Quasitwisted Codes

Chapter 9

Quasitwisted Codes This chapter follows [12,11] in order in sections 9.1 and 9.2, respectively.

9.1 ON QUASITWISTED CODES OVER FINITE FIELDS 9.1.1 Basic Theory Let Fq denote the finite field of q = p m elements, where p is a prime and m is a positive integer. Let C be a linear code of length n over Fq . Let λ ∈ F∗q and let l be a positive integer. For each codeword c = (c0 , c1 , . . . , cn−1 ) in C , if the vector (λcn−l , λcn−l+1 . . . , λcn−1 , c0 , . . . , cn−l−1 ) ∈ C , where the subscripts are taken modulo n, then the code C is called a (λ, l)-quasitwisted (QT) code. It is well known that a (λ, l)-QT code of length n = lθ over F [x] F [x] Fq is identified with (x θq−λ) -submodule of ( (x θq−λ) )l . F [x]

First, by decomposing the ring Rθ,λ := (x θq−λ) into a direct sum of local rings, it is shown that a (λ, l)-QT code of length lθ over Fq can be decomposed into a direct sum of linear codes Ci over these component rings. The decomposition of the ring involves the factorization of the polynomial x θ − λ over Fq . If gcd(θ, q) = 1 (nonrepeated-root case), then the polynomial x θ − λ is factorized into a product of distinct irreducible polynomials. It is shown that if gcd(θ, q) = p a with a ≥ 1 (repeated-root case), then all the irreducible factors of the polynomial x θ − λ are with multiplicity p a . In this chapter, we allow a ≥ 0 and then both cases are included. When a a a x θ − λ = (f1 (x))p (f2 (x))p · · · (fk (x))p , where fi (x)’s are irreducible polynomials over Fq , the ring Rθ,λ is decomposed into a direct sum of the local rings F [x] Ri :=  q pa  , 1 ≤ i ≤ k. (fi (x))

Since the dual code C ⊥Fq of a (λ, l)-QT code C is a (λ−1 , l)-QT code, a natural question that then arises is: given the decomposition of C , what is the decomposition of C ⊥Fq ? When λ = ±1, C and C ⊥Fq are modules over the F [x] same ring (x θq−λ) and hence, only in this case, self-dual QT codes make sense.

When λ = ±1, C and C ⊥Fq are modules over different rings: Rθ,λ and Rθ,λ−1 , respectively. Since the two rings are isomorphic by identifying x ∈ Rθ,λ with Codes and Rings. http://dx.doi.org/10.1016/B978-0-12-813388-0.00009-4 Copyright © 2017 Elsevier Inc. All rights reserved.

145

146 Codes and Rings

x −1 ∈ Rθ,λ−1 , we map C ⊥Fq into the module Rlθ,λ and get an isomorphic copy of C ⊥Fq . Based on the dual defined over two modules over the same ring, the decomposition of the dual code over Rθ,λ−1 is explicitly described. In particular, the decomposition of self-dual QT codes is given. An important tool to study algebraic codes is the discrete Fourier transform (DFT), also known as Mattson–Solomon polynomial [10, Chap. 8, §6]. When gcd(θ, p) = 1, i.e., in the nonrepeated-root case, the classical DFT of c(x) ∈ Fq [x] is defined to be the vector (x θ −λ) cˆ = [cˆ0 , cˆ1 , . . . , cˆθ−1 ], where cˆi = c(βξ i ), for 0 ≤ i ≤ θ − 1, β is a θ th root of λ, and ξ is a primitive θth root of unity. It is well-known that the DFT is invertible. However, in the repeated-root case, the classical DFT is not invertible because gcd(θ, p) > 1. Therefore, we adopt the Hasse derivative to develop the generalized discrete Fourier transform (GDFT). We also give the inversion formula of the GDFT. The GDFT also gives an explicit connection between a QT code and its component codes. Therefore, by the inversion formula of the GDFT, we produce a formula to construct a QT code from linear codes over component rings. It is further shown that the computation can be done in the field Fq instead of the extension fields. Let Fq denote the finite field of q = p m elements and let F∗q denote Fq \ {0}, where p is a prime and m is a positive integer. Denote by Fq [x] the polynomial ring in indeterminate x with coefficients from Fq . A linear code C of length n and dimension k over Fq is a k-dimensional subspace of the vector space Fnq . It is known as an [n, k]q code. The elements of the subspace are the codewords of C and written as row vectors: c = (c0 , c1 , . . . , cn−1 ). Definition 9.1. An [n, k]q code C is called cyclic provided that, for each codeword c = (c0 , c1 , . . . , cn−1 ) in C , the vector (cn−1 , c0 , . . . , cn−2 ) is also a codeword in C . Mapping a codeword (c0 , c1 , . . . , cn−1 ) ∈ C to a polynomial c0 + c1 x +· · ·+ cn−1 x n−1 ∈ Fq [x], a cyclic code C is an ideal in Fq [x]/(x n − 1). Generalized from cyclic codes, we have the following three classes of codes.

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Definition 9.2. Let C be a linear code of length n over Fq . Let λ ∈ F∗q . For each codeword c = (c0 , c1 , . . . , cn−1 ) in C , if the vector (λcn−1 , c0 , . . . , cn−2 ) ∈ C , then the code C is called a λ-constacyclic code and λ is called the constant of C . By the correspondence between codewords and polynomials, a λ-constacyclic code can be identified with an ideal in Fq [x]/(x n − λ). Definition 9.3. Let C be a linear code of length n over Fq . For each codeword c = (c0 , c1 , . . . , cn−1 ) in C , if the vector (cn−l , cn−l+1 , . . . , cn−1 , c0 , . . . , cn−l−1 ) ∈ C where the subscripts are taken modulo n and l is a positive integer, then the code C is called an l-quasicyclic (QC) code and l is called the index of C . It is easy to check that an l-QC code of length n is also a gcd(l, n)-QC code. Without loss of generality, we therefore assume that the index l always divides the length n. Let θ = nl . Reindexing the coordinates of a codeword (c0 , c1 , . . . , cn−1 ) of the l-QC code as c = (c0 , cl , . . . , c(θ−1)l , c1 , cl+1 , . . . , c(θ−1)l+1 , . . . , cl−1 , . . . , cθl−1 ), we have divided c to l parts, and each part consists of θ consecutive coordinates. It is observed that each part can be regarded as a codeword in a cyclic code of length θ over Fq . Therefore, representing each part of c by a polynomial in Fq [x]/(x θ − 1), the codeword c is equivalent to the vector in (Fq [x]/(x θ − 1))l : (c0 + cl x + · · · + c(θ −1)l x θ −1 , c1 + · · · + c(θ −1)l+1 x θ −1 , . . . , cl−1 + c2l−1 x + · · · + cθ l−1 x θ −1 )

(see [7]). Then an l-QC code is equivalent to a submodule of (Fq [x]/(x θ − 1))l over the ring Fq [x]/(x θ − 1). Definition 9.4. Let C be a linear code of length n over Fq . Let λ ∈ F∗q and let l be a positive integer. For each codeword c = (c0 , c1 , . . . , cn−1 ) in C , if the vector (λcn−l , λcn−l+1 , . . . , λcn−1 , c0 , . . . , cn−l−1 ) ∈ C , where the subscripts are taken modulo n, then the code C is called a (λ, l)-quasitwisted (QT) code. We define an action Tλ,l on the codewords as Tλ,l (c0 , c1 , . . . , cn−1 ) = (λcn−l , λcn−l+1 , . . . , λcn−1 , c0 , . . . , cn−l−1 ). A (λ, l)-QT code is invariant as a set under the action Tλ,l .

148 Codes and Rings

It is easy to check that a (λ, l)-QT code of length n is also a (λ, gcd(l, n))-QT code (see [1]). Thus we always assume l divides n. Let θ = nl . When λ = 1, a (λ, l)-QT code is an l-QC code. When l = 1, a (λ, l)-QT code is a λ-constacyclic code. When λ = l = 1, a (λ, l)-QT code is a cyclic code. From the above discussion about constacyclic codes and QC codes, a (λ, l)-QT code of length n is a submodule of (Fq [x]/(x θ − λ))l over the ring Fq [x]/(x θ − λ). For convenience, we use the same notation for both the code over Fq and its corresponding submodule of (Fq [x]/(x θ − λ))l over the ring Fq [x]/(x θ − λ).

9.1.2 Decomposition of QT Codes Let C be a (λ, l)-QT code of length n over Fq . Recall that C is a module over the ring Fq [x]/(x θ − λ). Denote the ring Fq [x]/(x θ − λ) by Rθ,λ . In order to know more about the algebraic structure of QT codes, we focus now on the ring Rθ,λ . a Let θ = p a θ¯ , where gcd(θ¯ , p) = 1. Since the map x → x p is a power of the Frobenius automorphism of Fq defined by x → x p , it is an automorphism of a Fq . Therefore, for any λ ∈ F∗q , there exists a unique λ¯ ∈ F∗q such that λ¯ p = λ. Therefore, we may write ¯

x θ − λ = (x θ − λ¯ )p . a

¯ Since gcd(θ¯ , p) = 1, the polynomial x θ − λ¯ is factorized into distinct irreducible polynomials over Fq as follows: ¯

x θ − λ¯ = f1 (x)f2 (x) · · · fk (x). Therefore, we have a

a

a

x θ − λ = (f1 (x))p (f2 (x))p · · · (fk (x))p .

(9.1)

By the Chinese Remainder Theorem, we have the following decomposition: Fq [x] Fq [x]  Fq [x]   Fq [x]    ··· a a a θ (x − λ) (f1 (x))p (f2 (x))p (fk (x))p  a a  r(x) ↔ r(x) + ((f1 (x))p ), . . . , r(x) + ((fk (x))p ) . For convenience, we denote the ring 

Fq [x]

(fi (x))p

a

 by Ri for 1 ≤ i ≤ k. It fol-

lows that Rlθ,λ 

k 

Rli .

i=1

This ring decomposition translates directly into a code decomposition.

(9.2)

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149

Theorem 9.5. Let C be a (λ, l)-QT code of length lθ over Fq . Then C is a linear code over Rθ,λ of length l and it can be decomposed as the direct sum C

k 

Ci ,

(9.3)

i=1

where Ci is a linear code over Ri of length l for each 1 ≤ i ≤ k.

9.1.3 Dual Codes of QT Codes In this subsection, we discuss the dual codes of QT codes. For our purpose, we give the following definition about dual codes. Definition 9.6. Let K be a commutative ring or a finite field and let N be a positive integer. Let u = (u0 , . . . , uN −1 ) and v = (v0 , . . . , vN −1 ) be two vectors over K. The inner product of u and v over K is denoted by

u, vK =

N −1 

ui vi .

i=0

Let C be a linear code of length N over K, then the dual code of C (with respect to the inner product over K), denoted by C ⊥K , is defined as C ⊥K = {v ∈ KN | v, uK = 0, for any u ∈ C }. In particular, if C = C ⊥K , then C is a self-dual code over K. Notice that when K = Fq , the inner product defined above is exactly the Euclidean inner product. Recall that the index l always divides the length n for a QT code. The following proposition follows directly from the definition of QT codes. Proposition 9.7. Let C be a (λ, l)-QT code of length n over Fq and let C ⊥Fq be the dual code of C . Then C ⊥Fq is a (λ−1 , l)-QT code of length n over Fq . By the above proposition, we know that C ⊥Fq is a submodule of Rlθ,λ−1 over Rθ,λ−1 , and hence a linear code over Rθ,λ−1 .

150 Codes and Rings

Notice that a (λ, l)-QT code is an Rθ,λ -module while its dual code is an Rθ,λ−1 -module. However, the two rings Rθ,λ and Rθ,λ−1 are isomorphic: Rθ,λ  Rθ,λ−1 x ↔ x −1 , where x −1 = λ−1 x θ−1 in the ring Rθ,λ and x −1 = λx θ−1 in the ring Rθ,λ−1 . By the above isomorphism, we define the map φ as follows. Definition 9.8. For all (r0 (x), r1 (x), . . . , rl−1 (x)) ∈ Rlθ,λ−1 , we define the map

φ : Rlθ,λ−1 → Rlθ,λ with

  φ (r0 (x), r1 (x), . . . , rl−1 (x)) = (r0 (x −1 ), r1 (x −1 ), . . . , rl−1 (x −1 )), Obviously, the map φ is bijective since it is induced from the isomorphism between Rθ,λ and Rθ,λ−1 . Therefore, it immediately follows that: Proposition 9.9. The map φ gives a one-to-one correspondence between the Rθ,λ -submodules of Rlθ,λ and the Rθ,λ−1 -submodules of Rlθ,λ−1 . Although the (λ, l)-QT code and its dual code are modules over different rings, by the above proposition, we can consider the image of the dual code of a (λ, l)-QT code under the map φ. Then φ(C ⊥Fq ) and C are modules over the same ring Rθ,λ . Similarly, we can also consider the following two modules over Rθ,λ−1 : C ⊥Fq and the preimage of C under the map φ. The following lemma studies the dual with respect to the inner product over Rθ,λ . Lemma 9.10. Let c and d be any two vectors in Fnq , where n = lθ . Let the vector c(x) ∈ Rlθ,λ be the polynomial representation corresponding to the vector c and let the vector d(x) ∈ Rlθ,λ−1 be the polynomial representation corresponding to

i (c), d > = 0 the vector d. Then < c(x), φ(d(x)) >Rθ,λ = 0 if and only if < Tλ,l Fq for each 0 ≤ i ≤ θ − 1.

Proof. Assume that < c(x), φ(d(x)) >Rθ,λ = 0. Then we have l−1  θ−1    θ−1  ( ci+j l x j )( di+kl x −k ) = 0. i=0

j =0

(9.4)

k=0

Since the above equation is in the ring Rθ,λ , the left hand side can be written as a unique polynomial over Fq of degree less than θ . Denote by [x i ] the term in x i in such a unique expression, where 0 ≤ i ≤ θ − 1.

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Since x θ = λ in the ring Rθ,λ , it immediately follows that x −j = λ−1 x θ x −j = λ−1 x θ−j , for 1 ≤ j ≤ θ − 1. Therefore, each term on the left hand side of (9.4) is as follows: [x 0 ] =

θ−1 l−1  

ci+j l di+j l

i=0 j =0

=

θl−1 

ci di

i=0

= c, dFq , [x k ] =

l−1  ((ci+kl di + · · · + ci+(θ−1)l di+(θ−1−k)l )x k i=0

+ (ci di+(θ−k)l + · · · + ci+(k−1)l di+(θ−1)l )x k−θ ) = λ−1

l−1  (λci+kl di + · · · + λci+(θ−1)l di+(θ−1−k)l i=0

+ ci di+(θ−k)l + · · · + ci+(k−1)l di+(θ−1)l )x k θ−k = λ−1 ( Tλ,l (c), dFq )x k , for each 1 ≤ k ≤ θ − 1.

Then the uniqueness of the expression of the left hand side of (9.4) implies i (c), d = 0 for that each term is 0. Thus, the above equations imply that Tλ,l Fq 0 ≤ i ≤ θ − 1. It is easy to observe that the converse is also true. Applying Lemma 9.10, we have the following theorem: Theorem 9.11. Let C be a (λ, l)-QT code of length n over Fq and D a (λ−1 , l)-QT code of length n over Fq . Then D is the dual code of C with respect to the inner product on Fnq if and only if φ(D) is the dual code of C with respect to the inner product on Rlθ,λ , i.e., φ(C ⊥Fq ) = C

⊥Rθ,λ

,

(9.5)

where C on the left is the code over Fq while C on the right means its corresponding module over Rθ,λ . Proof. Since C is a (λ, l)-QT code, for any codeword c ∈ C , we have i (c) ∈ C . Then for any codeword d ∈ C ⊥Fq , we have T i (c), d = 0. By Tλ,l Fq λ,l

152 Codes and Rings ⊥

Lemma 9.10, it follows c, φ(d)Rθ,λ = 0. Therefore, we have φ(d) ∈ C Rθ,λ . ⊥ Then by Definition 9.6, we have φ(C ⊥Fq ) ⊆ C Rθ,λ . ⊥ Assume that e ∈ C Rθ,λ . Then by Lemma 9.10, for any codeword c ∈ C , we ⊥ i −1 have < Tλ,l (c), φ (e) >Fq = 0. It follows that φ −1 (e) ∈ C ⊥Fq . Then C Rθ,λ ⊆ φ(C ⊥Fq ). Therefore, φ(C ⊥Fq ) = C

⊥Rθ,λ

.

By the decomposition of the ring Rθ,λ in (9.2), we have the following corollary. Corollary 9.12. Let C be a (λ, l)-QT code over Fq of length n = lθ . Suppose ⊥ that C is decomposed as in (9.2). Then C Rθ,λ is decomposed as follows: C

⊥Rθ,λ



k 

Di ,

(9.6)

i=1

where, for each 1 ≤ i ≤ k, Di is the dual code of Ci with respect to the inner product on Rli . In particular, C is self-dual if and only if Ci = Di for all 1 ≤ i ≤ k. By Theorem 9.11, the above corollary gives the decomposition of φ(C ⊥Fq ). Next we discuss the relationship between the decomposition of C ⊥Fq ⊆ Rlθ,λ−1 ⊥

and that of φ(C ⊥Fq ) = C Rθ,λ ⊆ Rlθ,λ . Assume that x θ − λ is factorized as in Equation (9.1). Then x θ − λ−1 = −λ−1 (f1∗ (x))p · · · (fk∗ (x))p , a

a

(9.7)

where fi∗ (x) := x deg fi (x) fi (x −1 ) is the reciprocal polynomial of fi (x) over Fq . It is easy to check that fi∗ (x) is also irreducible over Fq if fi (x) is irreducible. Therefore, we have the following decomposition of the ring Rθ,λ−1 :    Fq [x] Fq [x] Fq [x] Fq [x]  ∗  ∗  ∗ ··· a a a −1 p p −λ ) (f1 (x)) (f2 (x)) (fk (x))p  a a  r(x) ↔ r(x) + ((f1∗ (x))p ), . . . , r(x) + ((fk∗ (x))p ) .

(x θ

For simplicity, we denote the ring 

Fq [x]

(fi∗ (x))p

a

(9.8)

 by R∗i for 1 ≤ i ≤ k. It follows

that Rlθ,λ−1 

k  (R∗i )l .

(9.9)

i=1

Note that a (λ, l)-QT code is self-dual only if λ = ±1. If λ = ±1, then the polynomials x θ − λ and x θ − λ−1 are coprime over Fq . Therefore, the irreducible polynomials fi (x), fj∗ (x), 1 ≤ i, j ≤ k, are pairwise coprime where

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153

fi (x), fj∗ (x), 1 ≤ i, j ≤ k are as in Equations (9.2) and (9.7). Thus, no irreducible polynomial is an associate of its reciprocal polynomial and no reciprocal pair exists in the factorization of x θ − λ, which is different from the case when λ = ±1.

9.1.3.1 Case when λ = ±1 In this subsection, we focus on the case when λ = ±1. If λ = ±1, then x θ − λ = x θ − λ−1 and hence Rθ,λ = Rθ,λ−1 . With the proper permutation of the irreducible polynomial factors, x θ − λ is written as x θ − λ = (g1 (x))p · · · (gs (x))p (h1 (x))p (h∗1 (x))p · · · (ht (x))p (h∗t (x))p , a

a

a

a

a

a

where s + 2t = k,  ∈ F∗q and, for each 1 ≤ i ≤ s, gi (x) is an associate of its reciprocal polynomial, i.e., gi (x) = i gi∗ (x) over Fq for some unit i . a Throughout this subsection, we denoteFq [x]/ (gi (x))p by Gi for 1 ≤ i ≤ s,   a a Fq [x]/ (hj (x))p by Hj and Fq [x]/ (h∗j (x))p by H∗j for 1 ≤ j ≤ t . Then the decomposition of Rθ,λ = Rθ,λ−1 is Rθ,λ 

s 

Gi

t 

(Hj



 H∗j ) .

(9.10)

j =1

i=1

Therefore, when λ = ±1, the map φ is an automorphism of Rlθ,λ . We define the same isomorphisms between the component rings as follows. Definition 9.13. For 1 ≤ i ≤ s, define φi : (Gi )l → (Gi )l by a

a

φi ((r1 (x) + ((gi (x))p ), . . . , rl (x) + ((gi (x))p ))) = (r1 (x −1 ) + ((gi (x))p ), . . . , rl (x −1 ) + ((gi (x))p )). a

a

For 1 ≤ j ≤ t , define φj : (Hj )l → (H∗j )l by φj ((r1 (x) + ((hj (x))p ), . . . , rl (x) + ((hj (x))p ))) a

a

= (r1 (x −1 ) + ((h∗j (x))p ), . . . , rl (x −1 ) + ((h∗j (x))p )). a

a

Actually, when λ = ±1, the maps φ, φi and φj are exactly the conjugate maps defined in [9].

154 Codes and Rings

Lemma 9.14. Assume that λ = ±1 and the decomposition of the ring Rθ,λ = Rθ,λ−1 is as in Equation (9.10). Let r(x) ∈ Rθ,λ and let its decomposition in Rθ,λ be (r1 (x), . . . , rs (x), r1 (x), r1 (x), . . . , rt (x), rt (x))  a = r(x) + (gi (x))p ∈ G 1 ≤ j ≤ t, where for 1 ≤ i ≤ s, ri (x)  i , and for  a a rj (x) = r(x) + (hj (x))p ∈ Hj and rj (x) = r(x) + (h∗j (x))p ∈ H∗j . Then the decomposition of φ −1 (r(x)) ∈ Rθ,λ−1 is (r1 (x −1 ), . . . , rs (x −1 ), r1 (x −1 ), r1 (x −1 ), . . . , rt (x −1 ), rt (x −1 )). a

Proof. For 1 ≤ i ≤ s, since ri (x) = r(x) + ((gi (x))p ), ri (x −1 ) = r(x −1 ) + ((gi (x −1 ))p ). a

Since g(x) is an associate of its reciprocal polynomial, ((gi (x))p ) = ((gi (x −1 ))p ). a

a

Therefore, we have ri (x −1 ) = r(x −1 ) + ((gi (x))p ), a

i.e., the component of φ −1 (r(x)) = r(x −1 ) in Gi is ri (x −1 ). For 1 ≤ j ≤ t , we have rj (x −1 ) = r(x −1 ) + ((hj (x −1 ))p ). a

Then rj (x −1 ) = r(x −1 ) + ((h∗j (x))p ), a

i.e., the component of φ −1 (r(x)) = r(x −1 ) in H∗j is rj (x −1 ). Similarly, the component of φ −1 (r(x)) = r(x −1 ) in Hj is rj (x −1 ). The following theorem gives the algebraic structure of the dual code of a (λ, l)-QT code when λ = ±1. Theorem 9.15. Let C be a (λ, l)-QT code of length lθ over Fq with λ = ±1. Let the decomposition of the ring Rθ,λ be as in Equation (9.10) and let the corresponding decomposition of C be C

s  i=1

Ci

t    Cj ) . (Cj j =1

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155

Then the decomposition of its dual code C ⊥Fq is C ⊥Fq 

s 

⊥Gi

φi (Ci

)

t   ⊥H∗   ⊥ ((φj )−1 ((Cj ) j ) φj ((Cj ) Hj )) , j =1

i=1

where the duality on the left is the duality with respect to the inner product over Fq , while the dualities on the right are the dualities with respect to the inner products over the respective component rings. In particular, C is self-dual if and only if ⎧ ⎨C = φ (C ⊥Gi ), 1 ≤ i ≤ s, i i i (9.11) ⊥ ⎩  Cj = φj ((Cj ) Hj ), 1 ≤ j ≤ t. Proof. This theorem follows from Corollary 9.12 and Lemma 9.14. When λ = ±1, the map φi ’s are actually the conjugates defined in [9]. We can check that the above theorem is consistent with Theorem 4.2 in [9] which describes the dual with respect to the Hermitian inner product.

9.1.3.2 Case when λ = ±1 In this subsection, we assume that λ = ±1. Recall that φ is the isomorphism between Rθ,λ and Rθ,λ−1 . Let the decompositions of Rlθ,λ and Rlθ,λ−1 be as in Equations (9.2) and (9.9), respectively. Then the quotient rings Ri = a a Fq [x]/((fi (x))p ) and R∗i = Fq [x]/((fi∗ (x))p ) are isomorphic as rings. The corresponding isomorphism is defined as follows. Definition 9.16. The isomorphism is φi : (Ri )l → (R∗i )l given by φi ((r1 (x) + ((fi (x))p ), . . . , rl (x) + ((fi (x))p ))) a

a

= (r1 (x −1 ) + ((fi∗ (x))p ), . . . , rl (x −1 ) + ((fi∗ (x))p )). a

a

By Corollary 9.12, the following theorem immediately follows. Theorem 9.17. Let λ = ±1 and let the decompositions of Rlθ,λ and Rlθ,λ−1 be as in Equations (9.2) and (9.9), respectively. Let C be a (λ, l)-QT code of length lθ over Fq , i.e., an Rθ,λ -linear code. Suppose that the decomposition of C is as in (9.3): C

k  i=1

Ci .

156 Codes and Rings

Then the decomposition of its dual code C ⊥Fq ⊆ Rlθ,λ−1 is C ⊥Fq 

k 

⊥Ri

φi (Ci

).

i=1

Given the decomposition of the code C ⊆ Rlθ,λ , Theorems 9.15 and 9.17 give the decomposition of the dual code C ⊥Fq ⊆ Rlθ,λ−1 , for cases λ = ±1 and λ = ±1, respectively.

9.1.4 Discrete Fourier Transform In order to deal with the repeated-root case, we introduce a generalized discrete Fourier transform (GDFT) as in [7]. For our purpose, we define the Hasse derivative as follows. Definition 9.18. (see [5]) For a polynomial g(x) = i gi x i ∈ Fq [x], the j th Hasse derivative is defined as  i  g [j ] (x) = gi x i−j . j i

Using the Hasse derivative, we define the generalized discrete Fourier transform (GDFT). Recall that θ = p a θ¯ , where gcd(θ¯ , p) = 1. Definition 9.19. If c(x) = i∈Z/θZ ci x i ∈ Rθ,λ , then the generalized discrete Fourier transform (GDFT) of c(x) can be described in terms of a matrix ⎡ ⎤ cˆ0,1 ... cˆ0,θ−1 cˆ0,0 ¯ ⎢ ⎥ ⎢ cˆ1,0 ⎥ cˆ1,1 ... cˆ1,θ−1 ¯ ⎢ ⎥ cˆ = ⎢ (9.12) ⎥, .. .. .. .. ⎢ ⎥ . . . . ⎣ ⎦ cˆpa −1,0 cˆpa −1,1 . . . cˆpa −1,θ¯ −1 where  i  ci (βξ h )i−g , for 0 ≤ g ≤ p a − 1, 0 ≤ h ≤ θ¯ − 1, cˆg,h = g i∈Z/θZ

¯ root of λ¯ , and ξ is a primitive θth ¯ root of unity. β is a θth Notice that cˆg,h is exactly the value of the gth Hasse derivative at βξ h , a θ¯ th ¯ Let x θ − λ be decomposed as in (9.1). Then for each 1 ≤ h ≤ θ¯ , there root of λ. is an irreducible factor of x θ − λ, say fi (x), such that βξ h is a root of fi (x).

Quasitwisted Codes Chapter | 9

157

 a Then cˆg,h is an element in Fq [x]/ (fi (x))p . Mimicking the method in [9] and replacing the root ζ h in [7] by βξ h , then the explicit description of the inverse transform is given. We give the inverse transform in the following theorem and omit the proof. Theorem 9.20. The GDFT (9.12) is invertible. More precisely, the inverse formula of GDFT is p −1  θ¯ −1  g  1 h −jp a = (βξ ) (−βξ h )g−i cˆg,h , i θ¯ a

c

i+jp a

h=0

(9.13)

g=0

for 0 ≤ i ≤ pa − 1 and 0 ≤ j ≤ θ¯ − 1, where β is a θ¯ th root of λ¯ and ξ is a primitive θ¯ th root of unity. ¯ Since (β q−1 )θ = λ¯ q−1 = 1 for λ¯ ∈ F∗q , β q−1 is a θ¯ -th root of unity. Then β q−1 can be expressed as a power of the primitive θ¯ th root of unity ξ , say

β q−1 = ξ δ , where 0 ≤ δ ≤ θ¯ − 1. By the definition of cˆg,h , it is easy to verify that, for 0 ≤ g ≤ p a − 1 and 0 ≤ h ≤ θ¯ − 1,  i q q q ci [(βξ h )q ]i−g cˆg,h = g i∈Z/θZ  i  = ci (βξ qh+δ )i−g g i∈Z/θZ

= cˆg,qh+δ . Given an irreducible polynomial fi (x), if βξ zi is a root of fi (x), so is β q ξ qzi = βξ qzi +δ . Define a map τ : ¯ τ : Z/θ¯ Z → Z/θZ z → qz + δ. As gcd(θ¯ , q) = 1, it follows that the map τ is one-to-one. Therefore, the map τ defines an equivalence relation ∼ on Z/θ¯ Z where h1 ∼ h2 if and only if there exists an integer i such that h1 = τ i (h2 ). Therefore, there is a one-to-one correspondence between the equivalence classes and the irreducible factors fi s. For convenience, we call the equivalence classes orbits of τ . From each orbit Oi , we can choose a representative, say zi . Then there is a one-to-one correspondence

158 Codes and Rings

between the irreducible factors fi (x)’s and the representatives zi ’s. We say the representative zi is corresponding to the irreducible polynomial fi . In particular, when δ = 0, the equivalence classes are known as the q-cyclotomic cosets modulo θ¯ . Therefore, using the same notations above, the inversion formula of the GDFT can be further simplified as follows. Theorem 9.21. The GDFT (9.12) is invertible as follows: for 0 ≤ i ≤ p a − 1 and 0 ≤ j ≤ θ¯ − 1, p −1  k  1  g a  g−i = T rγ (cˆg,zγ (βξ zγ )g−i−jp ) , (−1) ¯θ i g=0 γ =1 a

c

i+jp a

(9.14)

¯ ξ is a primitive θ¯ -th root of unity, zγ is a represenwhere β is a θ¯ -th root of λ, tative in the orbit corresponding to fγ (x) and T rγ is the trace map on the field Fq [x]/(fγ (x)) down to Fq . Although the choices of β and ξ in the formula (9.14) are not unique, the result of the formula (9.14) is unique when cˆg,h ’s are given. The above theorem gives the trace description of QT codes.

9.1.5 Construction Formula Let C be a (λ, l)-QT code of length lθ . By Theorem 9.5, we know that C

k 

Ci ,

i=1

where Ci is a linear code over Ri of length l for each 1 ≤ i ≤ k. F [x] The ring Ri =  q pa  is a finite chain ring. Each element in Ri can be (fi (x))

written in the following canonical form: a0 (x) + a1 (x)fi (x) + · · · + apa −1 (x)(fi (x))p where aj (x) ∈

Fq [x] (fi (x))

a −1

,

for 0 ≤ j ≤ p a − 1. Therefore,

Fq [x] Fq [x] Fq [x] Fq [x] a + fi (x) + · · · + (fi (x)p −1 ) . a  (fi (x)) (fi (x)) (fi (x)) (fi (x))p



Let di = deg fi (x) and let βξ zi be a root of fi (x). Then we have the following field isomorphism:

Quasitwisted Codes Chapter | 9

159

Fq [x]  Fq + (βξ zi )Fq + · · · + (βξ zi )di −1 Fq , (fi (x)) r(x) ↔ r(βξ zi ). Then we have the following proposition. Proposition 9.22. The following map is a ring isomorphism: σ : Ri → (Fq + (βξ zi )Fq + · · · + (βξ zi )di −1 Fq ) + u(Fq + · · · + (βξ zi )di −1 Fq ) + · · · + up

a −1

(Fq + · · · + (βξ zi )di −1 Fq ),

r(x) → r(βξ zi + u), a

where up = 0 and βξ zi is a root of fi (x). Proof. For convenience, denote fi (x) by f (x), d = deg(f (x)) and βξ zi by η. a Suppose that f (x) = di=0 ai x i . Since η is a root of f (x) and up = 0, we have a

σ ((f (x))p ) = (f (η + u))p =

d 

pa

a

pa

ai

a

a

ai (ηp + up )i

i=0

=

d 

ai η p

i=0

= (f (η))p

a

= 0. Therefore, this map is well defined. ¯ Since η is a root of the irreducible polynomial f (x), ηθ r¯ = λ¯ r¯ = 1, where ∗ r¯ is the order of λ¯ ∈ Fq . Since r¯ divides q − 1, r¯ is coprime to p a . Since θ¯ is coprime to p a too, p a and θ¯ r¯ are coprime. Then there exist integers N1 and N2 such that p a N1 + N2 θ¯ r¯ = 1. a

Then we have ηp N1 = η. a a It follows that x p N1 is mapped to η and x − x p N1 is mapped to u. Hence, the map σ is a ring isomorphism. For simplicity, we denote by Ji the chain ring (Fq + (βξ zi )Fq + · · · + (βξ zi )di −1 Fq ) + u(Fq + · · · + (βξ zi )di −1 Fq ) + · · · + up

a −1

(Fq + · · · + (βξ zi )di −1 Fq ).

160 Codes and Rings

Then we have Rθ,λ 

k 

Ji ,

i=1

and C

k 

Ci ,

i=1

where Ci is a code over Ji of length l. Then a codeword xi of Ci over Ji can be written as   xi = xi,0,0 + (βξ zi )xi,0,1 + · · · + (βξ zi )di −1 xi,0,di −1   + u xi,1,0 + (βξ zi )xi,1,1 + · · · + (βξ zi )di −1 xi,1,di −1 + · · ·   a + up −1 xi,pa −1,0 + (βξ zi )xi,pa −1,1 + · · · + (βξ zi )di −1 xi,pa −1,di −1 , where, for each 1 ≤ i ≤ k, 0 ≤ j ≤ p a − 1 and 0 ≤ w ≤ di − 1, xi,j,w is a row vector over Fq of length l. We vertically join all the above row vectors xi,j,w as x˜i = (xi,0,0 , . . . , xi,0,di −1 , xi,1,0 , . . . , xi,1,di −1 , . . . , xi,pa −1,0 , . . . , xi,pa −1,di −1 )T . Then x˜i is a matrix of size p a di × l. We vertically join all the above matrices as x = (x˜1 , x˜2 , . . . , x˜k )T .

(9.15)

Then x is in fact a matrix of size θ × l because ki=1 p a di = θ . By Theorem 9.20, a codeword in a QT code can be given if the component codewords are known. With the same notations as above, we have the following result about the construction of a QT code. Theorem 9.23. Let θ = p a θ¯ with gcd(p, θ¯ ) = 1, where p is the characteristic of Fq . Then, for any positive integer l and any λ ∈ F∗q , the (λ, l)-QT codes over Fq of length lθ are precisely given as follows: a (i) Write λ = λ¯ p where λ¯ ∈ F∗q . ¯

(ii) Write x θ − λ¯ = f1 (x)f2 (x) · · · fk (x), where for 1 ≤ γ ≤ k, fγ (x) are monic irreducible polynomials over Fq .  a (iii) Write Fq [x]/ (fγ (x))p = Rγ and deg fγ (x) = dγ . (iv) Let Oγ denote the orbit corresponding to fγ (x) and fix zγ ∈ Oγ . (v) For each 1 ≤ γ ≤ k, let Cγ be a linear code of length l over Rγ . For xγ ∈ Cγ , write

Quasitwisted Codes Chapter | 9

161

  xγ = xγ ,0,0 + (βξ zγ )xγ ,0,1 + · · · + (βξ zγ )dγ −1 xγ ,0,dγ −1   + u xγ ,1,0 + (βξ zγ )xγ ,1,1 + · · · + (βξ zγ )dγ −1 xγ ,1,dγ −1 + · · ·  a + up −1 xγ ,pa −1,0 + (βξ zγ )xγ ,pa −1,1 + · · ·  + (βξ zγ )dγ −1 xγ ,pa −1,dγ −1 , where, for each 1 ≤ γ ≤ k, 0 ≤ g ≤ p a − 1 and 0 ≤ w ≤ dγ − 1, xγ ,g,w is a row vector over Fq of length l. (vi) For each 0 ≤ i ≤ p a − 1 and 0 ≤ j ≤ θ¯ − 1, let p −1  γ −1 k d  1  g a g−i = (xγ ,g,w T rγ ((βξ zγ )g−i−jp +w )))), (−1) ( ( ¯θ i g=0 γ =1 w=0 a

ci+jpa

(9.16) and hence the codewords xγ ∈ Cγ , 1 ≤ γ ≤ k, give a vector (c0 , c1 , . . . , cθ−1 ). Then when the codeword xγ runs through all the codewords in Cγ for each γ , the collection of all the vectors (c0 , c1 , . . . , cθ−1 ) given by Equation (9.16) C = {(c0 , c1 , . . . , cθ−1 )} is a (λ, l)-QT code over Fq of length lθ . Conversely, every QT code over Fq of length lθ is obtained through this construction. Moreover, the construction can be expressed as follows: (c0 , c1 , . . . , cθ−1 )T = A · x, where x is defined as in (9.15), A is a θ × θ matrix over Fq such that, for 0 ≤ i ≤ p a − 1, 0 ≤ j ≤ θ¯ − 1, 0 ≤ g ≤ p a − 1, 1 ≤ γ ≤ k, 0 ≤ w ≤ dγ − 1, the γ −1 entry in the (i + jp a + 1)th row and (p a h=1 dh + gdγ + w + 1)th column, i.e., the coefficient in front of xγ ,g,w is A(i + jpa + 1, p a

γ −1

dh + gdγ + w + 1)

h=1

 1 g a = (−1)g−i T rγ ((βξ zγ )g−i−jp +w ). i θ¯ Proof. By the isomorphism in Proposition 9.22, cˆg,γ in Equation (9.14) can dγ −1 be written as w=0 (βξ zγ )w xγ ,g,w , and the γ th component of c(x) is cˆ0,γ + a −1 p cˆpa −1,γ . Then the theorem follows from Equation (9.13). ucˆ1,γ + · · · + u Obviously, the matrix A is over Fq because the entries are obtained by the respective trace maps down to Fq .

162 Codes and Rings

9.1.6 Examples The examples in this subsection are computed in the computer language MAGMA [2]. The following example gives a self-dual (2, 2)-QT code of length 24 over F3 . We can see that its decomposition satisfies Equation (9.11) given in Theorem 9.15. Example 9.24. Factorize x 12 − 2 over F3 as follows: x 12 − 2 = (x 4 + 1)3 = (x 2 + x + 2)3 (x 2 + 2x + 2)3 := h(x)h∗ (x). F3 [x] ∗ 3 [x] Denote by H the ring ((x 2F+x+2) 3 ) , and denote by H the ring ((x 2 +2x+2)3 ) . Let C be a self-dual (2, 2)-QT code of length 24 over F3 with generator (h(x), h∗ (x)). Then C can be decomposed as the direct sum of the following two component codes, C1 and C2 , where:

(i) C1 is generated by (0, h∗ (x) mod h(x)) over H and (ii) C2 is generated by (h(x) mod h∗ (x), 0) over H∗ . Since h(x) and h∗ (x) are coprime, the vector (0, 1) is also a generator of C1 over H. For the same reason, (1, 0) is a generator of C2 over H∗ . ⊥ It is easy to observe that the dual code C1 H of C1 over H is with generator (1, 0) over H. Since the isomorphism between H2 and (H∗ )2 is φ  : H2 → (H∗ )2 (r1 (x) + (h(x)), r2 (x) + (h∗ (x))) → (r1 (x −1 ) + (h∗ (x)), r2 (x −1 ) + (h(x))), ⊥

the image of (1, 0) over H is (1, 0) over H∗ . Therefore, the image of C1 H under φ  is generated by (1, 0) over H∗ , which is exactly C2 over H∗ . Therefore, Equation (9.11) given in Theorem 9.15 is satisfied. The next example gives a QT code over F5 as well as that of its dual code where λ = ±1. We can see that they satisfy Equation (9.5) in Theorem 9.11 and their decompositions satisfy Equation (9.6) in Corollary 9.12. Example 9.25. Factorize x 15 − 2 over F5 as follows: x 15 − 2 = (x 3 + 3)5 = (x + 2)5 (x 2 + 3x + 4)5 := (f1 (x))5 (f2 (x))5 .

Quasitwisted Codes Chapter | 9

163

Then F5 [x] F5 [x]  F5 [x]   .  15 5 2 (x − 2) (x + 2) (x + 3x + 4)5 Denote the ring

F5 [x] (x 15 −2)

F5 [x]  (x 2 +3x+4)5 Since 2−1 = 3

ring 

by R15,2 , denote the ring  F5 [x]5  by R1 and denote the (x+2)

by R2 . in F5 , by Equation (9.8), we have  R15,3  R∗1 R∗2 ,

where F5 [x] , (x 15 − 3) F5 [x] , R∗1 :=  (x + 3)5 F5 [x] . R∗2 :=  2 (x + 2x + 4)5

R15,3 :=

Let G1 (x) = x 2 + 4x + 4 = (x + 2)2 , and G2 (x) = x 6 + 4x 5 + 4x 4 + 4x 3 + x 2 + 4x + 4 = (x 2 + 3x + 4)3 . Let C be a (2, 2)-QT code of length 30 over F5 with generator (G1 (x), G2 (x)). Then we can decompose C as the direct sum of the following two component codes, C1 and C2 , where (i) C1 is generated by (G1 (x) mod (f1 (x))5 , G2 (x) mod (f1 (x))5 ) and (ii) C2 is generated by (G1 (x) mod (f2 (x))5 , G2 (x) mod (f2 (x))5 ). Then C ⊥F5 is a (3, 2)-QT code of length 30 over F5 with generator (g1 (x), g2 (x)) (over the ring R15,3 ) where g1 (x) = 3x 12 + 3x 11 + 2x 10 + 4x 9 + 4x 8 + 2x 7 + 2x 6 + 2x 4 + 3x 3 + 4x 2 + 4x + 1 = (x 2 + 2x + 4)3 (x 6 + 2x 3 + 3), g2 (x) = 4x 8 + 4x 7 + 2x 4 + 2x 2 + 4 = 4(x + 3)2 (x 3 + x 2 + 4x + 1)(x 3 + 4x 2 + 3x + 4).

164 Codes and Rings

The generator (g1 (x), g2 (x)) of C ⊥F5 over R15,3 is mapped to (g1 (x), g2 (x)) over R15,2 under the isomorphism defined as in Definition 9.8, where g1 (x) = 2x 14 + 2x 13 + 4x 12 + x 11 + x 9 + x 8 + 2x 7 + 2x 6 + x 5 + 4x 4 + 4x 3 + 1, g2 (x) = x 13 + x 11 + 2x 8 + 2x 7 + 4. Then the image of C ⊥F5 can be decomposed as the direct sum of the following two component codes, D1 and D2 , where (i) D1 is generated by (g1 (x) mod (f1 (x))5 , g2 (x) mod (f1 (x))5 ) and

(ii) D2 is generated by (g1 (x) mod (f2 (x))5 , g2 (x) mod (f2 (x))5 ). Notice that

g1 (x)G1 (x) + g2 (x)G2 (x) ≡ x 19 + 4x 18 + 3x 4 + 2x 3 mod (x 15 − 2) ≡ 0 mod (x 15 − 2). Therefore, Equation (9.5) in Theorem 9.11 is satisfied. Since both (f1 (x))5 and (f2 (x))5 are divisors of (x 15 − 2) over F5 , we have < (g1 (x), g2 (x)), (G1 (x), G2 (x)) >R15,2 = (g1 (x)G1 (x) + g2 (x)G2 (x)) mod (x 15 − 2) = 0, < (g1 (x) mod (f1 (x))5 , g2 (x) mod (f1 (x))5 ), (G1 (x) mod (f1 (x))5 , G2 (x) mod (f1 (x))5 ) >R1 = (g1 (x)G1 (x) + g2 (x)G2 (x)) mod (f1 (x))5 = 0, < (g1 (x) mod (f2 (x))5 , g2 (x) mod (f2 (x))5 ), (G1 (x) mod (f2 (x))5 , G2 (x) mod (f2 (x))5 ) >R2 = (g1 (x)G1 (x) + g2 (x)G2 (x)) mod (f2 (x))5 = 0.

Therefore, the decomposition of the image of C ⊥F5 satisfies Equation (9.6) in Corollary 9.12. The following example shows the decomposition of a (2, 2)-QT code of length 30 over F3 using GDFT. Example 9.26. Factorize x 15 − 2 over F3 as follows: x 15 − 2 = (x 5 + 1)3 = (x + 1)3 (x 4 + 2x 3 + x 2 + 2x + 1)3 . Let G1 (x) = (x + 1)2 (x 4 + 2x 3 + x 2 + 2x + 1),

(9.17)

Quasitwisted Codes Chapter | 9

165

and G2 (x) = (x + 1)(x 4 + 2x 3 + x 2 + 2x + 1)2 . Therefore, F3 [x] F3 [x]  F3 [x]      15 3 4 3 (x − 2) (x + 1) (x + 2x + x 2 + 2x + 1)3  (F34 + uF34 + u2 F34 ).  (F3 + uF3 + u2 F3 ) For simplicity, denote

F3 [x] (x 15 −2)

by R, (F3 + uF3 + u2 F3 ) by J1 and (F34 + uF34 +

u2 F34 ) by J2 . Set a root of x 5 + 1: β = 2. Let ξ be a fifth primitive root of unity. Since β 3−1 = 1 = ξ 5 , the map τ : Z/5Z → Z/5Z, z → 3z + 5, defines two orbits: O1 = {0} and O2 = {1, 3, 4, 2}. It is easily checked that β is the root of x + 1 while βξ, βξ 2 , βξ 3 , βξ 4 are the roots of x 4 + 2x 3 + x 2 + 2x + 1. Therefore, the orbit O1 corresponds to the polynomial x + 1 while the orbit O2 corresponds to the polynomial x 4 + 2x 3 + x 2 + 2x + 1 in (9.17). Let C be the (2, 2)-QT code of length 30 over F3 and let the generator of its corresponding R-submodule of R2 be (G1 (x), G2 (x)). Then C can be decomposed as direct sum of a code over J1 and another code over J2 . For the codeword (G1 (x), G2 (x)) ∈ C , Gˆ1 , Gˆ2 are two matrices of size 3×5 as defined in Equation (9.12), where ⎤

⎡ 0 ⎢ Gˆ1 = ⎢ ⎣ 0 2

0

0

0

0

2 + 2(2ξ ) + (2ξ )2 + 2(2ξ )3

1 + (2ξ )3

1 + 2(2ξ )2

(2ξ )3

2ξ

1 + 2(2ξ ) + (2ξ )2 + 2(2ξ )3

⎥ 1 + (2ξ ) ⎥ ⎦ 2(2ξ )2

and ⎡

0 ⎢ Gˆ2 = ⎣ 1 2

0 0

0 0

0 0

0 0

1 + (2ξ ) + (2ξ )3

1 + (2ξ ) + 2(2ξ )2

2 + 2(2ξ ) + (2ξ )2

2 + 2(2ξ ) + 2(2ξ )3

Let C1 be the J1 -linear code of length 2 with the generator (2u2 , u + 2u2 ) over J1 and let C2 be the J2 -linear code of length 2 with the generator

⎤ ⎥ ⎦.

166 Codes and Rings

((2 + 2(2ξ ) + (2ξ )2 + 2(2ξ )3 )u + ((2ξ )3 )u2 , (1 + (2ξ ) + (2ξ )3 )u2 )  over J2 . Then C  C1 C2 . The following example shows the construction of C from C1 and C2 , where C , C1 and C2 are as in Example 9.26. Example 9.27. Given the generator (2u2 , u + 2u2 ) ∈ C1 , its associated matrix x˜1 defined as in (9.15) is ⎤ ⎡ 0 0 ⎥ ⎢ x˜1 = ⎣ 0 1 ⎦ . 2 2 The matrix x˜2 associated to the generator ((2 + 2(2ξ ) + (2ξ )2 + 2(2ξ )3 )u + ((2ξ )3 )u2 , (1 + (2ξ ) + (2ξ )3 )u2 ) ∈ C2 is

 0 0

x˜2 =

0 0

0 0

0 0

2 0

2 0

1 0

2 0

0 1

0 1

0 0

1 1

T .

Then  x=

0 0

0 1

2 2

0 0

0 0

0 0

0 0

2 0

2 0

1 0

2 0

By Theorem 9.23, the matrix A is given as follows ⎡ 2 2 2 2 2 1 2 1 2 1 2 1 ⎢ 0 2 1 0 0 0 0 2 2 1 2 2 ⎢ ⎢ ⎢ 0 0 2 0 0 0 0 0 0 0 0 2 ⎢ ⎢ 1 1 1 2 1 2 2 2 1 1 1 2 ⎢ ⎢ 0 1 2 0 0 0 0 2 1 2 2 1 ⎢ ⎢ ⎢ 0 0 1 0 0 0 0 0 0 0 0 2 ⎢ ⎢ 2 2 2 1 1 1 2 2 2 1 2 1 ⎢ ⎢ 0 2 1 0 0 0 0 1 1 1 2 1 ⎢ ⎢ 0 0 2 0 0 0 0 0 0 0 0 1 ⎢ ⎢ ⎢ 1 1 1 2 1 2 1 2 1 2 2 2 ⎢ ⎢ 0 1 2 0 0 0 0 2 1 2 1 1 ⎢ ⎢ 0 0 1 0 0 0 0 0 0 0 0 2 ⎢ ⎢ ⎢ 2 2 2 1 2 2 2 1 1 1 2 2 ⎢ ⎣ 0 2 1 0 0 0 0 1 2 2 2 2 0 0 2 0 0 0 0 0 0 0 0 1

0 1

0 1

2 1 2 2 2 1 2 1 1 1 2 1 2 2 2

0 0

1 2 1 2 2 2 1 2 1 1 1 2 1 2 2

1 1

1 1 2 1 2 2 2 1 2 1 1 1 2 1 2

T

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.

Quasitwisted Codes Chapter | 9

167

Then  Ax =

1 1

1 2

0 1

0 2

0 1

1 1

1 2

0 1

0 2

0 1

0 0

0 0

0 0

0 0

0 0

T ,

whose columns are exactly the coefficients of G1 (x) and G2 (x), respectively. (G1 (x), G2 (x)) is the generator of the quasitwisted code C in the previous example.

9.2 QUASITWISTED CODES WITH CONSTACYCLIC CONSTITUENT CODES In [8], Ling and Solé viewed each quasicyclic code as a code over a polynomial ring, and extracted a description of each quasicyclic code as a direct sum of linear codes of shorter lengths over larger alphabets. These codes are called the constituent codes of the quasicyclic code in question. In [3], quasicyclic codes of length 5 and index  over Fq were obtained from a pair of codes over, respectively, Fq and Fq 4 , by a combinatorial construction called there the quintic construction. They are shown to be cyclic when the constituent codes are cyclic of odd length coprime to 5. In [6], Lim considered the same problem for quasicyclic codes of general index. In [4], Güneri and Özbudak also considered the same case. If the constituent codes of a quasicyclic code C of length m and index  are cyclic, they proved that C can be viewed as a 2D cyclic code of size m ×  over Fq . Moreover, in case that m and  are also coprime to each other, C must be equivalent to a cyclic code. In view of the analogy between cyclic and constacyclic codes on one hand and quasicyclic and quasitwisted codes on the other hand, a natural question is to characterize quasitwisted codes with constacyclic constituents. In this section, we will apply an algebraic method to solve this problem and give the conditions for a quasitwisted code with constacyclic constituents to be equivalent to a constacyclic code.

9.2.1 The λ-Circulant Set Decomposition of a λ-Constacyclic Code Throughout this chapter we require that (m, q) = (, q) = (m, ) = 1 and λ+m−1 = 1, where λ ∈ F∗q , q = p k for some positive integer k and p is a prime number. In this subsection, we require that (n, q) = 1. Definition 9.28. Let λ ∈ F∗q , and let C be a λ-constacyclic code of length n over Fq , then a λ-circulant matrix A containing the codeword (a0 , a1 , . . . , an−1 ) is

168 Codes and Rings

defined as follows: ⎛ ⎜ ⎜ ⎜ A=⎜ ⎜ ⎜ ⎝

a0 λan−1 λan−2 .. . λa1

a1 a0 λan−1 .. . λa2

a2 a1 a0 .. . λa3

... ... ... .. . ...

an−1 an−2 an−3 .. . a0

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

Note that the row vectors of A are codewords of C. Let k be the order of λ in F∗q , then a λ-circulant set A containing  i (a0 , a1 , . . . , an−1 ) is defined as A = k−1 i=0 λ A, where we identify A with the set of its rows. Remark 9.29. A can be considered as a set of kn codewords of C and A is λ-constacyclic about (a0 , a1 , . . . , an−1 ), so A is λ-constacyclic about every codeword of A . In our case, codewords repetition in A are omitted if necessary. Lemma 9.30. A λ-constacyclic code C of length n over Fq can be decomposed into a finite disjoint union of λ-circulant sets. Proof. The orbits under the action of the constashift on the codes are exactly the λ-circulant sets. Following Definition 9.28, we can prove the following lemma, which plays an important role in obtaining our results. Lemma 9.31. Let λ ∈ F∗q , and let C be a λ-constacyclic code of length n over Fq , then A is a λ-circulant matrix if and only if A = Pn diag(f (β), f (βζ ), . . . , f (βζ n−1 ))Pn−1 , where ⎛ ⎜ ⎜ ⎜ ⎜ Pn = ⎜ ⎜ ⎜ ⎝

1

1

1

...

1

β

βζ

βζ 2

...

βζ n−1

β2 .. .

(βζ )2 .. .

(βζ 2 )2 .. .

... .. .

(βζ n−1 )2 .. .

β n−1

(βζ )n−1

(βζ 2 )n−1

...

(βζ n−1 )n−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

is a Vandermonde matrix, ζ is a primitive nth root of unity, β is a primitive nth root of λ, (a0 , a1 , a2 , . . . , an−1 ) is the first row of A, and f (x) = a0 +a1 x +a2 x 2 + · · · + an−1 x n−1 .

Quasitwisted Codes Chapter | 9

169

Proof. Pn is invertible since ζ is a primitive n-th root of unity and β is a primitive nth root of λ. It is easy to check that ⎞ ⎛ f (β) f (βζ ) ... f (βζ n−1 ) ⎟ ⎜ ⎟ ⎜ βf (β) βζf (βζ ) ... βζ n−1 f (βζ n−1 ) ⎟ ⎜ APn = ⎜ ⎟ .. .. .. .. ⎟ ⎜ . . . . ⎠ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

β n−1 f (β) 1

1

...

β

βζ

βζ 2

...

β2

(βζ )2

(βζ 2 )2

.. .

.. .

.. .

... .. .

(βζ n−1 )n−1 f (βζ n−1 ) ⎞ 1 ⎟ βζ n−1 ⎟ ⎟ n−1 2 (βζ ) ⎟ ⎟ ⎟ .. ⎟ ⎠ .

β n−1

(βζ )n−1

(βζ 2 )n−1

...

(βζ n−1 )n−1

1

(βζ )n−1 f (βζ )

...

× diag(f (β), f (βζ ), . . . , f (βζ n−1 )). Equivalently, A = Pn diag(f (β), f (βζ ), . . . , f (βζ n−1 ))Pn−1 . The converse part is straightforward.

9.2.2 Quasitwisted Codes with Constacyclic Constituent Codes In this subsection, we write im = i (mod m), i = i (mod ), and we assume that the four numbers  ≥ 2, m ≥ 2, i and j are all positive integers. A linear code C is called a (λ, )-quasitwisted code of length m if for each codeword c = (a00 , a01 , . . . , a0,−1 , a10 , . . . , a1,−1 , . . . , am−1,0 , . . . , am−1,−1 ) ∈ C , we have (λam−1,0 , λam−1,1 , . . . , λam−1,−1 , a00 , . . . , a0,−1 , . . . , am−2,0 , . . . , am−2,−1 ) ∈ C ,

where λ ∈ F∗q , in the case when λ = 1, it is called a quasicyclic code, and for  = 1 we have a constacyclic code. Definition 9.32. Let C be a linear code over Fq of length m whose codewords are viewed as m ×  arrays, i.e., c ∈ C is written as ⎞ ⎛ c00 ... c0,−1 ⎜ c ... c1,−1 ⎟ 10 ⎟ ⎜ ⎟. ⎜ c=⎜ .. .. ⎟ .. ⎠ ⎝ . . . cm−1,0 . . . cm−1,−1

170 Codes and Rings

We define a row action Rλ, on the codewords as ⎛ ⎜ ⎜ Rλ, (c) = ⎜ ⎜ ⎝

λcm−1,0 c00 .. . cm−2,0

... ... .. . ...

λcm−1,−1 c0,−1 .. . cm−2,−1

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

and a column action Cλ, on the codewords as ⎛ ⎜ ⎜ Cλ, (c) = ⎜ ⎜ ⎝

λc0,−1 λc1,−1 .. . λcm−1,−1

c00 c10 .. . cm−1,0

... ... .. . ...

c0,−2 c1,−2 .. .

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

cm−1,−2

If C is closed under row action Rλ, and column action Cλ, , then we call C is a 2D λ-constacyclic code of size m × . Define the ring R:= Fq [Y ]/(Y m − λ) and recall that a linear code of length  over R is nothing but an R-submodule of R . Now consider the map ⎛

c00 c10 .. .

⎜ ⎜ c=⎜ ⎜ ⎝

cm−1,0

... ... .. . ...

 ω : Fm q →R , ⎞

c0,−1 c1,−1 .. .

⎟ ⎟ ⎟ → (c0 (Y ), c1 (Y ), . . . , c−1 (Y )), ⎟ ⎠

cm−1,−1

where cj (Y ) =

m−1 

cij Y i = c0j + c1j Y + c2j Y 2 + · · · + cm−1,j Y m−1 ∈ R

i=0

for each 0 ≤ j ≤  − 1. The mapping ω induces a one-to-one correspondence between index  quasitwisted codes of length m over Fq and linear codes of length  over R. Next, since (m, q) = 1, the polynomial Y m − λ is separable, assume the polynomial Y m − λ can be factorized into s distinct irreducible polynomials in Fq as follows: Y m − λ = g1 (Y )g2 (Y ) · · · gs (Y ).

Quasitwisted Codes Chapter | 9

171

By the Chinese Remainder Theorem, we have the following decomposition: R=

s 

Fq [Y ]/ gi (Y ).

i=1

For convenience, we denote Fq [Y ]/ gi (Y ) by Ri for 1 ≤ i ≤ s. Since the polynomials are irreducible, each of the quotients above are finite fields. It follows that s  Ri . R = i=1

Hence, any linear code C of length  over R can be decomposed as C=

s 

Ci ,

i=1

where Ci are linear codes of length  over the fields Ri . These linear codes of length  over various extensions of Fq are called the constituent codes of C. Then a (λ, )-quasitwisted code C of length m is called (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes provided that the constituent codes of C are all λ-constacyclic. In [4], Güneri and Özbudak showed that a quasicyclic code of length m and index  with cyclic constituent codes can be viewed as a 2D cyclic code. In our case, we can obtain the following proposition. Proposition 9.33. Let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes and  being coprime to m over Fq , then C can be viewed as a 2D λ-constacyclic code. Proof. For all indices i = 1, . . . , s, every codeword in the λ-constacyclic constituent code Ci can be written as an element of (Fq [y]/(gi )[x])/ x  − λ in j the form ci = −1 j =0 ci,j (y)x . By the Chinese Remainder Theorem in the ring j Fq [y]/ y m − λ, every codeword c in C can be written as c = −1 j =0 cj (y)x , where cj (y) is the Chinese Remainder Theorem image of the ci,j . Thus c is an element of Fq [x, y]/ x  − λ, y m − λ. The code C being invariant under multiplication by x and by y in the said ring can thus be regarded as a 2D λ-constacyclic code. Now consider the ring Fq [z]/ zn − λ, with n = m. Since  and m are coprime, by the Chinese Remainder Theorem on integers every integer 1 ≤ k ≤ n − 1 can be written as k = a + bm. Thus zk = x b y a , where we have set x = zm , y = z . This shows a ring isomorphism between Fq [z]/ zn − λ and Fq [x, y]/ x  − λ, y m − λ. This completes the proof.

172 Codes and Rings

A linear code C of length m is a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes if (a00 , a01 , a02 , . . . , a0,−1 , a10 , . . . , a1,−1 , . . . , am−1,0 , . . . , am−1,−1 ) ∈ C implies that (λ2 am−1,−1 , λam−1,0 , . . . , λam−1,−2 , λa0,−1 , a00 , . . . , a0,−2 , . . . , λam−2,−1 , . . . , am−2,−2 ) ∈ C by Proposition 9.33. Definition 9.34. Let λ ∈ F∗q , and let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes over Fq , and let T act linearly on the codewords of C , such that T (a00 , a01 , a02 , . . . , a0,−1 , a10 , . . . , a1,−1 , . . . , am−1,0 , . . . , am−1,−1 ) = (λ2 am−1,−1 , λam−1,0 , . . . , λam−1,−2 , λa0,−1 , a00 , . . . , a0,−2 , . . . , λam−2,−1 , am−2,0 , . . . , am−2,−2 ), then a similar λ-circulant matrix A containing the codeword c = (a00 , a01 , a02 , . . . , a0,−1 , a10 , . . . , a1,−1 , . . . , am−1,0 , . . . , am−1,−1 ) is defined as follows: A = (c, T c, T 2 c, T 3 c, . . . , T +m−1 c)T , namely, ⎛

a00 ⎜ 2 ⎜ λ am−1,−1 ⎜ ⎜ λ2 a ⎜ m−2,−2 A = ⎜ ⎜ . ⎜ . ⎜ . ⎝ λ+m a11

a01

...

a0,−1

...

am−1,0

am−1,1

...

am−1,−1

λam−1,0

...

λam−1,−2

...

λam−2,−1

am−2,0

...

am−2,−2

λ2 am−2,−1

...

λam−2,−3

...

λam−3,−2

λam−3,l−1

...

am−3,−3

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

λ+m a12

...

λ+m−1 a10

...

λ+m−1 a01

λ+m−1 a02

...

λ+m−2 a00

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

Let k be the order of λ in F∗q , then a similar λ-circulant set A  containing the codeword (a00 , a01 , . . . , a0,−1 , . . . , am−1,0 , . . . , am−1,−1 ) is defined as A  =

k−1 i=0

λi A , where we identify A with the set of its rows.

Remark 9.35. Parallel to Remark 9.29, A  can be considered as a set of km codewords of C . Codewords repetition in A  are omitted if necessary. Note that A is a m × m matrix. Similar to the proof of Lemma 9.30, we have the following corollary.

Quasitwisted Codes Chapter | 9

173

Corollary 9.36. Let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes, then the code C can be decomposed into a finite disjoint union of similar λ-circulant sets. We denote by Sn the symmetric group of n elements. The following lemma will be clear from matrix theory. Lemma 9.37. Let D1 and D2 be n × n matrices, for σ ∈ Sn , σ (D1 ) represents the action σ on coordinates of every row of D1 , σ T (D1 ) represents the action σ on coordinates of every column of D1 , which means if ⎛ ⎜ ⎜ D1 = ⎜ ⎜ ⎝

d00 d10 .. .

d01 d11 .. .

d02 d12 .. .

dn−1,0

dn−1,1

dn−1,2

... ... .. . ...

d0,n−1 d1,n−1 .. .

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

dn−1,n−1

then we have ⎛ ⎜ ⎜ σ (D1 ) = ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ σ (D1 ) = ⎜ ⎜ ⎝ T

d0,σ (0) d1,σ (0) .. .

d0,σ (1) d1,σ (1) .. .

d0,σ (2) d1,σ (2) .. .

dn−1,σ (0)

dn−1,σ (1)

dn−1,σ (2)

dσ (0),0 dσ (1),0 .. .

dσ (0),1 dσ (1),1 .. .

dσ (0),2 dσ (1),2 .. .

dσ (n−1),0

dσ (n−1),1

dσ (n−1),2

... ... .. . ... ... ... .. . ...

d0,σ (n−1) d1,σ (n−1) .. . dn−1,σ (n−1) dσ (0),n−1 dσ (1),n−1 .. .

⎞ ⎟ ⎟ ⎟, ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

dσ (n−1),n−1

and D1 D2 = σ (D1 )σ T (D2 ). We introduce the following lemma and two propositions that allow us to obtain an important result. Lemma 9.38. Let gcd(, m) = 1, 0 ≤ i ≤ m − 1, 0 ≤ j ≤ m − 1, then X=

i + j  i + j  i + 1 i + 1 i + j + 1 i + j + 1 m  + + + − − = 0, m  m  m 

where · is a ceiling function and · is a floor function. Proof. For any nonnegative integers i, j , and positive integer m, it is easy to check that

174 Codes and Rings

! ! ! i j i +j = + , if m | i, m | j or im + jm > m, m m m ! ! ! i j i +j = + − 1, if m  i, m  j and im + jm ≤ m. m m m

(9.18) (9.19)

Case 1. In the case of m | (i + 1) or m | j . j 1. If m | j , according to (9.18), we have  i+jm+1  =  i+1 m  +  m , and j j  imm+j  =  m  = m  since 0 ≤ im < m. Consequently, X= =

" im + j # m " i + j # 

+

+

" i + j #

 $i + 1% 

+

−

$i + 1%

+

m $i + j + 1%

$i + 1%





−

$i + j + 1% $i + j + 1% − m 

.

i+j +1 i+1 (i) If  | (i + 1), namely, i =  − 1 <  and X =  i +j  +  −  = j i +j j j  i +j   −   . If j = 0, then    =    =   , equivalently, X = 0. j j Otherwise, j > 0, then  ≤ i + j < 2 − 1,  i +j   =    + 1 =   , consequently, X = 0. (ii) If   (i + 1), namely, i =  − 1 (⇒ i ≤  − 2). Moreover, if j = 0, then j j i+j +1 j i+1  i +j   =    =    and    =   +   by Equation (9.18), equivalently, X = 0. If j > 0 (⇒ j ≤  − 1), then we have the following two subcases: (a)  ≤ i + j ≤ 2 − 3, then  + 1 ≤ i + j + 1 ≤ 2 − 2, according j j i+1 to Equation (9.18),  i+j+1  =  i+1   +    = 2 +    +   , and j i+1 i+1  i +j   = 1 +   , because   (i + 1) and  ≥ 2, then    =    + i +j i+j +1 i i+1 1 =    + 1. Consequently, X =    +    −    = 0. (b) i + j < , then i + j + 1 ≤ , by Equation (9.19), we have  i+j+1  = j i +j j j  i+1   +    − 1 and    =    =    − 1. Consequently, X = i+j +1 i+1  i +j   +    −    = 0.

2. If m | (i + 1), we can obtain X = 0 by the similar discussion. Case 2. In the case of m  (i + 1) and m  j . 1. If  | (i + 1) or  | j , since m and  are symmetrical in the expression of X, then it follows from Case 1. 2. If   (i + 1) and   j , namely, 0 < i ≤  − 2, 0 < im ≤ m − 2, 0 < j ≤  − 1 and 0 < jm ≤ m − 1, then we have the following four subcases: (i) i + j ≥  and im + jm ≥ m; (ii) i + j ≥  and im + jm ≤ m − 1; (iii) i + j ≤  − 1 and im + jm ≥ m; and (iv) i + j ≤  − 1 and im + jm ≤ m − 1. Parallel to the discussion of Case 1, we have X = 0.

175

Quasitwisted Codes Chapter | 9

Proposition 9.39. Let A be the similar λ-circulant matrix containing the codeword c defined in Definition 9.34, and let the j th row of A be of the form aj = (λ00 a00 , . . . , λ0,−1 a0,−1 , . . . , λim ,0 aim ,0 , . . . , λim ,i aim ,i , (j ) (j )

(j )

(j )

(j )

(j )

(j )

(j )

(j )

(j )

(j )

(j )

(j )

(j )

. . . , λim ,−1 aim ,−1 , . . . , λm−1,0 am−1,0 , . . . , λm−1,−1 am−1,−1 ), where 1 ≤ j ≤ m and λi  ,j  is a power of λ, 0 ≤ i  ≤ m − 1, 0 ≤ j  ≤  − 1, then (j )

(j +1)

(1)

(j +1)

(j +1)

aim ,i = a(i+j )m ,(i+j ) and the coefficient of a(i+j )m ,(i+j ) is λ(i+j )m ,(i+j ) = λ

im +j m

+

i +j 



.

Proof. If we fix j , by the construction of the similar λ-circulant matrix A , we know that in the (j + 1)th row of A , (j +1)

(1)

aim ,i = a(i+j )m ,(i+j ) . (1)

In other words, aim ,i represents (i + 1)th element of (im + 1)th block of

a1 . In the similar λ-circulant matrix A , aim ,i is ((i + j ) + 1)th element of (1)

(j +1)

((i + j )m + 1)th block of aj +1 , denoted by a(i+j )m ,(i+j ) , without considering (1)

(1)

(j +1)

the coefficient of aim ,i , we have aim ,i = a(i+j )m ,(i+j ) . (j +1)

(j +1)

Now, we calculate the coefficient of a(i+j )m ,(i+j ) , denoted by λ(i+j )m ,(i+j ) , (j +1)

(1)

which is a power of λ. It is clear that λ(i+j )m ,(i+j ) is obtained by aim ,i under shift in A . Firstly, we just consider the shift of the blocks of the codeword, we have the following matrix: ⎞ ⎛ a1 a2 ... ai m . . . am−2 am−1 a0 ⎜ λa a0 a1 . . . aim −1 . . . am−3 am−2 ⎟ ⎟ ⎜ m−1 ⎟ ⎜ ⎜ λam−2 λam−1 a0 . . . aim −2 . . . am−4 am−3 ⎟  A =⎜ ⎟, ⎟ ⎜ .. .. .. .. .. .. .. .. ⎟ ⎜ . . . . . . . . ⎠ ⎝ λ  a1 λ a2 λ a3 . . . λ aim +1 . . . λ am−1 λ−1 a0 where ak is a vector of length , 0 ≤ k ≤ m − 1. It is easy to check that the vector (λaim , . . . , λam−1 , a0 , . . . , aim −1 ) is the (m − im + 1)th row of A , then the coefficient of aim is λ, and the vector (λ2 aim , . . . , λ2 am−1 , λa0 , . . . , λaim −1 ) is the (2m − im + 1)th row of A , then the coefficient of aim is λ2 and so on. Thus the coefficient of aim in the (j + 1)th row of A is λ Consequently,

j +im m 

(j +1)

. Similarly, the coefficient of a(i+j )m ,(i+j ) of aim is λ

(j +1) λ(i+j )m ,(i+j )

=λ

 imm+j

i +j +  



.

j +i  

.

176 Codes and Rings

Proposition 9.40. Use the notations in Proposition 9.39. Let λ ∈ F∗q and λ+m−1 = 1, and let m−1 

bj =

λ+m−

i+1 i+1 m −  

(j +1) (j +1)

(λim ,i aim ,i )(γ ξ )i ,

i=0

where 1 ≤ j ≤ m − 1, ε is a primitive m-th root of unity, ξ ∈ {1, ε, ε 2 , . . . , εm−1 } and γ is a primitive (m)-th root of λ, then bj = (γ ξ )j

m−1 

λ+m−

i+1 i+1 m −  

(1)

aim ,i (γ ξ )i .

i=0

Proof. According to Proposition 9.39, if we fix j , since 1 ≤ i + j ≤ 2m − 2, then bj =

m−1 

λ+m−

i+1 i+1 m −  

(j +1) (j +1)

(λim ,i aim ,i )(γ ξ )i

i=0

=

m+j −1

λ+m−

(i+j )m +1 (i+j )m +1 −  m 

λ+m−

(i+j )m +1 (i+j )m +1 − + imm+j m 

i+j =j

=

m+j −1

(j +1)

(j +1)

(λ(i+j )m ,(i+j ) a(i+j )m ,(i+j ) )(γ ξ )(i+j )m +

i +j 

i+j =j

 (j +1) a(i+j )m ,(i+j ) (γ ξ )(i+j )m .

(i) If 1 ≤ i + j ≤ m − 1, we have (γ ξ )(i+j )m = (γ ξ )i+j , from Lemma 9.38, )m +1 i+1 i+1  −  (i+j )m +1  +  imm+j  +  i +j − (i+j m   = − m  −   , then λ+m−

(i+j )m +1 (i+j )m +1 − + imm+j m 

= λ+m−

i+1 i+1 m −  

+

i +j 



(γ ξ )(i+j )m

(γ ξ )i+j .

(ii) If m ≤ i + j ≤ 2m − 1, since (γ ξ )m = λ, we have λ(γ ξ )(i+j )m = )m +1 i+1  −  (i+j )m +1  +  imm+j  +  i +j (γ ξ )i+j , then − (i+j m   = − m  − i+1 +m−1 = 1, then    +  + m by Lemma 9.38, since λ λ+m−

(i+j )m +1 (i+j )m +1 − + imm+j m 

+

i +j 



(γ ξ )(i+j )m

= λ+m−

i+1 i+1 m −  ++m

= λ+m−

i+1 i+1 m −  

λ+m−1 λ(γ ξ )(i+j )m

= λ+m−

i+1 i+1 m −  

(γ ξ )i+j .

(γ ξ )(i+j )m

Quasitwisted Codes Chapter | 9

177

From Proposition 9.39 and the discussion above, we have bj =

m+j −1

λ+m−

(i+j )m +1 (i+j )m +1 − + imm+j m 

+

i+j =j

=

m+j −1

λ+m−

i+1 i+1 m −  

i+j =j

= (γ ξ )

j

m+j −1

λ+m−

= (γ ξ )j

λ+m−

 (j +1) a(i+j )m ,(i+j ) (γ ξ )(i+j )m

(1)

aim ,i (γ ξ )i+j

i+1 i+1 m −  

i+j =j m−1 

i +j 

i+1 i+1 m −  

(1)

aim ,i (γ ξ )i

(1)

aim ,i (γ ξ )i .

i=0

This completes the proof. Definition 9.41. Let λ ∈ F∗q , and let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes, for c = (a00 , a01 , a02 , . . . , a0,−1 , a10 , a11 , . . . , a1,−1 , . . . , am−1,0 , . . . , am−1,−1 ) ∈ C ,

we define a linear mapping ψ which maps a codeword of C to another codeword of C  , namely, mt

ψ : c → ψ(c) = (λ+m−2 a00 , λ+m− m − λ

+m− (−1)mt − (−1)mt  m 

λ

i+1 +m− i+1 m −  

mt  

a01 , λ+m−

2mt 2mt m −  

a02 , . . . ,

a0,−1 , . . . ,

aim ,i , . . . , λ0 am−1,−1 ),

where 0 ≤ i ≤ m − 1 and t is the multiplicative inverse of m module . For d = (d00 , d01 , . . . , d0,−1 , . . . , dim ,i , . . . , dm−1,−1 ) ∈ C  , its preimage is given by mt

ψ −1 (d) = (λ2−−m d00 , λ m +

mt  −−m

d01 , . . . , λ

i+1 i+1 m +  −−m

dim ,i ,

. . . , dm−1,−1 ) ∈ C . Then it is clear that C  is a linear code and ψ is a linear one-to-one mapping. Lemma 9.42. Let Sm represent the symmetric group of m elements and ε be a primitive m-th root of unity, and let λ ∈ F∗q and λ+m−1 = 1, let γ be a primitive (m)-th root of λ, then there exists a permutation θ ∈ Sm such that

178 Codes and Rings −1 θ (ψ(A )) = Pm Pm , where

⎛

⎞

1

1

1

...

1

γ

γε

γ ε2

...

γ ε m−1

γ2 .. .

(γ ε)2 .. .

(γ ε 2 )2 .. .

... .. .

(γ ε m−1 )2 .. .

γ m−1

(γ ε)m−1

(γ ε 2 )m−1

...

(γ ε m−1 )m−1

⎜ ⎜ ⎜ ⎜ Pm = ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

is a Vandermonde matrix,  = diag(g(γ ), g(γ ε), g(γ ε2 ), . . . , g(γ ε m−1 )) is a diagonal matrix, ψ(A ) = (ψ(a1 ), ψ(a2 ), . . . , ψ(am ))T , where aj is a row vector of A (1 ≤ j ≤ m), and g(y) =

m−1 

λ+m−

i+1 i+1 m −  

aim ,i y i

i=0

= λ+m−2 a00 + λ+m−2 a11 y + · · · + λ+m−

i+1 i+1 m −  

aim ,i y i

+ · · · + λ0 am−1,−1 y m−1 , then the similar λ-circulant matrix ψ(A ) is equivalent to a λ-circulant matrix θ (ψ(A )). Proof. Use the notations in Proposition 9.39. Let ξ ∈ {1, ε, ε 2 , . . . , ε m−1 } and  be obtained by the matrix P Pm m under certain row shift, then there exists a  T permutation θ such that θ (Pm ) = Pm . Since gcd(, m) = 1, according to the Chinese Remainder Theorem, we can establish a one-to-one correspondence between the coefficient of the term (γ ξ )i in g(γ ξ ) and (γ ξ )i denoted by aim ,i ↔ (γ ξ )i , utilize this correspondence, in a position to make the calculation of g(y)  (γ ξ ) easy. Note that the term λ is not considered in this correspondence. Let Pm  , and ψ(A )P  (γ ξ ) = (b , b , . . . , b T be any column vector of Pm 0 1 m−1 ) . Set m b0 = g(γ ξ ), by this correspondence and the elements of the first row of A ,  (γ ξ ) = (1, (γ ξ )tm , (γ ξ )2tm , . . . , (γ ξ )i , . . . , (γ ξ )m−1 ), we can determine Pm where t is the multiplicative inverse of m module . Thus θ is determined by  (γ ξ ). Pm Next, we try to calculate bj (j = 1, 2, . . . , m − 1). From Proposition 9.39, we know that in the (j + 1)th row of A , (1)

(j +1)

aim ,i = a(i+j )m ,(i+j ) ↔ (γ ξ )(i+j )m .

Quasitwisted Codes Chapter | 9

179

According to Proposition 9.40, bj =

m−1 

λ+m−

i+1 i+1 m −  

(j +1) (j +1)

(λim ,i aim ,i )(γ ξ )i

i=0

= (γ ξ )j

m−1 

λ+m−

i+1 i+1 m −  

(1)

aim ,i (γ ξ )i

(9.20)

i=0

= (γ ξ ) g(γ ξ ). j

From Equation (9.20), we have  (γ ξ ) = (b0 , b1 , . . . , bm−1 )T θ (ψ(A ))T Pm

= g(γ ξ )(1, γ ξ, (γ ξ )2 , . . . , (γ ξ )m−1 )T .

(9.21)

When ξ runs over all elements of {1, ε, ε 2 , . . . , ε m−1 }, from Equation (9.21), we have      (γ ), Pm (γ ε), Pm (γ ε 2 ), . . . , Pm (γ ε m−1 )) = ψ(A )Pm , ψ(A )(Pm

then  ψ(A )Pm ⎛

⎜ ⎜ ⎜ =⎜ ⎜ ⎝

⎞

g(γ )

g(γ ε)

...

g(γ ε m−1 )

γ g(γ ) .. .

γ εg(γ ε) .. .

... .. .

γ εm−1 g(γ ε m−1 ) .. .

γ m−1 g(γ )

(γ ε)m−1 g(γ ε)

...

(γ ε m−1 )m−1 g(γ ε m−1 )

= Pm .

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (9.22)

 = P . From Lemma 9.37, we have Thus ψ(A )Pm m   = θ (ψ(A ))θ T (Pm ) = θ (ψ(A ))Pm = Pm . ψ(A )Pm −1 , then by Lemma 9.31 we know that Consequently, θ (ψ(A )) = Pm Pm  θ (ψ(A )) is a λ-circulant matrix, thus ψ(A ) is equivalent to a λ-circulant matrix θ (ψ(A )). Moreover, the coefficients of f (x) in Lemma 9.31 are determined by the first row of λ-circulant matrix, while the coefficients of g(y) are determined by the first row of θ (ψ(A )). Therefore, the λ-circulant matrix θ (ψ(A )) is none other than the λ-circulant i+1 i+1 matrix containing (λ+m−2 a00 , λ+m−2 a11 , . . . , λ+m− m −   aim ,i , . . . , λ0 am−1,−1 ). This completes the proof.

180 Codes and Rings

In connection with the preceding discussion, we can obtain the following main result. Theorem 9.43. Let λ ∈ F∗q , λ+m−1 = 1, the order of λ be k in F∗q and let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes over Fq , then ψ(C ) is equivalent to a λ-constacyclic code C. 

Proof. From Corollary 9.36, we can write C = ∪ki=1 A  i , then ψ(C ) =   j  ∪ki=1 ψ(A  i ), where A  i = k−1 j =0 λ Ai . From Lemma 9.42, let θ be the per mutation such that θ (ψ(A1 )) is a λ-circulant matrix. According to the proof of Lemma 9.42, the permutation θ is universally applicable for the matrices ψ(Ai ), thus θ (ψ(Ai )) (i = 1, 2, . . . , k  ) are all λ-circulant matrices. So do θ (ψ(λj Ai )) (i = 1, 2, . . . , k  ; j = 0, 1, . . . , k − 1). Now we prove that θ (ψ(C )) is a linear λ-constacyclic code. For θ (ψ(c)) ∈ θ (ψ(C )), then there exists i such that θ (ψ(c)) ∈ θ (ψ(Ai )). According to the construction of λ-circulant matrix and λ-circulant set, θ (ψ(C )) is λ-constacyclic. The linearity of θ (ψ(C )) is obtained by the linearity of ψ(C ). Therefore, θ (ψ(C )) is a linear λ-constacyclic code and ψ(C ) is equivalent to a λ-constacyclic code θ (ψ(C )). Let d = (d00 , d01 , d02 , . . . , d0,−1 , d10 , . . . , d1,−1 , . . . , dm−1,0 , . . . , dm−1,−1 ) ∈ ψ(C ), by the last statement of Lemma 9.42, the equivalence of θ is given by θ (d) = (d00 , d11 , d22 , . . . , dim ,i , . . . , dm−1,−1 ) ∈ θ (ψ(C )).

The following corollary gives the inverse θ −1 of θ . Corollary 9.44. Let λ ∈ F∗q and λ+m−1 = 1, and let  and m be coprime positive integers with m coprime to q, let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes over Fq , let t denote the multiplicative inverse of m module , then ψ(C ) is equivalent to a λ-constacyclic code C, the inverse θ −1 of θ is given by e = (e0 , e1 , . . . , em−1 ) ∈ C, its preimage θ −1 (e) in ψ(C ) is given by (e(0)tm+0 , etm+0 , e2tm+0 , . . . , e(−1)tm+0 , e(−1)tm+1 , e(0)tm+1 , . . . , e(−2)tm+1 , . . . , e(−m+1)tm+(m−1) , e(−m+2)tm+(m−1) , e(−m+3)tm+(m−1) , . . . , e(−m)tm+(m−1) ).

Proof. According to Lemma 9.42, the codeword (d00 , . . . , d0,−1 , d10 , . . . , d1,−1 , . . . , dm−1,0 , . . . , dm−1,−1 ) ∈ ψ(C ) is equivalent to the codeword (d00 , d11 , d22 , . . . , dim ,i , . . . , dm−1,−1 ) ∈ θ (ψ(C )). Let

Quasitwisted Codes Chapter | 9

181

(d00 , d11 , d22 , . . . , dim ,i , . . . , dm−1,−1 ) = (y0 , y1 , y2 , . . . , yi , . . . , ym−1 ), in such a way that dim ,i = yi , where 0 ≤ i ≤ m − 1. For any di,j , write km = i, k = j ⇔ k ≡ i (mod m), k ≡ j (mod ).

(9.23)

Note that mt ≡ 1 (mod ), and 0 ≤ k ≤ m − 1, it is easy to check that k = (j − i) mt + i is a solution of the congruence equation (9.23). Therefore (d00 , d01 , d02 , . . . , d0,−1 , d10 , . . . , d1,−1 , . . . , dm−1,0 , . . . , dm−1,−1 ) = (y(0)tm+0 , ytm+0 , y2tm+0 , . . . , y(−1)tm+0 , y(−1)tm+1 , y(0)tm+1 , . . . , y(−2)tm+1 , . . . , y(−m+1)tm+(m−1) , y(−m+2)tm+(m−1) , y(−m+3)tm+(m−1) , . . . , y(−m)tm+(m−1) ).

This completes the proof. Remark 9.45. Let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes over Fq . If the mapping ψ satisfies C = ψ(C ), then C is equivalent to a λ-constacyclic code. When λ = 1, it is clear that ψ is identity, in such a way that a (1, )-quasitwisted code of length m with cyclic constituent codes is equivalent to a cyclic code, namely, a quasicyclic code of length m and index  with cyclic constituent codes is equivalent to a cyclic code, which has been proved in [4,6].

9.2.3 The Generator Polynomial of θ (ψ(C )) In this section, we make an attempt to describe the generator polynomials of C and θ (ψ(C )) over Fq . For our purpose, we give the following definition with respect to an isomorphism mapping. Definition 9.46. Let λ ∈ F∗q , and if c = (a00 , a01 , a02 , . . . , a0,−1 , a10 , . . . , a1,−1 , . . . , am−1,0 , . . . , am−1,−1 ) ∈ C , we define an isomorphism mapping φ which maps the codeword c ∈ C to the bivariate polynomial ring Fq [x, y]/ x m − λ, y  − λ: φ : c → φ(c) =

m−1  −1 

aij x i y j

i=0 j =0

= a00 + a01 y + a02 y 2 + · · · + aij x i y j + · · · + am−1,−1 x m−1 y −1 ,

where 0 ≤ i ≤ m − 1, 0 ≤ j ≤  − 1.

182 Codes and Rings

Note that φ(ψ(c)) =

m−1 

λ+m−

i+1 i+1 m −  

aim ,i x im y i

i=0 mt

= λ+m−2 a00 + λ+m− m −

mt  

a01 y + · · · + am−1,−1 x m−1 y −1 ,

where t is the multiplicative inverse of m module . Theorem 9.47. Let λ ∈ F∗q and λ+m−1 = 1, then J = φ(C ) is a principal ideal of Fq [x, y]/ x m − λ, y  − λ if and only if C is a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes. Proof. According to Proposition 9.33, C can be viewed as a 2D λ-constacyclic codes. For c = (a00 , a01 , a02 , . . . , a0,−1 , a10 , . . . , a1,−1 , . . . , am−1,0 , . . . , am−1,−1 ) ∈ C , we write J = φ(C ), namely, if φ(c) = a00 + a01 y + a02 y 2 + · · · + aij x i y j + · · · + am−1,−1 x m−1 y −1 ∈ J , then we have xφ(c) = a00 x + a01 xy + a02 xy 2 + · · · + aij x i+1 y j + · · · + λam−1,−1 y −1 ∈ J . Therefore (λam−1,0 , λam−1,1 , λam−1,2 , . . . , λam−1,−1 , a00 , . . . , a0,−1 , . . . , am−2,0 , . . . , am−2,−1 ) ∈ C

(9.24)

and yφ(c) = a00 y +a01 y 2 +a02 y 3 +· · ·+aij x i y j +1 +· · ·+λam−1,−1 x m−1 ∈J , then (λa0,−1 , a00 , a01 , . . . , a0,−2 , λa1,−1 , . . . , a1,−2 , . . . , λam−1,−1 , . . . , am−1,−2 ) ∈ C .

(9.25)

Moreover, since J is a principal ideal, x i y j φ(c) ∈ J , then x i y j φ(c)’s preimage is φ −1 (x i y j φ(c)) ∈ C .

(9.26)

Furthermore, φ(C ) satisfies Equations (9.24)–(9.26), so that C is a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes. Next, we consider the converse part. From Theorem 9.43, θ (ψ(C )) is a λ-constacyclic code, then θ (ψ(C )) is a principal ideal of Fq [z]/ zm − λ due to (m, q) = 1. Let the generator polynomial of θ (ψ(C )) be g(z) =

m−1 

dim ,i zi ,

i=0

then θ (d) = (d00 , d11 , d22 , . . . , dim ,i , . . . , dm−1,−1 ) ∈ θ (ψ(C )). According to Theorem 9.43, we have

Quasitwisted Codes Chapter | 9

183

d = (d00 , d01 , d02 , . . . , d0,−1 , d10 , . . . , d1,−1 , . . . , dm−1,0 , . . . , dm−1,−1 ) ∈ ψ(C ),

so there exists c ∈ C such that ψ(c) = d. Now we claim that φ(C ) = φ(c). Clearly, φ(c) ∈ φ(C ). On the other hand, recall that φ(C ) is an ideal of Fq [x, y]/ x m − λ, y  − λ, then

φ(c) ⊆ φ(C ).

(9.27)

It is easy to check that φ(λ2 am−1,−1 , λam−1,0 , . . . , λam−1,−2 , . . . , λam−2,−1 , am−2,0 , . . . , am−2,−2 ) = xyφ(c)

and (λ2 am−1,−1 , λam−1,0 , . . . , λam−1,−2 , λa0,−1 , . . . , a0,−2 , . . . , λam−2,−1 , . . . , am−2,−2 )

is exactly the second row of the similar λ-circulant matrix A containing c in Definition 9.34. From Lemma 9.42, θ (ψ(A )) is equivalent to ψ(A ), and θ (ψ(A )) is the λ-circulant matrix containing the codeword i+1 i+1 (λ+m−2 a00 , λ+m−2 a11 , . . . , λ+m− m −   aim ,i , . . . , λ0 am−1,−1 ) with zg(z) being its second row. Thus ψ(xyφ(c)) is equivalent to zg(z), similarly, z2 g(z) is equivalent to ψ(x 2 y 2 φ(c)), and so on. We can define a mapping  which maps the polynomial (codeword) of θ (ψ(C )) to the corresponding polynomial (codeword) of φ(c) (the corresponding codeword of c ∈ C is θ (ψ(c)) ∈ θ (ψ(C ))). Namely,  : f (z)g(z) ∈ θ (ψ(C )) → f (xy)φ(c) ∈ φ(c) ⊆ φ(C ). Next we prove the mapping  is bijective. For θ (d ) ∈ θ (ψ(C )), since θ (ψ(C )) is a principal ideal, we can write θ (d ) = f1 (z)g(z), considering the equivalence between ψ(C ) and θ (ψ(C )), we can obtain (f1 (z)g(z)) = f1 (xy)φ(c). It is clear that  is injective. Now it is sufficient to prove that x i y j φ(c) has its preimage in θ (ψ(C )). Note that x m = λ and y  = λ in Fq [x, y]/ x m − λ, y  − λ, then λk1 +k2 x i y j = x k1 m+i y k2 +j , where k1 , k2 are nonnegative integers. It is clear that the equation k1 m + i = k2  + j has integer solution (k1 , k2 ), note that λ+m−1 = 1, it is sufficient to choose the pair (k1 , k2 ) such that k1 m + i is smallest. Set k1 m + i = k2  + j = e, then x i y j φ(c) has preimage λ−k1 −k2 ze g(z) ∈ θ (C ) for some positive integer e. Thus the mapping  is bijective. Since ψ and φ are bijective,

184 Codes and Rings

|φ(C )| = |θ (ψ(C ))| = |φ(ψ(C ))| = | φ(c)|

(9.28)

Combining (9.27) and (9.28), we obtain φ(c) = φ(C ). From the proof of Theorem 9.47, we have the following corollary. Corollary 9.48. Let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes over Fq , then φ(C ) is a principal ideal of Fq [x, y]/ x m − λ, y  − λ. Similar to the case of constacyclic codes, φ(c) = a00 + a01 y + a02 y 2 + · · · + aij x i y j + · · · + am−1,−1 x m−1 y −1 is a generator polynomial of C . Namely, C is a principal ideal of Fq [x, y]/ x m − λ, y  − λ. Moreover, if C has generator polynomial c = a00 + a01 y + a02 y 2 + · · · + aij x i y j + · · · + am−1,−1 x m−1 y −1 , then θ (ψ(C )) is a λ-constacyclic code i+1 +m− i+1  − m  ai ,i zi . with the generator polynomial g(z) = m−1 m  i=0 λ The following theorem is devoted to the study on whether ψ(C ) is a principal ideal of Fq [x, y]/ x m − λ, y  − λ or not. Theorem 9.49. Let C be a (λ, )-quasitwisted code of length m with λ-constacyclic constituent codes over Fq , where λ ∈ F∗q , λ+m−1 = 1 and λ = 1. Then ψ(C ) is not a principal ideal of Fq [x, y]/ x m − λ, y  − λ. Proof. According to Corollary 9.48, C is a principal ideal of Fq [x, y]/ x m − λ, y  − λ with generator f1 (x, y), then C = f1 (x, y). Suppose that ψ(C ) is a principal ideal of Fq [x, y]/ x m − λ, y  − λ, then ψ(f1 (x, y)) ∈ ψ(C ), thus

ψ(f1 (x, y)) ⊂ ψ(C ).

(9.29)

Let h(x, y) ∈ Fq [x, y]/ x m − λ, y  − λ, we can define a mapping ϕ from

f1 (x, y) to ψ(f1 (x, y)): ϕ : h(x, y)f1 (x, y) → ϕ(h(x, y)f1 (x, y)) = h(x, y)ψ(f1 (x, y)). Suppose that h1 (x, y)ψ(f1 (x, y)) = h2 (x, y)ψ(f1 (x, y)), then h1 (x, y) = h2 (x, y). Consequently, h1 (x, y)f1 (x, y) = h2 (x, y)f1 (x, y), in such a way that ϕ is injective. It is clear that ϕ is surjective. Thus we have |C | = | f1 (x, y)| = | ψ(f1 (x, y))|.

(9.30)

Since ψ is one-to-one, then |C | = |ψ(C )|.

(9.31)

185

Quasitwisted Codes Chapter | 9

Combining (9.29)–(9.31), we have ψ(C ) = ψ(f1 (x, y)). Let h(x, y) ∈ Fq [x, y]/ x m − λ, y  − λ. Since h(x, y)f1 (x, y) ∈ C , then ψ(h(x, y)f1 (x, y)) ∈ ψ(C ). Thus (9.32)

ψ(f1 (x, y))|ψ(h(x, y)f1 (x, y)).

In fact, ψ(c) can be considered as a vector multiplication, then Equation (9.32) doesn’t work, which is a contradiction. This completes the proof. Remark 9.50. From Theorem 9.49, we know that if λ = 1, then C = ψ(C ), namely, ψ is identity if and only if λ = 1. Consequently, this method fails in finding the equivalence between C and a constacyclic code if the equivalence does exist. However, this method may be developed, for example, we can use this method to investigate the (λk1 , )-quasitwisted code of length m with λk2 -constacyclic codes. Analogously, we can define the linear mapping ψk1 ,k2 , and then discuss the case when ψk1 ,k2 = id.

9.2.4 Examples In this section, we mainly give some examples to illustrate the obtained results in the previous sections. Example 9.51. Let C be a (λ, 3)-quasitwisted code of length 12 with λ-constacyclic constituent codes over Fq , where λ ∈ F∗q , (q, 6) = 1 and λ+m−1 = λ6 = 1. Let ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜  ⎜ A =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

a00

a01

a02

a10

a11

a12

a20

a21

a22

a30

a31

a32

λ2 a32

λa30

λa31

λa02

a00

a01

λa12

a10

a11

λa22

a20

a21

λ2 a21

λ2 a22

λa20

λ2 a31

λ2 a32

λa30

λa01

λa02

a00

λa11

λa12

a10

λ2 a10

λ2 a11

λ2 a12

λ2 a20

λ2 a21

λ2 a22

λ2 a30

λ2 a31

λ2 a32

λa00

λa01

λa02

λ3 a02

λ2 a00

λ2 a01

λ3 a12

λ2 a10

λ2 a11

λ3 a22

λ2 a20

λ2 a21

λ3 a32

λ2 a30

λ2 a31

λ4 a31

λ4 a32

λ3 a30

λ3 a01

λ3 a02

λ2 a00

λ3 a11

λ3 a12

λ2 a10

λ3 a21

λ3 a22

λ2 a20

λ4 a20

λ4 a21

λ4 a22

λ4 a30

λ4 a31

λ4 a32

λ3 a00

λ3 a01

λ3 a02

λ3 a10

λ3 a11

λ3 a12

λ5 a12

λ4 a10

λ4 a11

λ5 a22

λ4 a20

λ4 a21

λ5 a32

λ4 a30

λ4 a31

λ4 a02

λ3 a00

λ3 a01

λ5 a01

λ5 a02

λ4 a00

λ5 a11

λ5 a12

λ4 a10

λ5 a21

λ5 a22

λ4 a20

λ5 a31

λ5 a32

λ4 a30

λ6 a30

λ6 a31

λ6 a32

λ5 a00

λ5 a01

λ5 a02

λ5 a10

λ5 a11

λ5 a12

λ5 a20

λ5 a21

λ5 a22

λ7 a22

λ6 a20

λ6 a21

λ7 a32

λ6 a30

λ6 a31

λ6 a02

λ5 a00

λ5 a01

λ6 a12

λ5 a10

λ5 a11

λ7 a11

λ7 a12

λ6 a10

λ7 a21

λ7 a22

λ6 a20

λ7 a31

λ7 a32

λ6 a30

λ6 a01

λ6 a02

λ5 a00

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

be a similar λ-circulant matrix of C , where  = 3, m = 4, g(y) = λ5 a00 + λ5 a11 y + λ5 a22 y 2 + λ4 a30 y 3 + λ3 a01 y 4 + λ3 a12 y 5 + λ2 a20 y 6 + λ2 a31 y 7 + λ3 a02 y 8 + a10 y 9 + a21 y 10 + a32 y 11 . We have ψ(a00 , a01 , a02 , a10 , a11 , a12 , a20 , a21 , a22 , a30 , a31 , a32 ) = (λ5 a00 , λ3 a01 , λa02 , a10 , λ5 a11 , λ3 a12 , λ2 a20 , a21 , λ5 a22 , λ4 a30 , λ2 a31 , a32 ) by Definition 9.41, then ψ(A ) is given by

186 Codes and Rings ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

λ5 a00

λ3 a01

λa02

a10

λ5 a11

λ3 a12

λ2 a20

λa32

λ4 a30

λ2 a31

λa02

λ5 a00

λ3 a01

λ3 a12

λa21

λ5 a22

λ2 a20

λ2 a31

λa32

λ4 a30

λa10

λ5 a11

λ3 a12

λ2 a20

λa21

λ5 a22

λ2 a02

λ5 a00

λ3 a01

λ3 a12

λa10

λ3 a31

λa32

λ4 a30

λ3 a01

λ2 a02

λ3 a20

λa21

λ5 a22

λ4 a30

λ3 a31

λ4 a12

λa10

λ5 a11

λ5 a22

λ3 a20

λ4 a01

λ2 a02

λ5 a00

λ5 a11

λ4 a12

λ5 a30

λ3 a31

λa32

λ5 a00

a22

λ3 a20

λa21

a11

λ4 a12

λa10

⎞

a21

λ5 a22

λ4 a30

λ2 a31

a32

a10

λ5 a11

λ5 a22

λ2 a20

a21

λ3 a01

λa02

λ5 a00

λ5 a11

λ3 a12

a10

λ4 a30

λ2 a31

λa32

λ5 a00

λ3 a01

λa02

λ5 a11

λ5 a22

λ2 a20

λa21

λa32

λ4 a30

λ2 a31

λ5 a00

λ5 a11

λ3 a12

λa10

λa21

λ4 a22

λ2 a20

λa32

λ5 a00

λ3 a01

λ2 a02

λa10

λ5 a11

λ3 a12

λa21

λa32

λ4 a30

λ3 a31

λ2 a02

λ5 a00

λ3 a01

λa10

λa21

λ5 a22

λ3 a20

λ3 a31

λa32

λ4 a30

λ4 a01

λ2 a02

λa10

λ5 a11

λ4 a12

λ3 a20

λa21

λ5 a22

λa32

λ5 a30

λ3 a31

λ2 a02

λ5 a00

λ4 a01

λ4 a12

λa10

λ5 a11

λa21

a22

λ3 a20

λ3 a31

λa32

λ5 a30

λ4 a01

λ2 a02

λ5 a00

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

According to the proof of Lemma 9.42, the correspondence is a00 ↔ 1, a11 ↔ γ ξ , a22 ↔ (γ ξ )2 , a30 ↔ (γ ξ )3 , a01 ↔ (γ ξ )4 , a12 ↔ (γ ξ )5 , a20 ↔ (γ ξ )6 , a31 ↔ (γ ξ )7 , a02 ↔ (γ ξ )8 , a10 ↔ (γ ξ )9 , a21 ↔ (γ ξ )10 , a32 ↔ (γ ξ )11 , where γ is a primitive 12th root of λ, ξ ∈ {1, ε, ε 2 , . . . , ε 11 }, and ε is a primitive 12th root of unity. Write  ψ(A )P3×4 (ε) = (b0 , b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 , b9 , b10 , b11 )T .

Set b0 = g(γ ξ ), then we have  P3×4 (γ ξ ) = (1, (γ ξ )4 , (γ ξ )8 , (γ ξ )9 , (γ ξ ), (γ ξ )5 , (γ ξ )6 , (γ ξ )10 , (γ ξ )2 ,

(γ ξ )3 , (γ ξ )7 , (γ ξ )11 )T . Then  ψ(A )P3×4 (γ ξ ) = g(γ ξ )(1, γ ξ, (γ ξ )2 , (γ ξ )3 , (γ ξ )4 , (γ ξ )5 , (γ ξ )6 , (γ ξ )7 ,

(γ ξ )8 , (γ ξ )9 , (γ ξ )10 , (γ ξ )11 )T . Therefore θ (ψ(A )) is given by the matrix below: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ3 a12

λ2 a20

λ2 a31

λa02

a10

a21

a32

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ3 a12

λ2 a20

λ2 a31

λa02

a10

a21

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ3 a12

λ2 a20

λ2 a31

λa02

a10

λa10

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ3 a12

λ2 a20

λ2 a31

λa02

λ2 a02

λa10

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ3 a12

λ2 a20

λ2 a31

λ3 a31

λ2 a02

λa10

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ3 a12

λ2 a20

λ3 a20

λ3 a31

λ2 a02

λa10

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ3 a12

λ4 a12

λ3 a20

λ3 a31

λ2 a02

λa10

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ3 a01

λ4 a01

λ4 a12

λ3 a20

λ3 a31

λ2 a02

λa10

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

λ4 a30

λ5 a30

λ4 a01

λ4 a12

λ3 a20

λ3 a31

λ2 a02

λa10

λa21

λa32

λ5 a00

λ5 a11

λ5 a22

a22

λ5 a30

λ4 a01

λ4 a12

λ3 a20

λ3 a31

λ2 a02

λa10

λa21

λa32

λ5 a00

λ5 a11

a11

a22

λ5 a30

λ4 a01

λ4 a12

λ3 a20

λ3 a31

λ2 a02

λa10

λa21

λa32

λ5 a00

and the equivalence is given by θ = (2 5)(3 9)(4 10)(8 11) in S12 .

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Quasitwisted Codes Chapter | 9

187

Example 9.52. Let C be a (−1, 2)-quasitwisted code of length 6 with negacyclic constituent codes over F5 , where  = 2, m = 3, then (−1)2+3−1 = 1 and the generator polynomial of φ(C ) is φ(c) = −1 − xy + x 2 (−1, 0, 0, −1, 1, 0) ∈ F5 [x, y]/ x 3 + 1, y 2 + 1, where the codeword c = (−1, 0, 0, −1, 1, 0) is the corresponding polynomial −1 − xy + x 2 by Definition 9.46, and ψ(a00 , a01 , a10 , a11 , a20 , a21 ) = (−a00 , −a01 , a10 , −a11 , a20 , a21 ) by Definition 9.41. Equivalently, φ(C ) = φ(c), then from Corollary 9.48, θ (ψ(C )) = 1 + z + z2 (111000) ∈ F5 [z]/ z6 + 1. Set  : 1 + z + z2  → −1 − xy + x 2 . From the mapping , we have 1 → 1, z → xy, z2 → x 2 y 2 = −x 2 , z3 → x 3 y 3 = y, z4 → x 4 y 4 = −x, z5 → x 5 y 5 = −x 2 y. In more details: φ(c) = −1 − xy + x 2 (−1, 0, 0, −1, 1, 0) ⇔ g(z) = 1 + z + z2 (1, 1, 1, 0, 0, 0), xyφ(c) = −y − xy + x 2 (0, −1, 0, −1, 1, 0) ⇔ zg(z) = z + z2 + z3 (0, 1, 1, 1, 0, 0), −x 2 φ(c) = x − y + x 2 (0, −1, 1, 0, 1, 0) ⇔ z2 g(z) = z2 + z3 + z4 (0, 0, 1, 1, 1, 0), yφ(c) = −y + x + x 2 y (0, −1, 1, 0, 0, 1) ⇔ z3 g(z) = z3 + z4 + z5 (0, 0, 0, 1, 1, 1), −xφ(c) = x + x 2 y + 1 (1, 0, 1, 0, 0, 1) ⇔ z4 g(z) = −1 + z4 + z5 (−1, 0, 0, 0, 1, 1), −x yφ(c) = 1 + xy + x 2 y (1, 0, 0, 1, 0, 1) ⇔ z5 g(z) = −1 − z + z5 (−1, −1, 0, 0, 0, 1), 2

and f (z)g(z) → f (xy)φ(c) is given by the linearity of C and θ (ψ(C )). Then ψ(φ(c)) = (1, 0, 0, 1, 1, 0) ⇔ g(z) = 1 + z + z2 (1, 1, 1, 0, 0, 0), ψ(xyφ(c)) = (0, 1, 0, 1, 1, 0) ⇔ zg(z) = z + z2 + z3 (0, 1, 1, 1, 0, 0), ψ(−x 2 φ(c)) = (0, 1, 1, 0, 1, 0) ⇔ z2 g(z) = z2 + z3 + z4 (0, 0, 1, 1, 1, 0), ψ(yφ(c)) = (0, 1, 1, 0, 0, 1) ⇔ z3 g(z) = z3 + z4 + z5 (0, 0, 0, 1, 1, 1), ψ(−xφ(c)) = (−1, 0, 1, 0, 0, 1) ⇔ z4 g(z) = −1 + z4 + z5 (−1, 0, 0, 0, 1, 1), ψ(−x 2 yφ(c)) = (−1, 0, 0, −1, 0, 1) ⇔ z5 g(z) = −1 − z + z5 (−1, −1, 0, 0, 0, 1),

and the equivalence is given by θ = (24)(35) in S6 .

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