Constacyclic Codes over Finite Fields

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1 Constacyclic Codes over Finite Fields Bocong Chen, Yun Fan, Liren Lin, Hongwei Liu arxiv: v1 [cs.it] 3 Jan 2013 School of Mathematics and Statistics, Central China Normal University Wuhan, Hubei, , China Abstract An equivalence relation called isometry is introduced to classify constacyclic codes over a finite field; the polynomial generators of constacyclic codes of length l t p s are characterized, where p is the characteristic of the finite field and l is a prime different from p. Keywords: finite field, constacyclic code, isometry, polynomial generator Mathematics Subject Classification: 94B05; 94B15 1 Introduction Constacyclic codes constitute a remarkable generalization of cyclic codes, hence form an important class of linear codes in the coding theory. And, constacyclic codes also have practical applications as they can be encoded with shift registers. In [4], for any positive integer a and any odd integer n, Blackford used the discrete Fourier transform to show that Z 4 [X]/ X 2an + 1 is a principal ideal ring, where Z 4 denotes the residue ring of integers modulo 4, and to establish a concatenated structure of negacyclic codes of length 2 a n over Z 4. In [1] Abualrub and Oehmke classified the cyclic codes of length 2 k over Z 4 by their generators. Generalizing the result of [1], Dougherty and Ling in [12] classified the cyclic codes of length 2 k over the Galois ring GR(4,m). Let F q be a finite field with q = p m elements where p is a prime, and let λ F q where F q denotes the multiplicative group consisting of all non-zero elements of F q. Any λ-constacyclic code C of length n over F q is identified with an ideal of the quotient algebra F q [X]/ X n λ where X n λ denotes the ideal generated by X n λ of the polynomial algebra F q [X], hence C is generated by a factor polynomial of X n λ, called the polynomial generator of the λ-constacyclic code C. In order to obtain all λ-constacyclic codes of address: b c chen@yahoo.com.cn (B. C. Chen), yunfan02@yahoo.com.cn (Y. Fan), xiaomi @126.com (L. R. Lin), h w liu@yahoo.com.cn (H. Liu). 1

2 length n over F q, we need to determine all the irreducible factors of X n λ over F q. It is remarkable that, though all irreducible binomials over F q have been explicitly characterized by Serret early in 1866 (e.g. see [16, Theorem 3.75] or [19, Theorem 10.7]), no effective method were found to characterize the irreducible factors of X n λ over F q so far. It is a challenge to determine explicitly the polynomial generators of all constacyclic codes over finite fields. It iswellknownthat X n λisafactorofx N 1forasuitable integern, and the irreducible factors of X N 1 over F q with q = p m as above can be described bytheq-cyclotomiccosets. Recently, assumingthatpisoddandtheorderofλin the multiplicative groupf q is a powerof 2, Bakshi and Raka in [2] described the polynomial generators of λ-constacyclic codes of length 2 t over F q by means of recognizing the q-cyclotomic cosets which are corresponding to the irreducible factors of X 2t λ. In the same paper [2], Bakshi and Raka determined the polynomial generators of all the λ-constacyclic codes of length 2 t p s over F q, q = p m, foranynonzeroλin F q. Almost the sametime but inanotherapproach, assuming that p is odd, Dinh in [11] determined the polynomial generators of all constacyclic codes of length 2p s over F q in a very explicit form: the irreducible factors of the polynomial generators are all binomials of degree 1 or 2. In this paper, we are concerned with the constacyclic codes of length l t p s overf q, whereq = p m asbeforeandlisaprimedifferentfromp. Weintroducea concept isometry for the non-zeroelements of F q to classify constacyclic codes over F q such that the constacyclic codes belonging to the same isometry class have the same distance structures and the same algebraic structures. Then we characterize in an explicit way the polynomial generators of constacyclic codes of length l t p s over F q according to the isometry classes. It is notable that, except for the constacyclic codes which are isometric to cyclic codes, the irreducible factors of the polynomial generator of any constacyclic code of length l t p s over F q are either all binomials or all trinomials. The plan of this paper is as follows. The necessary notations and some known results to be used are provided in Section 2. In Section 3, we introduce precisely the concept of isometry, which is an equivalence relation on F q; and some necessary and sufficient conditions for any two elements of F q isometric to each other are established; as a consequence, the constacyclic codes isometric to cyclic codes are described. In Section 4, we classify the constacyclic codes of length l t p s over F q into isometry classes, characterize explicitly the polynomial generators of the constacyclic codes of each isometry class, and derive some consequences, including the main result of [11]. In Section 5, with the help of the GAP ([13]), the polynomial generators of all constacyclic codes of length 6 over F 2 4, all constacyclic codes of length 175 over F 5 2 and all constacyclic codes of length 20 over F 5 2 are computed. 2

3 2 Preliminaries Throughout this paper F q denotes a finite field with q elements where q = p m is a power of a prime p. Let Fq denote the multiplicative group of F q consisting of all non-zero elements of F q ; and for β Fq, let ord(β) denote the order of β in the group Fq; then ord(β) is a divisor of q 1, and β is called a primitive ord(β)-th root of unity. It is well-known that Fq is a cyclic group of order q 1, i.e. Fq is generated by a primitive (q 1)-th root ξ of unity, we denote it by Fq = ξ. For any integer k, it is known that ord(ξk q 1 ) = gcd(k,q 1), where gcd(k,q 1) denotes the greatest common divisor of k and q 1. Assume that n is a positive integer and λ is a non-zero element of F q. A linearcode C of length n overf q is said to be λ-constacyclic if for any code word (c 0,c 1,,c n 1 ) C we have that (λc n 1,c 0,c 1,,c n 2 ) C. We denote by F q [X], the polynomial algebra over F q, and denote by X n λ, the ideal of F q [X]generatedbyX n λ. Anyelement ofthe quotientalgebraf q [X]/ X n λ is uniquely represented by a polynomial a 0 +a 1 X + +a n 1 X n 1 of degree less than n, hence is identified with a word (a 0,a 1,,a n 1 ) of length n over F q ; so we have the corresponding Hamming weight and the Hamming distance on the algebra F q [X]/ X n λ. In this way, any λ-constacyclic code C of length n over F q is identified with exactly one ideal of the quotient algebra F q [X]/ X n λ, which is generated by a divisor g(x) of X n λ, and the divisor g(x) is determined by C uniquely up to a scale; in that case, g(x) is called a polynomial generator of C and write it as C = g(x). Specifically, the irreducible factorization of X n λ in F q [X] determines all λ-constacyclic codes of length n over F q. Note that the 1-constacyclic codes are just the usual cyclic codes, and there is a lot of literature to deal with the cyclic codes. In particular, the irreducible factorization of X n 1 in F q [X] can be described as follows. As usual, we adopt the notations: k n means that the integer k divides n; and, for a prime integer l, l e n means that l e n but l e+1 n. Remark 2.1. Assume that n = n p s with s 0 and p n. For an integer r with 0 r n 1, the q-cyclotomic coset of r modulo n is defined by C r = {r q j (mod n ) j = 0,1, }. A subset {r 1,r 2,,r ρ } of {0,1,,n 1} is called a complete set of representatives of all q-cyclotomic cosets modulo n if C r1,c r2,,c rρ are distinct and ρ i=1 C r i = {0,1,,n 1}. Take η to be a primitive n -th root of unity (maybe in an extension of F q ), and denote by M η (X), the minimal polynomial of η over F q. It is well-known that (e.g. see [15, Theorem 4.1.1]): X n 1 = M η r 1(X)M η r 2(X) M η rρ(x) (2.1) with M η r i(x) = j C ri (X η j ), i = 1,,ρ, 3

4 all being irreducible in F q [X], hence X n 1 = (X n 1) ps = M η r 1(X) ps M η r 2(X) ps M η rρ(x) ps (2.2) is the irreducible decomposition of X n 1 in F q [X]. In a very special case the irreducible factorization of X n λ in F q [X] has been characterized precisely, we quote it as the following remark. Remark 2.2. Assume that q 3 (mod 4) (in particular, q is a power of an odd prime), equivalently, 2 (q 1). Then X 2t +1 is factorized into irreducible polynomials over F q in [5, Theorem 1]. We should mention that, though [5, Theorem 1] is proved for a prime p with p 3 (mod 4), one can check in the same way as in [5] that it also holds for the present case when q is a power of a prime and q 3 (mod 4). We reformulate the result as follows. Note that 4 (q+1)in the present case, hence there is an integer e 2 such that 2 e (q+1). Set H 1 = {0}; recursively define { } H i = ±( h+1)q h Hi 1, for i = 2,3,,e 1; and set { } H e = ±( h 1)q h H e 1 = H e+1 = H e+2 =. Let t 1. Set b = t and c = 0 if 1 t e 1; while set b = e and c = 1 if t e. Then (see [5, Theorem 1] or [19, Theorem 10.13]): X 2t +1 = ( X 2 t b+1 2hX 2t b +( 1) c) (2.3) h H t with all the factors in the right hand side being irreducible over F q. Return to our general case. As we mentioned before, the irreducible nonlinear binomials over F q have been determined by Serret early in 1866 (see [16, Theorem 3.75] or [19, Theorem 10.7]), we restate it as a remark for later quotations. Remark 2.3. Assume that n 2. For any a Fq with ord(a) = k, the binomial X n a is irreducible over F q if and only if both the following two conditions are satisfied: (i) Every prime divisor of n divides k, but does not divide (q 1)/k; (ii) If 4 n, then 4 (q 1). 3 Isometries between constacyclic codes Let F q be a finite field of order q = p m and F q = ξ as before, where ξ is a primitive (q 1)-th root of unity. Let n be a positive integer. 4

5 Generalizing the usual equivalence between codes, we consider a kind of equivalences between the λ-constacyclic codes and the µ-constacyclic codes which preserve the algebraic structures of the constacyclic codes. Definition 3.1. Let λ,µ F q. We say that an F q-algebra isomorphism ϕ : F q [X]/ X n µ F q [X]/ X n λ is an isometry if it preserves the Hamming distances on the algebras, i.e. d H ( ϕ(a),ϕ(a ) ) = d H (a,a ), a,a F q [X]/ X n µ. And, if there is an isometry between F q [X]/ X n λ and F q [X]/ X n µ, then we say that λ is n-isometric to µ in F q, and denote it λ = n µ. Obviously, the n-isometry = n is anequivalence relationon F q, hence F q is partitioned into n-isometry classes. If λ = n µ, then all the λ-constacyclic codes oflength n are one to one correspondingto all the µ-constacycliccodes of length n such that the corresponding constacyclic codes have the same dimension and the same distance distribution, specifically, have the same minimum distance; at that case we say that, for convenience, the λ-constacyclic codes of length n are isometric to the µ-constacyclic codes of length n. So, it is enough to study the n-isometry classes of constacyclic codes. Theorem 3.2. For any λ,µ Fq, the following three statements are equivalent to each other: (i) λ = n µ. (ii) λ,ξ n = µ,ξ n, where λ,ξ n denotes the subgroup of F q generated by λ and ξ n. (iii) There is a positive integer k < n with gcd(k,n) = 1 and an element a F q such that an λ = µ k and the following map ϕ a : F q [X]/ X n µ k F q [X]/ X n λ, (3.1) which maps any element f(x)+ X n µ k of F q [X]/ X n µ k to the element f(ax)+ X n λ of F q [X]/ X n λ, is an isometry. In particular, the number of n-isometry classes of Fq is equal to the number of positive divisors of gcd(n,q 1). Proof. (i) (ii). By (i) we have an isometry ϕ between the algebras: ϕ : F q [X]/ X n µ F q [X]/ X n λ. Since ϕ preserves the Hamming distance, it must map X of weight 1 of the algebraf q [X]/ X n µ toanelement ofthe algebraf q [X]/ X n λ ofweight1, so there is an element b F q and an integer j with 0 j < n such that ϕ(x) = bx j. (3.2) 5

6 Consider ϕ(x i ) = (bx j ) i = b i X ji (mod X n λ) for i = 0,1,,n 1; since ϕ is a bijection, we see that any index e with 0 e n 1 must appear in the following sequence: ji (mod n), i = 0,1,,n 1; hence j (mod n) must be invertible, i.e. 0 < j < n and gcd(j,n) = 1. Note that X n = λ (mod X n λ); further, note that ϕ is an algebra isomorphism and µ F q, we see that ϕ(µ) = µ, and can make the following calculation in F q [X]/ X n λ (or equivalently, modulo X n λ): µ = ϕ(µ) = ϕ(x n ) = ϕ(x) n = (bx j ) n = b n X jn = b n λ j ; (3.3) i.e. as elements of F q we have µ = λ j b n. Obviously, ξ n = {a n a F q}. We have µ λ,ξ n, and hence µ,ξ n λ,ξ n. On the other hand, since gcd(j,n) = 1, there are integers k,h such that jk +nh = 1; so µ k = λ jk b nk = λ jk+nh λ nh b nk = λ(λ h b k ) n ; i.e. λ = µ k (λ h b k ) n µ,ξ n ; and we have that λ,ξ n µ,ξ n. Thus, we get the desired conclusion: λ,ξ n = µ,ξ n. (ii) (iii). Denote d = gcd(n,q 1). Then the subgroup ξ n = ξ d, and the quotient group F q/ ξ n = F q/ ξ d = ξ / ξ d is a cyclic group of order d. From the statement (ii) we have that λ,ξ n / ξ d = µ,ξ n / ξ d ; which implies that, in the cyclic group F q / ξd of order d, λ and µ generate the one and the same subgroup, in particular, they have the same order in the quotient group F q/ ξ d. Thus there are integers k,h such that λ = µ k ξ dh and gcd(k,d) = 1. Since d n, it is known that the natural map Z n Z d, z (mod n) z (mod d), is a surjective homomorphism, where Z n denotes the multiplicative group consisting of all reduced residue classes modulo n. We can take a positive integer k < n with gcd(k,n) = 1 and k k (mod d). Then there is an integer h such that k = k +dh. So λ = µ k ξ dh = µ k+dh ξ dh = µ k (µ h ξ h ) d. As (µ h ξ h ) d ξ d = ξ n, we have an a F q such that (µ h ξ h ) d = a n. In a word, we have an integer k coprime to n and an a F q such that an λ = µ k. Now we define an algebra homomorphism: ˆϕ a : F q [X] F q [X]/ X n λ, 6

7 by mapping f(x) F q [X] to ˆϕ a ( f(x) ) = f(ax) (mod X n λ); since a is non-zero, ˆϕ a is obviously surjective. Noting that X n = λ (mod X n λ), we have ˆϕ a (X n µ k ) = (ax) n µ k = a n X n µ k = a n λ µ k = 0 (mod X n λ). So the surjective algebra homomorphism ˆϕ a induces an algebra isomorphism ϕ a : F q [X]/ X n µ k F q [X]/ X n λ, which maps any element f(x)+ X n µ k of F q [X]/ X n µ k to the element f(ax)+ X n λ of F q [X]/ X n λ ; since ϕ a maps any element X i of weight 1 to an element a i X i of weight 1, the algebra isomorphism ϕ a preserveshamming distances of the algebras. we are done for the statement (iii). (iii) (i). Since the map (3.1) in the statement (iii) is an algebra isomorphism, we have that 0 = ϕ a (X n µ k ) = (ax) n µ k = a n λ µ k (mod X n λ); that is, λa n = µ k. By (iii) it is assumed that gcd(k,n) = 1, i.e. there are integers j,h such that kj +nh = 1, which also implies that gcd(j,n) = 1; so µ = µ kj+nh = (µ k ) j µ nh = (λa n ) j µ hn = λ j (a j µ h ) n. Set b = a j µ h, then b F q and b n λ j = µ. Since F q [X] is a free F q -algebra with X as a free generator, by mapping X to bx j, we can define an algebra homomorphism: ˆϕ : F q [X] F q [X]/ X n λ, which maps any f(x) F q [X] to ˆϕ ( f(x) ) = f(bx j ) (mod X n λ). Since j is coprime to n, the following ˆϕ(X i ) = b i X ji (mod X n λ), i = 0,1,,n 1, form a basis of the algebra F q [X]/ X n λ ; so ˆϕ is a surjective algebra homomorphism. Further, we have ˆϕ(X n µ) = (bx j ) n µ = b n X nj µ = b n λ j µ = 0 (mod X n λ). Thus the surjective algebra homomorphism ˆϕ induces an algebra isomorphism: ϕ : F q [X]/ X n µ F q [X]/ X n λ, which maps any element f(x) + X n µ of F q [X]/ X n µ to the element f(bx j )+ X n λ of F q [X]/ X n λ ; in particular, ϕ maps any element X i of weight 1 to an element b i X ji (mod X n λ) of weight 1, hence ϕ preserves the Hamming distances. That is, (i) holds. Finally, by the equivalence of (i) and (ii), the number of the n-isometry classes of Fq is equal to the number of the subgroups of the quotient group 7

8 F q / ξd where d = gcd(n,q 1). The quotient F q / ξd is a cyclic group of order d, so, for any divisor d d it has a unique subgroup of order d. Then the number of the subgroups of F q / ξd is equal to the number of the positive divisors of d. In conclusion, the number of the n-isometry classes of F q is equal to the number of the positive divisors of gcd(n,q 1). Remark 3.3. Though the statement (i) of Theorem 3.2 states that there is an isometry ϕ : F q [X]/ X n µ F q [X]/ X n λ, the statement (iii) of Theorem 3.2 exhibits a specific isometry ϕ a such that ϕ a (X) = ax, which outperforms ϕ in (3.2) and provides an easy way to connect the polynomial generators of the λ-constacyclic codes with those of the µ k -constacyclic codes. In particular, taking µ = 1, we see that λ = n 1 implies that there is an isometry ϕ a : F q [X]/ X n 1 F q [X]/ X n λ such that ϕ(x) = ax. Thus for the constacyclic codes n-isometric to the cyclic codes, we have the following consequence which is closely related to [14, Lemma 3.1]. Corollary 3.4. Let n be a positive integer, and λ Fq. The λ-constacyclic codes of length n are isometric to the cyclic codes of length n if and only if a n λ = 1 for an element a Fq ; further, in that case the map ϕ a : F q [X]/ X n 1 F q [X]/ X n λ, (3.4) which maps f(x) to f(ax), is an isometry, and X n λ = λ M η r 1(aX) ps M η r 2(aX) ps M η rρ(ax) ps (3.5) is an irreducible factorization of X n λ in F q [X], where n = n p s with s 0 and p n, M η i(x) and {r 1,,r ρ } are defined in the formula (2.2); in particular, any λ-constacyclic code C has a polynomial generator as follows: ρ M η r i(ax) ei, 0 e i p s, i = 1,,ρ. (3.6) i=1 Proof. By Theorem 3.2, λ =n 1 if and only if λ,ξ n = 1,ξ n = ξ n ; i.e. λ = n 1 if and only if λ ξ n. However, ξ n = {a n a F q }; so λ = n 1 if and only if λ = b n for an element b F q. Assume that it is the case, i.e. a n λ = 1. By the statement (iii) of Theorem 3.2, the map (3.4) is an isometry between the algebras. And, as in the formula (2.2), we have the irreducible decomposition of X n 1 in F q [X]: X n 1 = M η r 1(X) ps M η r 2(X) ps M η rρ(x) ps ; hence the following is an irreducible decomposition of (ax) n 1 in F q [X]: (ax) n 1 = M η r 1(aX) ps M η r 2(aX) ps M η rρ(ax) ps. However, since a n = λ 1, we have that (ax) n = a n X n = λ 1 X n ; thus we get the irreducible decomposition of X n λ in F q [X] in the formula (3.5). Finally, the polynomialgeneratorofany λ-constacycliccode is a divisorofx n λ, hence has the form in (3.6). 8

9 Corollary 3.5. If n is a positive integer coprime to q 1, then there is only one n-isometry class in F q; in particular, for any λ F q the λ-constacyclic codes of length n are isometric to the cyclic codes of length n, i.e. a n λ = 1 for an a F q and all the (3.4), (3.5) and (3.6) hold. Proof. Since gcd(n,q 1) = 1, the conclusion is obtained immediately. It is an automorphism of the group Fq which maps any a F q to an Fq ; thus there is a b Fq such that λ = bn. Let n = n p s as in Corollary 3.4. If n = 1, then n = p s is coprime to q 1 and X ps 1 = (X 1) ps, and we get the following result at once. Corollary 3.6. For any λ F q the λ-constacyclic codes of length p s are isometric to the cyclic codes of length p s ; in particular, there is an a F q such that a ps λ = 1 and X ps λ = λ(ax 1) ps is an irreducible factorization in F q [X]; in particular, any λ-constacyclic code C of length p s has a polynomial generator (X a 1 ) i with 0 i p s. Remark 3.7. Taking λ = 1, Corollary 3.6 implies that negacyclic codes of length p s are isometric to cyclic codes of length p s. This generalizes [10, Theorem 3.3] which showed that, in our terminology, λ-constacyclic codes of length p s over F p m are isometric to the negacyclic codes of length p s over F p m. 4 Constacyclic codes of length l t p s Let F q be a finite field of order q = p m and F q = ξ be generated by a primitive (q 1)-th root ξ of unity as before. In this section, we consider constacyclic codes of length l t p s over F q, where l is a prime integer different from p and s, t are non-negative integers. We will showthat anyλ-constacycliccodeoflengthl t p s with λ =l t ps 1has apolynomial generator with irreducible factors all being binomials of degrees equal to powers of the prime l except for the case when l = 2, t 2 and 2 (q 1); and in the exceptional case the polynomial generator with irreducible factors all being trinomials corresponding to the factorization (2.3). As we did in Remark 2.1, take a complete set {r 1,,r ρ } of representatives of q-cyclotomic cosets modulo l t ; take a primitive l t -th root η of unity (maybe in an extension of F q ), and denote M η (X) the minimal polynomial of η over F q ; by the formula (2.2), X lt p s 1 = (X lt 1) ps = M η r 1(X) ps M η r 2(X) ps M η rρ(x) ps (4.1) is the irreducible factorization of X lt p s 1 in F q [X]. Further, assume that l u (q 1), ζ = ξ q 1 l u, v = min{t,u}. (4.2) 9

10 Theorem 4.1. With notations as above, for any λ Fq there is an index j with 0 j v such that λ = l t p s ζlj and one of the following two cases holds: (i) j = v, then λ = l t p s 1, alt p s λ = 1 for an a F q and Xlt p s λ = λ ρ i=1 M η r i(ax) ps with {r 1,,r ρ } and M η r i(x) s defined in (4.1). (ii) 0 j v 1, then a lt p s λ = ζ klj for an a F q and a positive integer k coprime to l t p s ; there are two subcases: (ii.a) if l = 2, t 2 and 2 (q 1), then j = 0, a lt p s λ = 1 and, setting H t, b and c to be as in Remark 2.2, we have that X 2t p s λ = ( λ) h H t ( a 2 t b+1 X 2t b+1 2a 2t b hx 2t b +( 1) c) p s (4.3) with all the factors in the right hand side being irreducible over F q ; (ii.b) otherwise, taking an integer s with 0 s < m and s s (mod m), we have that X lt p s λ = l j 1 i=0 (X lt j a lt j ζ ilu j +kp m s ) p s (4.4) with all the factors in the right hand side being irreducible over F q. Proof. As q 1 = p m 1, it is clear that gcd(p s,q 1) = 1. From the notation (4.2), we have ζ F q is a primitive l u -th root of unity, ζ is the Sylow l-subgroup of F q, and ζlu j for 0 j u is a primitive l j -th root of unity; l v = gcd(l t p s, q 1), so ord(ξ lt p s q 1 ) = gcd(l t p s,q 1) = q 1 l = ord(ξ lv ), v hence in the multiplicative group Fq we have that ξ lt p s = ξ lt = ξ lv (4.5) which is a subgroup of F q of order q 1 l v. Thus the quotient group F q/ ξ lv is a cyclic group of order l v ; and for each positive divisor l v j of l v, where j = 0,1,,v, ζ lj,ξ lv / ξ lv is the unique subgroup of order l v j of the quotient group F q / ξlv. By the equivalence (i) (ii) of Theorem 3.2, the number of the l t p s -isometry classes of Fq is equal to v + 1; precisely, for any λ F q there is exactly one index j with 0 j v such that λ = l t p s ζlj. We continue the discussion in two cases. Case (i): j = v, i.e. λ = l t p s ζlv ; by the equality (4.5), we see that λ,ξ lt p s = ζ lv,ξ lv = 1,ξ lt p s, in other words, λ = l t ps 1. By Corollary 3.4, 10

11 a lt p s λ = 1 k = 1 for an a Fq, and from the irreducible factorization (4.1) we get the irreducible factorization X lt p s λ = λ ρ i=1 M η r i(ax) ps. Case (ii): 0 j v 1. Then by (4.2) we have 0 j v 1 < v = min{t,u}, (4.6) in particular,v 1, i.e. l (q 1); further, sinceλ = l t p s ζlj, by Theorem3.2(iii) there is an a F q and a positive integer k such that a lt p s λ = ζ klj, gcd(k, l t p s ) = 1. (4.7) We discuss it in the two subcases (ii.a) and (ii.b) as described in the theorem. Subcase (ii.a). Since l = 2, t 2 and 2 (q 1), we have that q is odd, t > u = v = 1, ζ = 1 and j = 0; and, from (4.7) we see that l = 2 k and a 2t p s λ = ( 1) k = 1. From the formula (2.3), we have the following irreducible factorization in F q [X]: X 2t p s +1 = ( X 2 t b+1 2hX 2t b +( 1) c) p s ; h H t thus the following is an irreducible factorization of (ax) 2t p s +1 in F q [X]: (ax) 2t p s +1 = ( a 2 t b+1 X 2t b+1 2a 2t b hx 2t b +( 1) c) p s. h H t However,sincea 2t p s = λ 1,wehavethat(aX) 2t p s = a 2t p s X 2t p s = λ 1 X 2t p s ; thus we get the irreducible factorization (4.3) of X 2t p s λ in F q [X]. Subcase (ii.b). Remember that the conclusion in Remark 2.3 is applied in this subcase. By the choice of s, m s +s 0 (mod m), so (p m 1) (p m s +s 1), i.e. p m s +s 1 (mod q 1); in particular, β pm s +s = β for any β F q. Obviously, ζ lu j is a primitive l j -th root of unity in F q. Therefore, hence ( X lt j ζ kpm s ( X lt j ζ kpm s ) l j 1 = l j 1 i=0 ) l j p s ( ) ( X l t j ) l j p s 1 = 1 = ζ kpm s ( ) X lt j ζ il u j, ζ kpm s l j 1 i=0 Noting that ζ kpm s p s = (ζ k ) pm s +s = ζ k, we get that X lt p s ζ klj = l j 1 i=0 ( X lt j ζ il u j ζ kpm s ) p s ) p s (X lt j ζ ilu j +kp m s. (4.8) 11.

12 From (4.7) and (4.6), we see that u > j, l (p m 1) and l k; hence l il u j but l kp m s. So l (il u j +kp m s ), hence, in the multiplicative group F q we have that ord(ζ ilu j +kp m s ) = l u. By Remark 2.3, all the polynomials X lt j ζ ilu j +kp m s, i = 0,1,,l j 1, are irreducible polynomials in F q [X], and (4.8) is a irreducible factorization of X lt p s ζ klj in F q [X]. Replacing X by ax, we get (ax) lt p s ζ klj = l j 1 i=0 ) p s ((ax) lt j ζ ilu j +kp m s. But a lt p s λ = ζ klj, i.e. a lt p s ζ klj = λ. We get the irreducible factorization of X lt p s λ in F q [X] as follows: l j 1 ) p s X lt p s λ = a lt p ((ax) s lt j ζ ilu j +kp m s. i=0 Finally, noting that a lt p s = ( (a lt j ) ps ) l j, from the above we get the desired irreducible factorization (4.4) of X lt p s λ in F q [X]. Remark 4.2. With the same notation as in Theorem 4.1, we can describe the polynomial generator g(x) of any λ-constacyclic code C of length l t p s over F q for the two cases as follows. (i): j = v, then g(x) = ρ M η r i(ax) ei, i=1 0 e i p s i = 1,,ρ. By the way, we show an easy subcase of this case: if j = v = t, then ζ lu t = ξ q 1 l t F q is a primitive l t -th root of unity, hence X lt 1 = l t 1 i=0 (X ); thus the polynomial generator g(x) looks simple: ζilu t g(x) = (X a 1 ζ ilu t ) ei, 0 e i p s i = 0,,l t 1. (4.9) l t 1 i=0 (ii): 0 j < v t, there are two subcases: (ii.a): if l = 2, t 2 and 2 (q 1), then g(x) = (a 2 t b+1x 2t b+1 2a 2t b hx 2t b +( 1) c ) ei h H t with 0 e i p s for i = 0,1,,2 b

13 (ii.b): otherwise, g(x) = l j 1 i=0 with 0 e i p s for i = 0,1,,l j 1. (X lt j a lt j ζ ilu j +kp m s ) ei It is a special case for Theorem 4.1 that t = v = 1, i.e. l (q 1) and t = 1; at that case, as stated in the following corollary, there are only two lp s -isometry classes in F q, and any constacyclic code of length lp s over F q has a polynomial generator with all irreducible factors being binomials. Corollary 4.3. Assume that l is a prime such that l u (q 1) with u 1, ζ F q is a primitive l u -th root of unity, and λ Fq. Let C be a λ-constacyclic code of length lp s over F q. Then either λ ξ l, a lps λ = 1 for an a F q, and we have C = l 1 ( X a 1 ζ ilu 1) e i, 0 e i p s, i = 0,1,,l 1; i=0 or λ / ξ l, a lps λ = ζ k for an a Fq and an integer k coprime to lp s, and, taking s such that 0 s < m and s s (mod m), we have (X C = l a l ) e ζ kpm s, 0 e p s. Proof. It follows from Remark 4.2 immediately. We just remark that ζ lu 1 is a primitive l-th root of unity, while ζ kpm s is a primitive l u -th root of unity. More specifically, if l = 2 in the above corollary, we reobtain the main result of [11], as stated below in our notation. Corollary 4.4. Assume that 2 u (q 1) with u 1, ζ F q is a primitive 2 u -th root of unity, and λ F q. Let C be a λ-constacyclic code of length 2p s over F q. Then either λ ξ 2, a 2ps λ = 1 for an a F q, and we have C = ( X a 1) e 0 ( X +a 1 ) e 1, 0 ei p s, i = 0,1; or λ / ξ 2, a 2ps λ = ζ k for an a Fq and an integer k coprime to 2ps, and, taking an integer s such that 0 s < m and s s (mod m), we have (X C = 2 a 2 ) e ζ kpm s, 0 e p s. Proof. Just note that ζ 2u 1 is a primitive square root, i.e. ζ 2u 1 = 1. 13

14 5 Examples By Theorem 4.1, the polynomial generators of all constacyclic codes of length l t p s over the finite field F p m are easy to be established, where l,p are different primes and s, t are non-negative integers. In this section, some examples are given to illustrate the result. Example 5.1. Consider all constacyclic codes of length 6 = 3 2 over F 2 4. Here, l = 3, t = 1, p = 2 and s = 1. Let ξ be a primitive 15th root of unity in F 2 4. Since 3 (2 4 1), it follows that there exists primitive 3rd root of unity in F 2 4. Therefore, X 3 1 = (X 1)(X ξ 5 )(X ξ 10 ). By Theorem 4.1, the number of the 6-isometry classes of F 2 4 is 2. Hence, all the constacyclic codes are divided into two parts. The polynomial generators of all constacyclic codes are given in Table 1 and Table 2. λ a λ-constacyclic codes: 0 j 0,j 1,j 2 2 sizes 1 1 (X 1) j0 (X ξ 5 ) j1 (X ξ 10 ) j jo j1 j2 ξ 3 ξ 7 (ξ 7 X 1) j0 (ξ 7 X ξ 5 ) j1 (ξ 7 X ξ 10 ) j jo j1 j2 ξ 6 ξ 4 (ξ 4 X 1) j0 (ξ 4 X ξ 5 ) j1 (ξ 4 X ξ 10 ) j jo j1 j2 ξ 9 ξ (ξx 1) j0 (ξx ξ 5 ) j1 (ξx ξ 10 ) j jo j1 j2 ξ 12 ξ 3 (ξ 3 X 1) j0 (ξ 3 X ξ 5 ) j1 (ξ 3 X ξ 10 ) j jo j1 j2 Table 1: λ-constacyclic codes of length 6 over F 2 4, λ =6 1, a 6 λ = 1 λ k a λ-constacyclic codes: 0 j 2 sizes ξ 5 ξ 4 (X 3 ξ 8 ) j j ξ 4 5 ξ 6 (X 3 ξ 2 ) j j ξ 7 5 ξ 8 (X 3 ξ 11 ) j j ξ 10 5 ξ 5 (X 3 ξ 5 ) j j ξ 13 5 ξ 2 (X 3 ξ 14 ) j j ξ 2 1 ξ 3 (X 3 ξ) j j ξ (X 3 ξ 10 ) j j ξ 8 1 ξ 2 (X 3 ξ 4 ) j j ξ 11 1 ξ 4 (X 3 ξ 13 ) j j ξ 14 1 ξ (X 3 ξ 7 ) j j Table 2: λ-constacyclic codes of length 6 over F 2 4, λ = 6 ξ 5, a 6 λ = ξ 5k Example 5.2. Consider all constacyclic codes of length 175 = over F 5 2. Here, l = 7, t = 1, p = 5 and s = 2. Let ξ be a primitive 24th root of unity in F 5 2. Since gcd(175,5 2 1) = 1, by Corollary 3.5, all the constacyclic codes of length 175 are isometric to the cyclic codes of length 175. By [13], it follows that X 7 1 = (X 1)(X 3 +ξx 2 +ξ 17 X 1)(x 3 +ξ 5 X 2 +ξ 13 X 1) is the factorization 14

15 λ a λ-constacyclic codes: 0 i,j,k 25 sizes 1 1 (X 1) i g(x) j h(x) k i 3j 3k ξ ξ 17 (ξ 17 X 1) i g(ξ 17 X) j h(ξ 17 X) k i 3j 3k ξ 2 ξ 10 (ξ 10 X 1) i g(ξ 10 X) j h(ξ 10 X) k i 3j 3k ξ 3 ξ 3 (ξ 3 X 1) i g(ξ 3 X) j h(ξ 3 X) k i 3j 3k ξ 4 ξ 20 (ξ 20 X 1) i g(ξ 20 X) j h(ξ 20 X) k i 3j 3k ξ 5 ξ 13 (ξ 13 X 1) i g(ξ 13 X) j h(ξ 13 X) k i 3j 3k ξ 6 ξ 6 (ξ 6 X 1) i g(ξ 6 X) j h(ξ 6 X) k i 3j 3k ξ 7 ξ 23 (ξ 23 X 1) i g(ξ 23 X) j h(ξ 23 X) k i 3j 3k ξ 8 ξ 16 (ξ 16 X 1) i g(ξ 16 X) j h(ξ 16 X) k i 3j 3k ξ 9 ξ 9 (ξ 9 X 1) i g(ξ 9 X) j h(ξ 9 X) k i 3j 3k ξ 10 ξ 2 (ξ 2 X 1) i g(ξ 2 X) j h(ξ 2 X) k i 3j 3k ξ 11 ξ 19 (ξ 19 X 1) i g(ξ 19 X) j h(ξ 19 X) k i 3j 3k ξ 12 ξ 12 (ξ 12 X 1) i g(ξ 12 X) j h(ξ 12 X) k i 3j 3k ξ 13 ξ 5 (ξ 5 X 1) i g(ξ 5 X) j h(ξ 5 X) k i 3j 3k ξ 14 ξ 22 (ξ 22 X 1) i g(ξ 22 X) j h(ξ 22 X) k i 3j 3k ξ 15 ξ 15 (ξ 15 X 1) i g(ξ 15 X) j h(ξ 15 X) k i 3j 3k ξ 16 ξ 8 (ξ 8 X 1) i g(ξ 8 X) j h(ξ 8 X) k i 3j 3k ξ 17 ξ (ξx 1) i g(ξx) j h(ξx) k i 3j 3k ξ 18 ξ 18 (ξ 18 X 1) i g(ξ 18 X) j h(ξ 18 X) k i 3j 3k ξ 19 ξ 11 (ξ 11 X 1) i g(ξ 11 X) j h(ξ 11 X) k i 3j 3k ξ 20 ξ 4 (ξ 4 X 1) i g(ξ 4 X) j h(ξ 4 X) k i 3j 3k ξ 21 ξ 21 (ξ 21 X 1) i g(ξ 21 X) j h(ξ 21 X) k i 3j 3k ξ 22 ξ 14 (ξ 14 X 1) i g(ξ 14 X) j h(ξ 14 X) k i 3j 3k ξ 23 ξ 7 (ξ 7 X 1) i g(ξ 7 X) j h(ξ 7 X) k i 3j 3k Table 3: λ-constacyclic codes of length 175 over F 5 2, λ =175 1, a 175 λ = 1 of X 7 1 into irreducible factors over F 5 2. Let g(x) = X 3 +ξx 2 +ξ 17 X 1 and h(x) = x 3 +ξ 5 X 2 +ξ 13 X 1. The polynomial generators of constacyclic codes are given in Table 3. Example 5.3. Consider all constacyclic codes of length 20 = over F 5 2. Here, l = 2, t = 2, p = 5 and s = 1. Let ξ be a primitive 24th root of unity in F 5 2. Since 4 (5 2 1), it follows that there exists a primitive 4th root of identity in F 5 2. Therefore, X 4 1 = (X 1)(X ξ 6 )(X ξ 12 )(X ξ 18 ). By Theorem 4.1, the number of the 20-isometry classes of F 2 4 is 3. The polynomial generators of constacyclic codes are given in Table

16 λ a λ-constacyclic codes: 0 j 0,j 1,j 2,j 3 5 sizes 1 ξ 6 (ξ 6 X 1) j 0(ξ 6 X ξ 6 ) j 1(ξ 6 X ξ 12 ) j 2(ξ 6 X ξ 18 ) j jo j 1 j 2 j 3 ξ 4 ξ (ξx 1) j 0(ξX ξ 6 ) j 1(ξX ξ 12 ) j 2(ξX ξ 18 ) j jo j 1 j 2 j 3 ξ 8 ξ 2 (ξ 2 X 1) j 0(ξ 2 X ξ 6 ) j 1(ξ 2 X ξ 12 ) j 2(ξ 2 X ξ 18 ) j jo j 1 j 2 j 3 ξ 12 ξ 3 (ξ 3 X 1) j 0(ξ 3 X ξ 6 ) j 1(ξ 3 X ξ 12 ) j 2(ξ 3 X ξ 18 ) j jo j 1 j 2 j 3 ξ 16 ξ 4 (ξ 4 X 1) j 0(ξ 4 X ξ 6 ) j 1(ξ 4 X ξ 12 ) j 2(ξ 4 X ξ 18 ) j jo j 1 j 2 j 3 ξ 20 ξ 5 (ξ 5 X 1) j 0(ξ 5 X ξ 6 ) j 1(ξ 5 X ξ 12 ) j 2(ξ 5 X ξ 18 ) j jo j 1 j 2 j 3 Table 4: λ-constacyclic codes of length 20 over F 5 2, λ =20 1, a 20 λ = 1 λ k a λ-constacyclic codes: 0 j 5 sizes ξ 3 ξ 4 (X 4 ξ 5 ) j j ξ 5 3 ξ 23 (X 4 ξ 5 ) j j ξ 9 3 ξ 6 (X 4 ξ 21 ) j j ξ 13 3 ξ (X 4 ξ 14 ) j j ξ 17 3 ξ 2 (X 4 ξ 13 ) j j ξ 21 3 ξ 3 (X 4 ξ 9 ) j j ξ (X 4 ξ 15 ) j j ξ 7 1 ξ (X 4 ξ 11 ) j j ξ 11 1 ξ 2 (X 4 ξ 7 ) j j ξ 15 1 ξ 3 (X 4 ξ 3 ) j j ξ 19 1 ξ 4 (X 4 ξ 23 ) j j ξ 23 1 ξ 5 (X 4 ξ 19 ) j j Table 5: λ-constacyclic codes of length 20 over F 5 2, λ = 20 ξ 3, a 20 λ = ξ 3k λ k a λ-constacyclic codes: 0 j 0,j 1 5 sizes ξ 2 1 ξ 23 (X 2 ξ 5 ) j0 (X 2 +ξ 5 ) j jo 2j1 ξ 6 1 ξ 6 (X 2 ξ 15 ) j0 (X 2 +ξ 15 ) j jo 2j1 ξ 10 1 ξ (X 2 ξ) j0 (X 2 +ξ) j jo 2j1 ξ 14 1 ξ 2 (X 2 ξ 23 ) j0 (X 2 +ξ 23 ) j jo 2j1 ξ 18 1 ξ 3 (X 2 ξ 18 ) j0 (X 2 +ξ 18 ) j jo 2j1 ξ 22 1 ξ 4 (X 2 ξ 11 ) j0 (X 2 +ξ 11 ) j jo 2j1 Table 6: λ-constacyclic codes of length 20 over F 5 2, λ = 20 ξ 6, a 20 λ = ξ 6k Acknowledgements This work was supported by NSFC, Grant No , and Research Funds of CCNU, Grant No. 11A The authors would like to thank the anonymous referees for their many helpful comments. 16

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