c.p.. On the other hand if C has h(X ) as p.c.p., then G(X ) and h(X ) are coprime. Therefore there exist s(X ), t (X ) ∈ Sn such that s(X )G(X ) + t (X )h(X ) = 1. Multiplying both sides by θ = p(X )g(X ), we have θ = p(X )s(X )g(X )G(X ) implying G(X ) ∈ G.   In the following result, we investigate the enumeration of all different 1-generator λ-QT codes, which have h(X ) as p.c.p. Theorem 3 Let M be the set of the different 1-generator λ-QT codes of length n with index over Fq , which have h(X ) as p.c.p.. Let h(X ) = h 1 (X )h 2 (X ) . . . h m (X ), where each h j (X ) is an irreducible polynomial with degree e j over Fq , j = 1, 2, . . . , m. Then m  q e j −1 . |M| = qe j − 1 j=1

Proof Let G = (g(X ))∗ , the group of units of the ideal g(X ) of Rn . Let M be the set of the different 1-generator λ-QT codes of length n with index over Fq , which have h(X ) as p.c.p., where G(X ) = (g0 (X ), g1 (X ), . . . , g −1 (X )) ∈ Rn and X n − λ = g(X )h(X ). Define a map as follows η : G/G → M G(X )G → Rn G(X ),  −1 η is a well-defined map. For where G(X ) = ε=0 gε (X )αε . One can verify that  −1 any G 1 (X )G and G 2 (X )G in G/G , let G 1 (X ) = ε=0 g1,ε (X )αε and G 2 (X ) =  −1 ε=0 g2,ε (X )αε . Assume that G1 (X ) = (g1,0 (X ), g1,1 (X ), . . . , g1, −1 (X )) and G2 (X ) = (g2,0 (X ), g2,1 (X ), . . . , g2, −1 (X )). If Rn G1 (X ) = Rn G2 (X ), by Lemma 1, there exists u(X ) such that G1 (X ) = u(X )G2 (X ) and gcd(u(X ), h(X )) = 1. Since

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θ is the identity of the subring g(X ) Sn , it follows that G 1 (X ) = θ u(X )G2 (X ). Moreover θ and u(X ) are coprime with h(X ), respectively. Therefore θ u(X ) are coprime to h(X ). From Lemma 1, θ u(X ) ∈ G , which implies that G 1 (X )G = G 2 (X )G . Thus η is an injective map. On the other hand, for any Rn G(X ), by Lemma 1, G(X ) ∈ G. Moreover, η(G(X )G ) = Rn G(X ). Therefore η is a surjective. Let X n −λ = g(X )h(X ). Then, by Proposition 1, Fq [X ]/h(X ) and Fq [X ]/h(X ) are isomorphic to g(X ) and g(X ) Sn , respectively. If we assume that h(X ) = h 1 (X )h 2 (X ) . . . h m (X ), where each h j (X ) is an monic irreducible polynomial with degree e j over Fq and Fq , j = 1, 2, . . . , m, then Fq [X ]/h(X )  ⊕mj=1 Fq [X ]/h j (X ) and Fq [X ]/h(X )  ⊕mj=1 Fq [X ]/h j (X ). Therefore |G/G |  | ⊕mj=1 (Fq [X ]/h j (X ))∗ |/| ⊕mj=1 (Fq [X ]/h j (X ))∗ |. Thus |M| =

m  q e j − 1 . qe j − 1 j=1

  In the rest of this section, we discuss how to find the one and the only one generator for each 1-generator λ-QT code of length n with index over Fq , which has h(X ) as p.c.p. From the proof of Theorem 3, if we find the representation of the quotient group G/G , we can determine the generator of each 1-generator λ-QT code. By Proposition 1, we have G/G  (Fq [X ]/h(X ))∗ /(Fq [X ]/h(X ))∗

 ⊕mj=1 (Fq [X ]/h j (X ))∗ /(Fq [X ]/h j (X ))∗ ,

which implies that G(X ) =

m 

T j (X )θ j ,

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j=1

h j (X ) Sn and T j (X ) is the representation of where θ j is the identity of the subring  the quotient group (Fq [X ]/h j (X ))∗ /(Fq [X ]/h j (X ))∗ for each j = 1, 2, . . . , m.

e j

−1 Lemma 2 For each j = 1, 2, . . . , m, let the set D = {ζ a | a = 0, 1, . . . , qq e j −1 }, where ζ is a primitive element of Fq [X ]/h j (X ). Then D is the complete set of representation of the quotient group (Fq [X ]/h j (X ))∗ /(Fq [X ]/h j (X ))∗ .

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Note on quasi-twisted codes

e j

−1 Proof Clearly, |D| = qq e j −1 = |(Fq [X ]/h j (X ))∗ /(Fq [X ]/h j (X ))∗ |. It means that we only need to prove different elements in D belongs to different cosets of the quotient group. For any A = ζ a and B = ζ b of D, if A(Fq [X ]/h j (X ))∗ = ej B(Fq [X ]/h j (X ))∗ , then ζ a−b ∈ (Fq [X ]/h j (X ))∗ . Therefore ζ (a−b)(q −1) = 1,

which implies that q e j − 1 is a divisor of (a − b)(q e j − 1). Thus of a − b, which is a contradiction. It follows that a = b.

e

q j −1 e q j −1

is a divisor  

4 An application In this section, as an interesting application of Sect. 3, we describe some structural properties of λ-QT codes over the finite non-chain ring Fq + vFq , where q is a power of some prime p and v 2 = v. Let R = Fq + vFq = {a + bv| a, b ∈ Fq } with v 2 = v. Then (i) (ii) (iii) (iv)

R is a principal ideal ring and has ideals 0, v, 1 − v, 1; v and 1 − v are maximal ideals of R; R is a finite commutative ring, but not a chain ring; For any λ = α + βv ∈ R, λ is a unit if and only if α = 0 and α + β = 0.

Let λ = α + βv. Define a Gray map  from R to Fq2 as follows  : R → Fq2 c = a + bv → (b, a + b). Furthermore, for any element c ∈ R, the Gray weight of c is defined as WG (c) = W H (b) + W H (a + b), where W H () denote the Hamming weight of  over Fq . ∈ R n to be the rational Define a Gray weight of a vector c = (c0 , c1 , . . . , cn−1 )  n−1 sum of the Gray weight of its components, i.e. WG (c) = t=0 WG (ct ). For any n elements c1 , c2 ∈ R , the Gray distance is given by dG (c1 − c2 ) = WG (c1 − c2 ). A code C of length n over R is a subset of R n . C is a linear code if and only if C is an R-submodule of R n . The minimum Gray distance of C is the smallest nonzero Gray distance between all pairs of distinct codewords. The minimum Gray weight of C is the smallest nonzero Gray weight among all codewords. If C is a linear code, then the minimum Gray distance is the same as the minimum Gray weight. Similarly, the Gray map on R n can be defined as  : R n → Fq2n (c0 , c1 , . . . , cn−1 ) → (b0 , a0 + b0 , b1 , a1 + b1 , . . . , bn−1 , an−1 + bn−1 ), where ci = ai + bi v for i = 0, 1, . . . , n − 1. Then one can verify that the Gray map  is a distance preserving map from R n (Gray distance) to Fq2n (Hamming distance) and it is also Fq -linear.

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Let C be a linear code of length n over R. Define

and

  C1 = a ∈ Fqn | ∃b ∈ Fqn , va + (1 − v)b ∈ C

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  C2 = b ∈ Fqn | ∃a ∈ Fqn , va + (1 − v)b ∈ C .

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Clearly, C1 and C2 are linear codes of length n over Fq . Moreover, C can be uniquely expressed as C = vC1 ⊕ (1 − v)C2 . Lemma 3 Let C = vC1 ⊕ (1 − v)C2 be a linear code over R and λ = α + vβ be a unit of R. Then C is a λ-QT code of length n with index over R if and only if C1 and C2 are (α + β)-QT and α-QT codes of length n with index over Fq , respectively. Proof Let (r0 , r1 , . . . , rn −1 ) be an element of C1 and (q0 , q1 , . . . , qn −1 ) be an element of C2 . Assume that ct = vrt + (1 − v)qt for t = 0, 1, . . . , n − 1, then the vector (c0 , c1 , . . . , cn −1 ) belongs to C . Since C is a λ-QT code of length n with index

over R, it follows that (λc(n−1) , λc(n−1) +1 , . . . , λcn −1 , c0 , . . . , c(n−1) −1 ) ∈ C . Note that λc j = (α + βv)(vr j + (1 − v)q j ) = v(α + β)r j + (1 − v)αq j for j = (n − 1) , (n − 1) + 1, . . . , n − 1. Then (λc(n−1) , λc(n−1) +1 , . . . , λcn −1 , c0 , . . . , c(n−1) −1 ) = v((α + β)r(n−1) , (α + β)r(n−1) +1 , . . . , (α + β)rn −1 , r0 , . . . , r(n−1) −1 ) +(1 − v)(αq(n−1) , αq(n−1) +1 , . . . , αqn −1 , q0 , . . . , q(n−1) −1 ) ∈ C . Hence ((α + β)r(n−1) , (α + β)r(n−1) +1 , . . . , (α + β)rn −1 , r0 , . . . , r(n−1) −1 ) ∈ C1 and (αq(n−1) , αq(n−1) +1 , . . . , αqn −1 , q0 , . . . , q(n−1) −1 ) ∈ C2 , which implies that C1 and C2 are (α + β)-QT and α-QT codes of length n with index

over Fq , respectively. Conversely, suppose that C1 and C2 are (α + β)-QT and α-QT codes of length n

with index over Fq , respectively. Let (c0 , c1 , . . . , cn −1 ) ∈ C , where ct = vrt +(1− v)qt for t = 0, 1, . . . , n −1. Then (r0 , r1 , . . . , rn −1 ) ∈ C1 and (q0 , q1 , . . . , qn −1 ) ∈ C2 . Note that

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Note on quasi-twisted codes

(λc(n−1) , λc(n−1) −1 , . . . , λcn −1 , c0 , . . . , c(n−1) −1 ) = v((α + β)r(n−1) , (α + β)r(n−1) +1 , . . . , (α + β)rn −1 , r0 , . . . , r(n−1) −1 ) +(1 − v)(αq(n−1) , αq(n−1) +1 , . . . , αqn −1 , q0 , . . . , q(n−1) −1 ) ∈ vC1 ⊕ (1 − v)C2 = C . Hence C is a λ-QT code of length n with index over R.

 

Let C be a λ-QT code of length n with index over R. Denote Rn as R[X ]/X n − λ. Similar to the λ-QT code over Fq , the λ-QT code of length n with index over R can be viewed as an Rn -submodule of R n . Lemma 4 Let C be a λ-QT code of length n with index over R. Then C is a 1generator λ-QT code if and only if C1 and C2 are 1-generator (α + β)-QT and α-QT codes over Fq , respectively. Moreover, if G1 (X ), G2 (X ) ∈ Rn are the generators of C1 and C2 , respectively, then the generator of C is G(X ) = vG1 (X ) + (1 − v)G2 (X ). Proof Let C be a 1-generator λ-QT code of length n with index over R. Then, by Lemma 3, C1 and C2 are (α + β)-QT and α-QT codes of length n with index over Fq , respectively. From the definitions of C1 and C2 , one can check that C1 and C2 are all 1-generator. On the other hand, suppose that C1 and C2 are all 1-generator over Fq generated by G1 (X ) and G2 (X ), respectively. Then, by C = vC1 ⊕ (1 − v)C2 , we could assume that C is generated by the generating set {vG1 (X ), (1 − v)G(X )}. Let G(X ) = vG1 (X ) + (1 − v)G2 (X ). Then the 1-generator λ-QT code Rn G(X ) ⊆ C . Notice that vG1 (X ) = vG(X ) and (1 − v)G2 (X ) = (1 − v)G(X ), which implies that   C ⊆ Rn G(X ). Therefore C is a 1-generator λ-QT code generated by G(X ). Particularly, by Lemma 4, if = 1, i.e. C is a λ-constacyclic code over R, then C is generated by the polynomial v g1 (X ) + (1 − v) g2 (X ), where  g1 (X ) and  g2 (X ) are the generator polynomials of (α + β)-constacyclic code and α-constacyclic code over Fq , respectively. It means that the quotient ring R[X ]/X n − λ is principal. Let C be a 1-generator λ-QT code of length n with index over R generated by the set G(X ) = (g0 (X ), g1 (X ), . . . , g −1 (X )) ∈ R n . Define Ann(C ) = {a(X ) ∈ Rn | a(X )gi (X ) = 0, i = 0, 1, . . . , − 1}. Clearly, Ann(C ) is an ideal of Rn and is called the annihilator of C . Consider a map χ from Rn to C , which sends a(X ) ∈ Rn to a(X )G(X ) ∈ C . It is easy to verify that χ is a well-defined surjective R-module homomorphism and the kernel of χ is Ann(C ), which implies that Rn /Ann(C ) is isomorphic to C . Thus |C | = |Rn |/|Ann(C )|.

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Lemma 5 Let C = vC1 ⊕ (1 − v)C2 be a 1-generator λ-QT code of length n with index over R generated by G = (g0 (X ), g1 (X ), . . . , g −1 (X )) ∈ R n . Let h 1 (X ) and h 2 (X ) be the p.c.p. of C1 and C2 , respectively. Then the annihilator of C is Ann(C ) = vh 1 (X ) + (1 − v)h 2 (X ).

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Proof Let I = g0 (X ), g1 (X ), . . . , g −1 (X ) be an ideal of Rn and Ann(I ) be the annihilator of I . Then Ann(I ) = Ann(C ). Clearly, vh 1 (X )+(1−v)h 2 (X ) ∈ Ann(I ). On the other hand, since I is isomorphic to Rn /Ann(I ), it follows that |Ann(I )| = |Rn |/|I | = q 2n−degh 1 (X )−degh 2 (X ) = |vh 1 (X )+(1−v)h 2 (X )|. Therefore Ann(I ) =   Ann(C ) = vh 1 (X ) + (1 − v)h 2 (X ). Let δ and ϑ be the nth root of α + β and α, respectively. Let Fq l1 and Fq l2 be the smallest Galois extension fields of Fq , which contain α, ξ and ϑ, ξ , respectively. Suppose that gcd(l1 , ) = gcd(l2 , ) = 1, then by Theorem 3 and Lemma 4, we have the following results immediately. Theorem 4 Let M be the set of 1-generator λ-QT codes of length n with index and the annihilator vh 1 (X ) + (1 − v)h 2 (X ) over R. Let h 1 (X ) = f 1,1 (X ) f 1,2 (X ) . . . f 1,r (X ) and h 2 (X ) = f 2,1 (X ) f 2,2 (X ) . . . f 2,s (X ), where each f 1, j (X ) and f 2,k (X ), j = 1, 2, . . . , r and k = 1, 2, . . . , s, are monic irreducible polynomials with degree e1, j and e2,k over Fq , respectively. Then M=

r s  q e1, j − 1  q e2,k − 1 . q e1, j − 1 q e2,k − 1 j=1

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k=1

In the following, we give an algorithm to construct all distinct 1-generator λ-QT codes of length n with index and the annihilator vh 1 (X ) + (1 − v)h 2 (X ) over R. Algorithm 1 Let R = Fq + vFq and λ = α + βv, where v 2 = v. Let C = vC1 ⊕ (1 − v)C2 be a 1-generator λ-QT code of length n with index and annihilator vh 1 (X ) + (1 − v)h 2 (X ) over R, where h 1 (X ) and h 2 (X ) are p.c.p. of C 1 and C2 over Fq , respectively. Assume that h 1 (X ) = rj=1 f 1, j (X ) and h 2 (X ) = sk=1 f 2,k (X ), where each f 1, j (X ) and f 2,k (X ), j = 1, 2, . . . , r and k = 1, 2, . . . , s, are monic irreducible polynomials with degree e1, j and e2,k over Fq , respectively. Step 1 For j = 1, 2, . . . , r , calculate u 1, j (X ) and v1, j (X ) by Euclidean Algorithm f 1, j (X ) + v1, j (X ) f 1, j (X ) = 1. Let θ1, j (X ) = u 1, j (X )  f 1, j (X ) in such that u 1, j (X )  n Fq [X ]/X − λ. Similarly, for k = 1, 2, . . . , s, calculate u 2,k (X ) and v2,k (X ) and let θ2,k (X ) = u 2,k (X )  f 2,k (X ) in Fq [X ]/X n − λ. Step 2 For j = 1, 2, . . . , r and k = 1, 2, . . . , s, use the Gauss’s Algorithm to find out primitive elements ξ1, j of Fq [X ]/ f 1, j (X ) and ξ2,k of Fq [X ]/ f 2,k (X ), respectively. Then get the complete sets of the representative of the quotient groups (Fq [X ]/ f 1, j (X ))∗ /(Fq [X ]/ f 1, j (X ))∗ and (Fq [X ]/ f 2,k (X ))∗ /(Fq [X ]/ f 2,k (X ))∗ , respectively. They are   q e1, j − 1 w D1, j = ξ1, 1,j j | w1, j = 0, 1, . . . , e1, j q −1

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Note on quasi-twisted codes

and

  q e2,k − 1 w . D2,k = ξ2,k2,k | w2,k = 0, 1, . . . , e q 2,k − 1

Step 3 For any B1, j ∈ D1, j and B2,k ∈ D2,k , calculate

G 1 (X ) =

r 

B1, j θ1, j (X ) (modX n − (α + β))

j=1

and G 2 (X ) =

s 

B2,k θ2,k (X ) (modX n − α).

k=1

Step 4 Let G 1 (X ) =

−1 

g1,i (X )αi = G1 (X )(α0 , α1 , . . . , α −1 )T

i=0

and G 2 (X ) =

−1 

g2,i (X )αi = G2 (X )(α0 , α1 , . . . , α −1 )T ,

i=0

where (α0 , α1 , . . . , α −1 ) is an Fq -basis of Fq . Calculate G(X ) = vG1 (X ) + (1 − v)G2 (X ). Then all distinct 1-generator λ-QT codes of length n with index and annihilator vh 1 (X ) + (1 − v)h 2 (X ) over R can be constructed by C = Rn G(X ). At the end, we give an example to illustrate the main work in this paper. Example 3 Consider the 1-generator (4 + 4v)-QT codes of length 18 with index 3 and annihilator v(X 2 + 2X + 3) + (1 − v)(X + 3) over R = F5 + vF5 . From Theorem 4, we have that there are

|M| =

53×2 − 1 53 − 1 × = 20181 52 − 1 5−1

different 1-generator (4 + 4v)-QT codes of length 18 with index 3 and annihilator v(X 2 + 2X + 3) + (1 − v)(X + 3) over R. We can use Algorithm 1 to get the one and the only one generator for each of these 20181 different (4 + 4v)-QT codes. 6 −3 6 −4 = X 4 + 3X 3 + X 2 + 4X + 4 and  f 2 (X ) = XX +3 = Step 1 Let  f 1 (X ) = X 2X+2X +3 5 4 3 2 X + 2X + 4X + 3X + X + 2. Then, by the Euclidean Algorithm, we have u 1 (X ) = X + 3 and u 2 (X ) = 3, which implies that

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θ1 (X ) = (X + 3)(X 4 + 3X 3 + X 2 + 4X + 4) = X 5 + X 4 + 2X 2 + X + 2 and θ2 (X ) = 3(X 5 + 2X 4 + 4X 3 + 3X 2 + X + 2) = 3X 5 + X 4 + 2X 3 + 4X 2 + 3X + 1. Step 2 Let f (X ) = X 3 + 3X + 3. Then f (X ) is a primitive polynomial over F5 . Denote ξ = X +  f (X ). Then ξ is a primitive element of F53 = F5 [X ]/ f (X ). From F53 = F53 [X ]/X +3, we have ξ is also a primitive element of F53 [X ]/X +3. Using the Gauss’s Algorithm, we have ω = ξ 20 X + ξ 95 is a primitive element of F53 [X ]/X 2 + 2X + 3. Thus D1 = {ωi | i = 0, 1, . . . , 651} and D2 = {ξ j | j = 0, 1, . . . , 31}. Step 3 For any B1 ∈ D1 and B2 ∈ D2 , calculate G 1 (X ) = B1 θ1 (X ) (modX 6 − 3) and G 2 (X ) = B2 θ2 (X ) (modX 6 − 4). Step 4 Let G 1 (X ) =

2 

g1,i (X )ξ i = G1 (X )(1, ξ, ξ 2 )T

i=0

and G 2 (X ) =

2 

g2,i (X )ξ i = G2 (X )(1, ξ, ξ 2 )T ,

i=0

where (1, ξ, ξ 2 ) is an F5 -basis of F53 . Calculate G(X ) = vG1 (X )+(1−v)G2 (X ). Then all distinct 1-generator (4 + 4v)-QT codes of length 18 with index 3 and annihilator v(X 2 + 2X + 3) + (1 − v)(X + 3) over R can be constructed by C = (R[X ]/X 6 − (4 + 4v))3 G(X ). For example, let B1 = ω = (4ξ 2 + 2ξ + 4)X + 3ξ 2 ∈ D1 and B2 = ξ ∈ D2 . Then G 1 (X ) = (4X 5 + 3X 3 + 4X 2 + 3X + 2) + (2X 5 + 4X 3 + 2X 2 + 4X + 1)ξ +(2X 5 + 3X 4 + 3X 3 + X + 3)ξ 2 and G 2 (X ) = (3X 5 + X 4 + 2X 3 + 4X 2 + 3X + 1)ξ,

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Note on quasi-twisted codes

which implies that G(X ) = vG1 (X ) + (1 − v)G2 (X ) = (4v X 5 + 3v X 3 + 4v X 2 + 3v X + 2, (3 − v)X 5 + (1 − v)X 4 + (2 + 2v)X 3 +(4 − 2v)X 2 + (3 + v)X + 1, 2X 5 + 3X 4 + 3X 3 + X + 3). As stated above, G(X ) generates a (4 + 4v)-QT code C = (R[X ]/X 6 − (4 + 4v))3 G(X ), which is of length 18 with index 3 and annihilator v(X 2 + 2X + 3) + (1 − v)(X + 3) over R. By the Eq.(24), we have |C | = 53 . Furthermore, by the definitions of C1 and C2 , we have that the minimum Hamming distance of C is d(C ) = min{d(C1 ), d(C2 )} = 6, where C1 and C2 are 3-QT code and 4-QT code of length 18 with index 3 generated by G1 (X ) and G2 (X ) over F5 , respectively. Moreover, by the definition of the Gray map  and the fact that  is a weight preserving map from R 18 to F36 5 , (C ) is a linear code with parameters [36, 3, 5] over F5 . 5 Conclusion In this paper, we consider some useful properties of QT codes over finite fields, including the lower Hamming distance bounds, the enumerators and the searching generator algorithm. As an interesting application, we also discuss QT codes over the finite non-chain ring Fq + vFq . At the end, we give a searching algorithm for generators of QT codes with a fixed annihilator. Some previous works have shown that some special linear codes over the ring Fq + vFq , such as constacyclic codes, quadratic residue codes and so on, could produce good linear codes over finite fields. We think that the QT codes over the ring Fq + vFq will also produce some good or new codes over finite fields by some useful linear maps, though we have not considered this issue in this paper. Moreover, the algorithm given in this paper may effectively reduce the searching time for constructing linear codes over finite fields. This is one of motivations to introduce QT codes over Fq + vFq . Acknowledgments The authors are deeply indebted to the referees and wish to thank them for their important suggestions and comments. This research is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204), and the National Natural Science Foundation of China (Grant Nos. 61171082, 61301137).

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