Left dihedral codes over Galois rings GR(p 2, m)

Transcription

1 Left dihedral codes over Galois rings GR(p 2, m) Yonglin Cao a, Yuan Cao b, Fang-Wei Fu c a School of Sciences, Shandong University of Technology, Zibo, Shandong , China b College of Mathematics and Econometrics, Hunan University, Changsha , China c Chern Institute of Mathematics and LPMC, Nankai University, Tianjin , China Abstract Let D 2n = x, y x n = 1, y 2 = 1, yxy = x 1 be a dihedral group, and R = GR(p 2, m) be a Galois ring of characteristic p 2 and cardinality p 2m where p is a prime. Left ideals of the group ring R[D 2n ] are called left dihedral codes over R of length 2n, and abbreviated as left D 2n -codes over R. Let gcd(n, p) = 1 in this paper. Then any left D 2n -code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A i and outer codes C i, where A i is a cyclic code over R of length n and C i is a skew cyclic code of length 2 over an extension Galois ring or principal ideal ring of R, and a generator matrix and basic parameters for each outer code C i is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D 2n -code is determined and all self-dual left D 2n - codes and self-orthogonal left D 2n -codes over R are presented, respectively. Keywords: Left dihedral code; Galois ring; Concatenated structure; Cyclic code; Dual code; Self-orthogonal code Mathematics Subject Classification (2000) 94B05, 94B15, 11T71 1. Introduction After the celebrated results in the 1990s ([14], [24], [31]) that many important yet seemingly non-linear codes over finite fields are actually closely related to corresponding author. addresses: ylcao@sdut.edu.cn (Yonglin Cao), yuan cao@hnu.edu.cn (Yuan Cao), fwfu@nankai.edu.cn (F. W. Fu). Preprint submitted to Elsevier September 13, 2016

2 linear codes over the ring of integers modulo four, codes over Z 4 in particular, and codes over finite commutative chain rings in general, have received a great deal of attention. Examples of finite commutative chain rings include the ring Z p k of integers modulo p k for a prime p, and the Galois rings GR(p k, m), i.e. the Galois extension of degree m of Z p k. These classes of rings have been used widely as an alphabet for codes. Let R be a finite chain ring with identity, R be the multiplicative group of invertible elements of R, N a positive integer and R N = {(r 1,..., r N ) r j R, j = 1,..., N} be the free R-module of rank N with the coordinate component addition and scalar multiplication by elements of R. Then linear codes over R (or linear R-codes) of length N are defined as R-submodules of R N. For any fixed λ R, a linear R-code C of length N is said to be λ-constacyclic if (λc n 1, c 0, c 1,..., c n 2 ) C for all (c 0, c 1,..., c n 1 ) C, and C is said to be cyclic (negacyclic) when λ = 1 (λ = 1). Readers are referred to [17], [20], [21], [25], [27], [32] and [34] for results on linear codes, cyclic codes and constacyclic codes over finite chain rings. Moreover, various decoding schemes for codes over Galois rings have been considered in [11 13]. Now, let N = ln. A linear code C over R of length N is called a l-quasicyclic code of length ln if (c 1,n 1, c 1,0, c 1,1,..., c 1,n 2,..., c l,n 1, c l,0, c l,1..., c l,n 2 ) C for all (c 1,0, c 1,1,..., c 1,n 1,..., c l,0, c l,1,..., c l,n 1 ) C. Quasi-cyclic codes and their generalizations over finite chain rings have been received a great deal of attention. For example, see [1], [4], [15], [16] and [19]. In this paper, let D 2n = x, y x n = 1, y 2 = 1, yxy = x 1 = {x i y j 0 i n 1, j = 0, 1} be a dihedral group of order n. The group ring R[D 2n ] is a free R-module with basis D 2n. Addition, multiplication with scalars c R and multiplication are defined by: for any a g, b g R where g D 2n, a g g + b g g = (a g + b g )g, c( a g g) = g D 2n g D 2n g D 2n g D 2n ( a g g)( b g g) = g D 2n g D 2n ( g D 2n uv=g a u b v )g. g D 2n ca g g, 2

3 Then R[D 2n ] is a noncommutative ring with identity 1 = 1 R 1 D2n where 1 R and 1 D2n is the identity elements of R and D 2n respectively. Readers are referred to [33] for more details on group rings. For any a = (a 0,0, a 1,0,..., a n 1,0, a 0,1, a 1,1,..., a n 1,1 ) R 2n, we define n 1 n 1 Ψ(a) = a i,0 x i + a i,1 x i y. Then Ψ is an R-module isomorphism from R 2n onto R[D 2n ]. As a natural generalization of [23], a nonempty subset C of the R-module R 2n is called a left dihedral code (or left D 2n -code for more clear) over R if Ψ(C) is a left ideal of R[D 2n ]. As usual, we will identify C with Ψ(C) in this paper. Since Ψ(a n 1,0, a 0,0, a 1,0,..., a n 2,0, a n 1,1, a 0,1, a 1,1,..., a n 2,1 ) = xψ(a) for all a = (a 0,0, a 1,0,..., a n 1,0, a 0,1, a 1,1,..., a n 1,1 ) R 2n, we see that every left D 2n -code over R is a 2-quasi-cyclic code over R of length 2n. When R = F q is a finite field of cardinality q, there have been many research results on codes as two-sided ideals and left ideals in a finite group algebra over finite fields. For example, Dutra et al [22] investigated codes that are given as two-sided ideals in a semisimple finite group algebra F q [G] defined by idempotents constructed from subgroups of a finite group G, and given a criterion to decide when these ideals are all the minimal two-sided ideals of F q [G] in the case when G is a dihedral group. McLoughlin [30] provided a new construction of the self-dual, doubly-even and extremal [48,24,12] binary linear block code using a zero divisor in the group ring F 2 [D 48 ] where D 48 is a dihedral group of order 24. Recently, Brochero Martínez [10] shown all central irreducible idempotents and their Wedderburn decomposition of the dihedral group algebra F q [D 2n ], in the case when every divisor of n divides q 1. This characterization depends to the relation of the irreducible idempotents of the cyclic group algebra F q [C n ] and the central irreducible idempotents of the group algebras F q [D 2n ]. Gabriela and Inneke [23] provided algorithms to construct minimal left group codes. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra F q [G] for a large class of groups G. More importantly, Bazzi and Mitter [3] shown that for infinitely many block lengths a random left ideal in the binary group algebra of the dihedral group is an asymptotically good rate-half code with a high probability. 3

4 Martínez-Pérez and Willems [29] proved that the class of binary self-dual doubly even 2-quasi-cyclic transitive codes is asymptotically good. A system theory for left D 2n -codes over finite fields was given [18]. Let R = GR(p 2, m) be a Galois ring of characteristic p 2 and cardinality p 2m and gcd(n, p) = 1, we try to achieve the following two goals: Develop a system theory for left D 2n -codes over R using an elementary method. Precisely, only finite field theory, Galois ring theory and basic theory of cyclic codes and skew cyclic codes are used, and it does not involve any group algebra language and technique except the definition for left D 2n -codes. Provide a precise expression for all distinct left D 2n -codes, their dual codes, self-orthogonal and self-dual left D 2n -codes over R. Using the expression provided, one can present left D 2n -codes, self-orthogonal and self-dual left D 2n -codes over R for specific n, m and p (not too big) conveniently and easily, and design left D 2n -codes for their requirements directly. The present paper is organized as follows. In section 2, we present every left D 2n -code and its dual code over R by their direct sum decomposition, where each direct summand is a concatenated code, respectively. In particular, we determine the inner code and give a generator matrix and basic parameters for the outer code of each concatenated code precisely. As a corollary, we obtain a formula to count the number of all left D 2n -codes over R. In section 3, we give a proof for the main results (Theorem 2.5) of this paper by use of the known results for left dihedral codes over finite fields and Galois ring theory. Then we present all self-dual and all self-orthogonal left D 2n -codes over R in Section 4. Finally, we list 60 optimal Z 4 -linear codes of length 30 obtained from left D 30 -codes over Z Concatenated structure of left D 2n -codes over R In this section, we give the concatenated structure of every left D 2n -code and its Euclidian dual code over R explicitly, and obtain a formula to count the number of all left D 2n -codes over R precisely. In this paper, we adopt the following notations. Let R = GR(p 2, m) be a fixed Galois ring of characteristic p 2 and cardinality p 2m, F p m = R/pR be the residue class field of R modulo its unique maximal ideal pr and : R F p m be the natural surjective homomorphism of rings from R onto F p m defined by a = a (mod p) for all a R. 4

5 Example Let Z p 2 = {0, 1,..., p 2 1} be the ring of integers modulo p 2, h(x) = x m + h m 1 x m h 1 x + h 0 be a fixed basic irreducible monic polynomial of degree m in Z p 2[x], i.e. h(x) is a monic polynomial of degree m in Z p 2[x] such that h(x) (mod p) is irreducible in Z p [x], and let R = {a 0 + a 1 z a m 1 z m 1 a 0, a 1,..., a m 1 Z p 2} = Z p 2[z] in which z m = h 0 h 1 z... h m 1 z m 1. Then R is a Galois ring of characteristic p 2 and cardinality p 2m. Denote a = a (mod p) for any a Z p 2. It is known that F p m = Z p [x]/ h(x) = Z p [z], where h(x) = x m +h m 1 x m h 1 x + h 0 Z p [x] and z m = h 0 h 1 z... h m 1 z m 1, and the natural surjective homomorphism of rings : R F p m is given by a 0 + a 1 z a m 1 z m 1 a 0 + a 1 z a m 1 z m 1 for any a 0, a 1,..., a m 1 Z p 2 (cf. [36] Theorem 14.1). The ring homomorphism : R F p m can be extended to a surjective homomorphism of polynomial rings from R[x] onto F p m[x] in the natural way: g(x) g(x) = k g i x i, g(x) = k g i x i R[x] with g i R. We will denote this homomorphism by as well. Then a monic polynomial g(x) R[x] is said to be basic irreducible (basic primitive) if g(x) is an irreducible (primitive) polynomial in F p m[x]. Denote n 1 A = R[x]/ x n 1 = {a(x) a(x) = a j x j, a 0, a 1,..., a n 1 R} where the arithmetic is done modulo x n 1. For any positive integer l, by a linear A-code of length l we mean an A-submodule of A l = {(α 1,..., α l ) α 1,..., α l A}. If l = 1, linear A-codes are exactly ideals of the ring A which are the well-known cyclic codes over R of length n (cf. [37] Proposition 1.1). If l 2, every linear A-code of l can be seen as an R-linear code of length nl by replacing each a(x) = n 1 j=0 a jx j A with (a 0, a 1,..., a n 1 ) R n, and these R-linear codes of length nl are the well known l-quasi-cyclic codes over R of length nl. 5 j=0

6 Let Ω n = {1, x,..., x n 1 } be the cyclic subgroup of D 2n generated by x. Then we can identify the group ring R[Ω n ] with A in the natural way. Hence every element of the group ring R[D 2n ] is uniquely expressed as: a 0 (x) + a 1 (x)y, a 0 (x), a 1 (x) A. Now, we define a map Ξ : A 2 R[D 2n ] by (a 0 (x), a 1 (x)) a 0 (x) + a 1 (x)y ( a 0 (x), a 1 (x) A). It is routine to check that Ξ is an isomorphism of A-modules from A 2 onto R[D 2n ]. As stated above, A-submodules of A 2 are 2-quasi-cyclic codes over R of length 2n. We will identify R[D 2n ] with A 2 under Ξ in this paper. Then by yx = x 1 y in D 2n, we deduce the following conclusion. Theorem 2.1 Let = C A 2. Then C is a left ideal of R[D 2n ] if and only if C is an A-submodule of A 2 satisfying the following condition (a 1 (x 1 ), a 0 (x 1 )) C, (a 0 (x), a 1 (x)) C. For any nonzero polynomial g(x) = d a ix i R[x] of degree d, recall that the reciprocal polynomial of g(x) is defined by g (x) = (g(x)) = x d g( 1 x ) = a d + a d 1 x a 0 x d. g(x) is said to be self-reciprocal if g (x) = ug(x) for some u R. As gcd(p, n) = 1, we have x n 1 = r+t f i(x), where f 0 (x), f 1 (x),..., f r+t (x) are pairwise coprime monic polynomials in R[x] such that f 0 (x) = x 1, f i (x) is a self-reciprocal and basic irreducible polynomial of degree d i, i = 1,..., r, f i (x) = ρ i (x)ρ i (x) where ρ i (x) is a monic basic irreducible polynomial of degree d i such that ρ i (x) and ρ i (x) are coprime in R[x], i = r+1,..., r+t. Then it is clear that n = r d i + 2 r+t i=r+1 d i where d 0 = 1. Denote A i = R[x]/ f i (x) where we consider elements of A i as polynomials in R[x] of degree < deg(f i (x)) and the arithmetic is done modulo f i (x), for all i = 0, 1,..., r + t. Υ i,1 = R[x]/ ρ i (x) and Υ i,2 = R[x]/ ρ i (x) where we consider elements of Υ i,1 (resp. Υ i,2 ) as polynomials in R[x] of degree < d i and the arithmetic is done modulo ρ i (x) (resp. ρ i (x)), for i = r + 1,..., r + t. 6

7 As a direct corollary of Wan [36] Theorem and Theorem 14.27, we have the following lemma. Lemma 2.2 (i) For each integer i, 0 i r, A i is a Galois ring of characteristic p 2 and cardinality p 2md i which is a Galois extension of R with degree d i, and there is an invertible element ζ i (x) in A i of multiplicative order p md i 1. (ii) For each integer i, r + 1 i r + t, Υ i,1 is a Galois ring of characteristic p 2 and cardinality p 2md i which is a Galois extension of R with degree d i, and there is an invertible element ζ i (x) in Υ i,1 of multiplicative order p md i 1. (iii) A i is a free R-module of rank deg(f i (x)) with an R-basis {1, x,..., x deg(fi(x)) 1 } for any 0 i r + t. For each 0 i r + t, denote F i (x) = xn 1 f i R[x]. Then F (x) i (x) and f i (x) are coprime polynomials in R[x]. Hence there are polynomials u i (x), v i (x) R[x] such that u i (x)f i (x) + v i (x)f i (x) = 1. (1) In the rest of this paper, let ε i (x) A satisfying ε i (x) u i (x)f i (x) = 1 v i (x)f i (x) (mod x n 1). Then from classical ring theory, we deduce the following lemma. Lemma 2.3 (cf. [37] Theorem 2.7 and its proof) (i) r+t ε i(x) = 1, ε i (x) 2 = ε i (x) and ε i (x)ε j (x) = 0 for all 0 i j r + t in the ring A. (ii) A = r+t A i, where A i = ε i (x)a with ε i (x) as its multiplicative identity. Moreover, this decomposition is a ring direct sum in that A i A j = {0} for all 0 i j r + t. (iii) For each 0 i r + t, let A i = R[x]/ f i (x). Then the map φ i : a(x) ε i (x)a(x) (mod x n 1), a(x) A i is an isomorphism of rings from A i onto A i. As usual, we will identify each (a 0, a 1,..., a n 1 ) R n with a 0 + a 1 x a n 1 x n 1 A. Then we have the following properties for A i = ε i (x)a. Corollary 2.4 Let 0 i r + t. Then (i) (cf. [37] Proposition 4.3) A i is a cyclic code over R of length n with parity check polynomial f i (x) and generating idempotent ε i (x). 7

8 (ii) A i is a free R-submodule of A = R[x]/ x n 1 with an R-basis {ε i (x), xε i (x),..., x deg(fi(x)) 1 ε i (x)}. Hence A i = p 2m deg(fi(x)). Proof. (ii) As f i (x) is a monic divisor of x n 1 in R[x], A i = R[x]/ f i (x) is a free R-module with an R-basis {1, x,..., x deg(fi(x)) 1 }, which implies A i = R deg(fi(x)) = p 2m deg(fi(x)). From this and by Lemma 2.3(iii), we see that {ε i (x), xε i (x),..., x deg(fi(x)) 1 ε i (x)} is a free R-basis of A i. Now, let C i be an A i -linear code of length 2, i.e., C i is an A i -submodule of A 2 i = {(b 0 (x), b 1 (x)) b 0 (x), b 1 (x) A i }. For each ξ = (b 0 (x), b 1 (x)) A 2 i, we denote by w (A i) H (ξ) = {j b j(x) 0 in A i, j = 0, 1} the Hamming weight of ξ. Then the minimum Hamming distance of C i is given by d (A i) H (C i) = min{w (A i) H (ξ) ξ 0, ξ C i}. As a natural generalization of the concept for concatenated codes over finite field (cf. [35, Definition 2.1]), we define the concatenated code of the inner code A i and the outer code C i as following A i φi C i = {(φ i (b 0 (x)), φ i (b 1 (x))) (b 0 (x), b 1 (x)) C i } = {(ε i (x)b 0 (x), ε i (x)b 1 (x)) (b 0 (x), b 1 (x)) C i } A 2 i. By Lemma 2.3(iii), we see that A i φi C i is a 2-quasi-cyclic code over R of length 2n with cardinality A i φi C i = C i. Moreover, the minimum Hamming distance of A i φi C i satisfies d (R) H (A i φi C i ) d (R) H (A i) d (A i) H (C i) where d (R) H (A i) is the minimum Hamming weight of A i as a linear R-code of length n. As the end of this section, we list all distinct left D 2n -codes over R and their Euclidean dual codes by the following theorem. Theorem 2.5 Using the notations above, all distinct left D 2n -codes C over R and their Euclidean dual codes C E are given by C = r+t A i φi C i and C E = r+t A i φi V i respectively, where C i and V i are A i -linear codes of length 2 with generator matrices G i and H i respectively, given by the one of following three tables. (I) Let 0 i r. We have one of the following two subcases. (i) Let d i = 1. Then (i-1) If p is odd, (G i, H i ) is given by the following table. 8

9 N i G i C i d H i I 2 2 p j I 2 (j = 0, 1) p 2(2 j)m 1 [ p 2 j I 2 ] w 1 2 (wp, p) p m 2 [ ] 0 p w 1 2 p 0 p 1 ( wp, p) 2 (w, 1) p 2m 2 ( w, 1) where w {1, 1}, d = d (A i) H (C i), N i is the number of C i in the same line and I 2 is the identity matrix of order 2 (the same below). (i-2) If p = 2, (G i, H i ) is given by the following table. N i G i C i d H i I j I 2 (j = 0, 1) 2 2(2 j)m 1 [ 2 2 j I 2 ] (2, 2) 2 m 2 [ ] (2, 2) 2 m (1 + 2u, 1) (u T ) 2 2m 2 ( 1 + 2u, 1) where T is a Teichmüller set of R with T = F 2 m = 2 m and T = F 2 m. (ii) Let d i 2 and ζ i (x) be an invertible element of A i with multiplicative order p md i 1. Then d i is even and (G i, H i ) is given by the following table. N i G i C i d H i I 2 2 p j I 2 (j = 0, 1) p 2(2 j)m 1 p 2 j I 2 [ ] (pw(x), p) p md i w(x) 1 2 [ ] 0 p w(x) 1 p 0 p 3md i 1 ( pw(x), p) p md i p md i p md i + p md i 2 (w(x)(1 + pϑ(x)), 1) p 2md i 2 ( w(x)(1 + pϑ(x)), 1) where w(x) W i and ϑ(x) V i. Here W i = {ζ i (x) (p md i 2 1)s s = 0, 1,..., p md i 2 }; 9

10 V i = {0} {ζ i (x) (2 md i 2 +1)l l = 0, 1,..., 2 md i 2 2} when p = 2, and V i = {0} {ζ i (x) 1 2 (p md i 2 +1)+(p md i 2 +1)l l = 0, 1,..., p md i 2 2} when p is odd. (II) Let r + 1 i r + t and ζ i (x) be an invertible element of Υ i,1 with multiplicative order p md i 1. Find ϕ i (x), ψ i (x) R[x] satisfying ϕ i (x)ρ i (x) + ψ i (x)ρ i (x) = 1, (2) and set ϵ i,1 (x), ϵ i,2 (x) A i = R[x]/ f i (x) = R[x]/ ρ i (x)ρ i (x) as follows ϵ i,1 (x) ϕ i (x)ρ i (x) (mod f i (x)) and ϵ i,2 (x) ψ i (x)ρ i (x) (mod f i (x)). Then (G i, H i ) is given by the following table: N i G i C i d H i I 2 2 p j I 2 (j = 0, 1) p 2(2 j)m 1 p 2 j I 2 [ ] Mi,1 pm i,2 1 pm i,1 p 2mdi 1 [ ] Mi,1 1 p pm 6mdi 1 pm i,1 i,2 [ ] Mi,2 pm i,1 1 pm i,2 p 2mdi 1 [ ] Mi,2 1 p pm 6md i 1 pm i,2 i,1 [ ] p md i 1 (pw(x), p) p 2md i w(x) 1 2 [ ] 0 p p md i w(x) 1 1 p 0 p 6md i 1 ( pw(x), p) [ ] [ ] p md ϵ i i,1 (x) pb i,1 (x) pb i,1 (x 1 p ) ϵ i,2 (x) 4md ϵ i 1 i,1 (x) pb i,1 (x) pb i,1 (x 1 ) ϵ i,2 (x) p 2md i p md i (w(x) + pϑ(x), 1) p 4md i [ ] 2 [ ( w(x) pϑ(x), 1) ] ϵ p mdi i,2 (x) pb i,2 (x) ϵ pb i,2 (x 1 p ) ϵ i,1 (x) 4mdi 1 i,2 (x) pb i,2 (x) pb i,2 (x 1 ) ϵ i,1 (x) [ ϵi,1 (x) 0 where M i,1 = 0 ϵ i,2 (x) for j = 1, 2, ϑ(x) V (w(x)) i ], M i,2 = and w(x) W i. Here [ ϵi,2 (x) 0 0 ϵ i,1 (x) ], b i,j (x) K i,j U i,1 = {0} {ϵ i,1 (x)ζ i (x) k k = 0, 1,..., p md i 2} (mod f i (x)); K i,1 = {0} {ϵ i,1 (x)ζ i (x) k k = 0, 1,..., p md i 2} = U i,1 (mod p); K i,2 = {0} {ϵ i,2 (x)ζ i (x n 1 ) k k = 0, 1,..., p md i 2} (mod f i (x)); W i = {u(x) + 1 u(x 1 ) 0 u(x) U i,1}; 10

11 V (w(x)) i = {v(x) ( 1 u(x 1 ) )2 v(x 1 ) v(x) U i,1 } (mod p) for any w(x) = u(x) + 1 u(x 1 ) W i where 0 u(x) U i,1. Remark In Theorem 2.5(II), we have that v(x 1 ) = 0 if v(x) = 0. Now, let u(x) = ϵ i,1 (x)ζ i (x) k where 0 k p md i 2. Then u(x 1 ) = ϵ i,2 (x)ζ i (x 1 ) k = ϵ i,2 (x)ζ i (x n 1 ) k (mod f i (x)). 1 u(x 1 ) = ϵ i,2(x)ζ i (x 1 ) pmd i 1 k = ϵ i,2 (x)ζ i (x n 1 ) pmd i 1 k (mod f i (x)). b i,1 (x 1 ) = ϵ i,2 (x)ζ i (x n 1 ) k K i,2, if b i,1 (x) = ϵ i,1 (x)ζ i (x) k K i,1 where 0 k p md i 2. b i,2 (x 1 ) = ϵ i,1 (x)ζ i (x) k K i,1, if b i,1 (x) = ϵ i,2 (x)ζ i (x n 1 ) k K i,2 where 0 k p md i 2. Finally, by Theorem 2.5 we obtain a formula to count the number of all left D 2n -codes over R. Corollary 2.6 Denote λ = {i d i = 1, 0 i r}. Let N (n,p 2,p 2m ) be the number of left D 2n -codes over R. If p is odd, N (n,p 2,p 2m ) = 9 λ and if p = 2, N (n,4,4 m ) = (2 m +5) λ d i 2,1 i r d i 2,1 i r 3. Proof of Theorem 2.5 ( ) r+t p md i + 3p md i i=r+1 ( ) r+t 2 md i md i ( p 2md i + 3p md i + 5 ) ; i=r+1 ( 4 md i md i + 5 ). In this section, we give a proof for Theorem 2.5. First, for basic irreducible (basic primitive) polynomials in R[x] we have the following conclusion. Lemma 3.1 (cf. [36] Theorem 14.22) For any integer κ 1 there exist monic basic irreducible (and monic basic primitive) polynomials of degree κ over R and dividing x pmκ 1 1 in R[x]. Since n is a positive integer satisfying gcd(p, n) = 1, there is an integer κ such that κ = min{s Z + (p m ) s 1 (mod n)}. Then n is a divisor 11

12 of p mκ 1. Using Lemma 3.1, we choose a fixed monic basic irreducible polynomial ς(z) of degree κ in R[z], and set R = R[z]/ ς(z) which is a Galois ring of characteristic p 2 and cardinality p 2mκ (cf. [36] Theorem 14.23). Now, we choose an invertible element ζ of R with multiplicative order p mκ 1 (cf. [36] Theorem 14.27) and set ω = ζ pmκ 1 n. Then ord(ω) = n and n 1 x n 1 = (x ω i ). For each 0 i n 1, let r i = min{l Z + i(p m ) l i (mod n)}; J (pm ) i = {i, ip m, ip 2m,..., ip (ri 1)m }. Then J (pm ) i is the p m -cyclotomic coset modulo n containing i, and the minimal polynomial of ω i over R is given by M i (x) = s J (pm ) i r i 1 (x ω s ) = (x ω iplm ) which is a monic basic irreducible polynomial in R[x] and satisfies M i (x) (x n 1). Moreover, we have r i 1 Mi (x) = x r i ( 1 r x i 1 ωiplm ) = u (x ω iplm ) = um j (x), l=0 where 0 j n 1 satisfying j i (mod n) and u = ω i( r i 1 l=0 plm) R (cf. [17] Lemma 2.3(ii)). Then we have the following conclusion. Lemma 3.2 M i (x) is a self-reciprocal polynomial if and only if J (pm ) i = J (pm ) i (mod n), i.e., i(plm + 1) 0 (mod n) for some integer l, 0 l n 1. From now on, we define θ(a(x)) = a(x 1 ) = a(x n 1 ) (mod x n 1), a(x) A = R[x]/ x n 1. Then one can easily verify that θ is a ring automorphism on A satisfying θ 2 = id A and θ(r) = r for any r R. Lemma 3.3 Using the notations in Lemma 2.3 and Section 2, in the ring A the following hold. l=0 l=0 12

13 (i) θ(ε i (x)) = ε i (x), θ(a i ) = A i and the restriction θ Ai of θ on A i is a ring automorphism on A i, for all 0 i r + t. (ii) For each 0 i r + t, define θ i : A i A i by θ i (b(x)) = b(x n 1 ) (mod f i (x)). Then the following diagram for ring isomorphisms commutes A i φ i A i θ i θ Ai Ai φ i Ai i.e., θ i = φ 1 i (θ Ai )φ i. Hence θ 2 i = id Ai. Proof. (i) As f i (x) is self-reciprocal, by Lemma 3.2 and the assumption of f i (x) it follows that f i (x) = j J (x ωj ), where J = J (pm ) s J (pm ) s and J (pm ) s = { j (mod n) j J (pm ) s } for some 0 s n 1. From this, by Equations (1) and the definition of ε i (x) in Section 2 we deduce that ε i (ω j ) = 1 if j J and ε i (ω j ) = 0 otherwise, 0 j n 1, which implies ε i (x) = 1 n n 1 k=0( j J ω jk )x k. Then by J = J (mod n) and x n = 1 in A, it follows that θ(ε i (x)) = ε i (x 1 ) = 1 n = 1 n 1 ( n k =0 n 1 j = j, j J k=0( j J ω jk )x k = 1 n ω j k )x k = ε i (x). n 1 k=0( j J ω ( j)( k) )x k Hence θ(a i ) = θ(ε i (x)a) = θ(ε i (x))θ(a) = ε i (x)a = A i. Therefore, the restriction θ Ai of θ on A i is a ring automorphism of A i. (ii) By the definition of φ i in Lemma 2.3(iii), we see that the inverse φ 1 i of φ i is a ring isomorphism from A i onto A i satisfying φ 1 i (w(x)) w(x) (mod f i (x)), w(x) A i, which implies φ 1 i (ε i (x)) 1 v i (x)f i (x) 1 (mod f i (x)), i.e., φ 1 i (ε i (x)) = 1 in A i. Then for any a(x) A i, by (i) and Lemma 2.3(iii) it follows that φ 1 i (θ Ai )φ i (a(x)) = φ 1 i (θ Ai (φ i a(x))) = φ 1 i (θ Ai (ε i (x)a(x))) 13

14 = φ 1 i = a(x 1 ) (mod f i (x)) = θ i (a(x)). (θ(ε i (x))a(x 1 )) = φ 1 i (ε i (x))φ 1 i (a(x 1 )) Hence θ i = φ 1 i (θ Ai )φ i. Now, we define the skew polynomial ring over A with indeterminate y by A[y; θ] = { k j=0 α jy j α j A, j = 0, 1,..., k, k = 0, 1,...}, where the multiplication is determined by ya(x) = θ(a(x))y = a(x 1 )y, a(x) A. Since y 2 a(x) = a(x)y 2 for all a(x) A, y 2 1 generates a two-sided ideal y 2 1 of A[y; θ]. In this paper, we denote R = A[y; θ]/ y 2 1 = {a(x) + b(x)y a(x), b(x) A} (y 2 = 1) which is the residue class ring of A[y; θ] modulo its ideal y 2 1. Similarly, for each 0 i r + t by Lemma 3.3(ii) we can define the skew polynomial ring over A i by A i [y; θ i ] = { k j=0 a j(x)y j a j (x) A i, j = 0, 1,..., k, k = 0, 1,...}, where the multiplication is determined by yb(x) = θ i (b(x))y = b(x n 1 )y (mod f i (x)), b(x) A i. As f i (x) is a minic divisor of x n 1 in R[x], x n 1 = x 1 in A i and y 2 1 generates a two-sided ideal y 2 1 of A i [y; θ i ]. In this paper, we denote R i = A i [y; θ i ]/ y 2 1 = {b 0 (x) + b 1 (x)y b 0 (x), b 1 (x) A i } (y 2 = 1) which is the residue class ring of A i [y; θ i ] modulo its ideal y 2 1. Lemma 3.4 Let 0 i r + t. We extend the ring isomorphism φ i from A i onto A i A to the following map b 0 (x) + b 1 (x)y φ i (b 0 (x)) + φ i (b 1 (x))y = ε i (x)(b 0 (x) + b 1 (x)y) (mod x n 1) ( b 0 (x) + b 1 (x)y R i ), and also denote this map by φ i. Then (i) The map φ i is an injective ring isomorphism from R i to R. (ii) The mapping Φ : R 0 R 1... R r+t R defined by r+t Φ(β 0, β 1,..., β r+t ) = φ i (β i ) ( β i R i, i = 0, 1,..., r + t) 14

15 is an isomorphism of rings. Proof. (i) For any b(x) A i, by Lemma 3.3(ii) it follows that φ i (yb(x)) = φ i (θ i (b(x)y)) = φ i (θ i (b(x))y = (φ i θ i )(b(x))y = (θ Ai φ i )(b(x))y = θ(φ i (b(x)))y = yφ i (b(x)). Since φ i is a ring isomorphism from A i onto A i by Lemma 2.3(iii), we conclude that φ i is an injective homomorphism of rings from R i to R. (ii) By (i) and its proof, we conclude that Φ is a ring homomorphism from R 0 R 1... R r+t to R. For any β i = b i0 (x) + b i1 (x)y R i where b i0 (x), b i1 (x) A i, it is clear that (β 0, β 1,..., β r+t ) Ker(Φ) if and only if r+t φ i(β i ) = 0 in R, where r+t φ i (β i ) = r+t ε i (x)(b i0 (x) + b i1 (x)y) (by (i)) r+t r+t = ( ε i (x)b i0 (x)) + ( ε i (x)b i0 (x))y and ε i (x)b i0 (x), ε i (x)b i1 (x) A i by Lemma 2.3(iii). As A = r+t A i by Lemma 2.3(ii), we deduce that r+t φ i (β i ) = 0 r+t r+t ε i (x)b i0 (x) = ε i (x)b i1 (x) = 0 in A ε i (x)b i0 (x) = ε i (x)b i1 (x) = 0 ( i = 0, 1,..., r + t), where ε i (x)b i0 (x) = φ i (b i0 (x)), ε i (x)b i1 (x) = φ i (b i1 (x)) A i for all i = 0, 1,..., r + t. Since φ i is a ring isomorphism from A i onto A i by Lemma 2.3(iii), we conclude that (β 0, β 1,..., β r+t ) Ker(Φ) if and only if b i0 (x) = b i1 (x) = 0, i.e. β i = 0, for all i = 0, 1,..., r + t. So Ker(Φ) = {(0, 0,..., 0)} and hence Φ is injective. Moreover, by Lemma 2.3 (iii) and (ii) we have R 0 R 1... R r+t = r+t r+t r+t r+t R i = A i 2 = A i 2 = ( A i ) 2 = A 2 = R. Therefore, φ i is a ring isomorphism from R i onto R. 15

16 Let 0 i r + t and C i be an A i -submodule of A 2 i = {(b 0 (x), b 1 (x)) b 0 (x), b 1 (x) A i }. Recall that C i is called a skew θ i -cyclic code if (θ i (b 1 (x)), θ i (b 0 (x))) C i, (b 0 (x), b 1 (x)) C i. From now on, we will identify each (b 0 (x), b 1 (x)) C i with b 0 (x) + b 1 (x)y R i = A i [y; θ i ]/ y 2 1. Then it is clear that C i is a skew θ i -cyclic code over A i of length 2 if and only if C i is a left ideal of R i (See [6, Theorem 1]). For more details on skew cyclic codes, readers are referred to [5] [9] and [26]. Now, we give a direct sum decomposition for each left D 2n -code over R by the following lemma. Lemma 3.5 For any α = n 1 a i,0x i + n 1 a i,1x i y R[D 2n ] where a i,0, a i,1 R for all i = 0, 1,..., n 1, we identify α with a 0 (x) + a 1 (x)y R where a 0 (x) = n 1 a i,0x i, a 1 (x) = n 1 a i,1x i A. Then the following statements are equivalent: (i) C is a left D 2n -code over R; (ii) C is a left ideal of the ring R = A[y; θ]/ y 2 1 ; (iii) For each 0 i r + t, there is a unique skew θ i -cyclic code C i over A i of length 2 such that Therefore, C = r+t C i. C = r+t (A i φi C i ). Proof. (i) (ii). As y 2 = 1 and yx = x 1 y in the group D 2n, we see that the identification of R[D 2n ] with R is a ring isomorphism. Hence C is a left ideal of R[D 2n ] if and only if C is a left ideal of R. (ii) (iii). By Lemma 3.4(ii), we deduce that C is a left ideal of R if and only if for each 0 i r + t, there is a unique left ideal C i of R i such that C = Φ(C 0 C 1... C r+t ) = r+t φ i(c i ) = r+t {φ i(b i,0 (x)) + φ i (b i,0 (x))y b i,0 (x) + b i,1 (x)y C i } = r+t {(φ i(b i,0 (x)), φ i (b i,0 (x)) (b i,0 (x), b i,1 (x)) C i } = r+t (A i φi C i ). In this case, we have C = C 0 C 1... C r+t = r+t C i. By Lemma 3.5, in order to list all left D 2n -codes over R it is sufficiency to determine all left ideals of the ring R i = A i [y; θ i ]/ y 2 1 where A i = 16

17 R[x]/ f i (x) for all i = 0, 1,..., r + t. Using the notations of Section 2, we have the following two cases: (I) 0 i r. In this case, f i (x) is a monic basic irreducible and self-reciprocal polynomial in R[x]. By Lemma 2.2(i), A i = R[x]/ f i (x) is a Galois ring of characteristic p 2 and cardinality p 2md i, and there exists ζ i (x) = d i 1 j=0 ζ ijx j A i with ζ ij R for all 0 j d i 1 such that ord(ζ i (x)) = p md i 1 in A i. Hence each element β of A i has a unique p-expansion: β = b 0 (x) + pb 1 (x), b 0 (x), b 1 (x) ζ i (x) {0} (3) where ζ i (x) {0} is a Teichmüller set of A i with ζ i (x) = {ζ i (x) k k = 0, 1,..., p md i 2} A i (cf. [36] Theorem 14.27). By [36] Section 14.6 and Equation (3), the generalized Frobenius automorphism ϕ i of A i over R is defined by ϕ i (b 0 (x) + pb 1 (x)) = b 0 (x) pm + pb 1 (x) pm, b 0 (x), b 1 (x) ζ i (x) {0}. (4) The set of all automorphism of A i over R form a group with respect to the map composition of maps, which is called the Galois group of A i over R and is denoted by Gal(A i /R). It is known that the multiplicative order of ϕ i is equal to d i and Gal(A i /R) = ϕ i = {ϕ 0 i, ϕ i, ϕ 2 i,..., ϕ d i 1 i }. (5) As is a ring homomorphism from R onto F p m defined at the beginning of Section 2, it follows that f i (x) is a monic irreducible and self-reciprocal polynomial in F p m[x] of degree d i. From now on, we denote K i = F p m[x]/ f i (x) and ζ i (x) = d i 1 j=0 ζ ijx j K i with ζ ij F p m. Then K i is an extension field of F p m with degree d i, which implies K i = p md i. By Equation (3), the surjective ring homomorphism : R F p m can be extended to a surjective homomorphism of rings from A i onto K i, which is also denoted by, as follows { ζi (x) β = b 0 (x) + pb 1 (x) β = k (mod f i (x)) if b 0 (x) = ζ i (x) k ; 0 if b 0 (x) = 0. 17

18 where b 0 (x), b 1 (x) ζ i (x) {0}. Hence ζ i (x) is a primitive element of K i, i.e, ord(ζ i (x)) = p md i 1. Therefore, K i = {0} {ζ i (x) k k = 0, 1,..., p md i 2}. By [36] Section 7.1, the Frobenius automorphism σ i of K i over F p m is defined by σ i (α) = α pm, α K i. The set of all automorphism of K i over F p m form a group with respect to the map composition of maps, which is called the Galois group of K i over F p m and is denoted by Gal(K i /F p m). It is known that the multiplicative order of σ i is equal to d i and Gal(K i /F p m) = σ i = {σi 0, σ i, σi 2,..., σ d i 1 i }. Lemma 3.6 Using the notations above, we have the following conclusions. (i) (cf. [36] Theorem 14.32(iii)) The following diagram commutes: A i K i ϕ i Ai. σ i Ki (ii) Let d i > 1 where 1 i r. Then d i is even and θ i = ϕ d i 2 i. Hence θ i (ξ) = ξ p md i 2, ξ ζ i (x) {0}. (iii) Let d i > 1, 1 i r, and w(x) = ζ i (x) (p md i 2 1)s where 0 s p md i 2. Then w(x) θ i (w(x)) = 1, i.e., θ i (w(x)) = w(x) 1. Proof. (ii) As d i > 1 and 1 i r, by Lemma 3.2 there exists a lest positive integer j such that i(p m ) j i (mod n), which implies i(p m ) 2j i(p m ) j i (mod n), and so d i = min{l Z + i(p m ) l i (mod n)} being a divisor of 2j. As i(p m ) j i (mod n), d i is not a divisor of j. Hence d i = 2j. As stated above, by Equation (5) we see that Gal(A i /R) is a cyclic group generated by ϕ i with even order d i. Since θ i is an automorphism of A i over R of order 2, we conclude that θ i = ϕ d i 2 i. From this and by Equation (4), we deduce that θ i (ξ) = ϕ d i 2 i (ξ) = ξ (pm ) di 2 = ξ p md i 2 for all ξ ζ i (x) {0}. (iii) It follows that w(x) θ i (w(x)) = ζ i (x) ((p md i 2 1)+p md i 2 (p md i 2 1))s = 1, by (ii) and ζ i (x) pmdi 1 = 1. 18

19 In the following, we adopt the following notation. Let θ i : K i K i be defined by θ i (c(x)) = c(x n 1 ) (mod f i (x)), for all c(x) = d i 1 j=0 c jx j K i with c j F p m. Then θ i is a ring automorphism of K i such that the following diagram commutes: A i K i θ i Ai θ i Ki, i.e., θ i (β) = θ i (β), β A i. (6) Moreover, by Lemma 3.6(ii) we deduce that θ i = σ d i 2 i when d i > 1 and 1 i r. Hence ζ i (x) (p md i 2 1)s θ i (ζ i (x) (p md i 2 1)s ) = 1 for any integer 0 s p md i 2 by Lemma 3.6(iii). Let K i [y; θ i ] be the skew polynomial ring over the finite field K i determined by θ i. Denote Γ i = K i [y; θ i ]/ y 2 1 which is the residue class ring of K i [y; θ i ] modulo its two-sided ideal y 2 1 generated by y 2 1. Then we extend the surjective ring homomorphism : A i K i to a map from R i onto Γ i, which is denoted by as well, by the natural way: β 0 + β 1 y = β 0 + β 1 y, β 0, β 1 A i. For any β A i, by Equation (6) it follows that yβ = θ i (β)y = θ i (β)y = θ i (β)y = yβ. From this, it can be verify easily that is a surjective ring homomorphism from R i onto Γ i = K i [y; θ i ]/ y 2 1. By W i = {ζ i (x) (p md i 2 1)s 0 s p md i 2 } A i (see Section 2), we have W i = {w(x) w(x) W i } = {ζ i (x) (p md i 2 1)s 0 s p md i 2 } K i. Then W i is the unique subgroup of K i with order p md i For ideals of Γ i = K i [y; θ i ]/ y 2 1, we known the following conclusion. Lemma 3.7 Let d i > 1. We have the following conclusions: (i) (cf. [18, Theorem 3.3]) Using the notations above, all distinct left ideals of the ring Γ i = F p m[y; θ i ]/ y 2 1 are given by the following: 0, Γ i, Γ i (w(x) + y) where w(x) W i. 19

20 Therefore, the number of left ideals of Γ i is equal to p md i (ii) (cf. [18, Theorem 3.4 and Corollary 3.5]) For any w(x) W i, Γ i (w(x)+y) is a minimal left ideal of Γ i, Γ i (w(x)+y) = p md i, the minimum Hamming weight w (K i) H (Γ 2 i(w(x) + y)) of Γ i (ζ i (x) 1)s + y) over K i is e- qual to 2 and Γ i = Γ i (w 1 (x) + y) Γ i (w 2 (x) + y) for any w 1 (x), w 2 (x) W i satisfying w 1 (x) w 2 (x). (p mdi Using the notations above, we give the following conclusions for left ideals of the ring R i = A i [y; θ i ]/ y 2 1. Theorem 3.8 Let 0 i r and deg(f i (x)) = d i = 1. Then R i = R[y]/ y 2 1 which is a commutative ring. (i) (cf. [32] Definition 4.1 and Theorem 4.4)If p is odd, there are 9 distinct ideals of R i which are given by: {0}, R i p, R i p(y 1), R i p(y+1), R i, R i (y 1), R i (y + 1), R i (y 1) + R i p, R i (y + 1) + R i p. Moreover, we have {0} = 1, R i = p 4m, R i p(y 1) = R i p(y+1) = p m, R i (y 1) = R i (y + 1) = p 2m, R i (y 1) + R i p = R i (y + 1) + R i p = p 3m. (ii) (cf. [28] Theorem 3.8) If p = 2, there are 2 m + 5 distinct ideals of R i which are given by: {0}, 2R i, 2R i (y 1), R i, R i (y 1) + 2R i, R i ((y 1) + 2u) with u T, where T is a Teichmüller set of R. Moreover, we have {0} = 1, R i = 2 4m, 2R i = R i ((y 1)+2u) = 2 2m, 2R i (y 1) = 2 m and R i (y 1) + 2R i = 2 3m. Theorem 3.9 Let 0 i r and deg(f i (x)) = d i 2. Then all skew θ i -cyclic code C i over A i of length 2, i.e., all left ideals of the ring R i = A i [y; θ i ]/ y 2 1, are given by the following table: case N i C i (left ideals of R i ) C i d (1) 1 {0} 0 0 (2) 2 R i p j (j = 0, 1) p 2(2 j)md i 1 (3) p md i R i p(w(x) + y) (w(x) W i ) p md i 2 (4) p md i + p md i 2 R i (w(x)(1 + pϑ(x)) + y) p 2md i 2 (w(x) W i, ϑ(x) V i ) (5) p md i R i (w(x) + y) + R i p (w(x) W i ) p 3md i 1 where N i is the number of ideals in the same line of the table, d = w (A i) H (C i) is the minimum Hamming weight of C i as a linear code over A i of length 2. 20

21 Therefore, the number of left ideals in R i is equal to p md i + 3p md i Proof. For any nonzero left ideal C of R i, let C = {α α C}. Then it is clear that C is a left ideal of Γ i and τ i : α α ( α C) is an A i -linear homomorphism from C onto C. Hence Im(τ i ) = C. Let (C : p) = {α R i pα C}. It is known that (C : p) is a left ideal of R i satisfying C (C : p), which implies that (C : p) is a left ideal of Γ i satisfying C (C : p). Moreover, by Ker(τ i ) = {α C α = 0} = p(c : p) it follows that C = Im(τ i ) Ker(τ i ) = C (C : p). (7) Then by Lemma 3.7, we have one of the following cases. (i) C = {0}. As C {0}, by (7) it follows that (C : p) is a nonzero left ideal of Γ i. Then by Lemma 3.7 we have one of the following subcases: (i-1) (C : p) = Γ i. Then C R i p and 1 (C : p). The latter implies 1 + pα (C : p) for some α R i. Then by p 2 = 0 and the definition of (C : p), we have p = p(1 + pα) C, which implies R i p C. Hence C = R i p. By Lemma 3.7(ii) and Equation (7), we deduce that C = Γ i = p 2md i and w (A i) H (C) = w(k i) H ((C : p)) = w(k i) H (Γ i) = 1. (i-2) (C : p) = Γ i (w(x) + y) where w(x) W i. Then w(x) + y = w(x) + y (C : p), which implies w(x) + y + pα (C : p) for some α R i, and so p(w(x) + y) = p(w(x) + y + pα) C. Hence R i p(w(x) + y) C. Conversely, let ξ C. By C = {0}, there exists β R i such that ξ = pβ, which implies β (C : p), and so β (C : p). Then there exists γ, δ R i such that β = γ(w(x) + y + pα) + pδ, which implies ξ = p(γ(w(x) + y + pα) + pδ) = γ p(w(x) + y) R i p(w(x) + y). Hence C R i p(w(x) + y) and so C = R i p(w(x) + y). Then by Lemma 3.7(ii) and Equation (7), we have that C = Γ i (w(x) + y) = p md i and w (A i) H (C) = w(k i) H ((C : p)) = w(k i) H (Γ i(w(x) + y)) = 2. (ii) C = Γ i. Then 1 + pα C for some α R i. As C (C : p), we have (C : p) = Γ i. From this and by the proof of (i-2), we know that p C. Hence 1 = ( pα) α p C. Therefore C = R i = R i p 0 and hence C = R i = p 4md i. It is obvious that w (A i) H (C) = 1 in this case. (iii) C = Γ i (w(x) + y) where w(x) W i. By (y + w(x)) = w(x) + y C, there exists α R i such that y + w(x) + pα C. As C (C : p), by Lemma 3.8(ii) we have one of the following two subcases: 21

22 (iii-1) (C : p) = Γ i. In this case, by the proof of (i-1), we know that p C. Hence y +w(x) = (y +w(x)+pα) α p C, and so R i (y +w(x))+r i p C. Conversely, let ξ C. By ξ C = Γ i (y + w(x)), there exists γ, δ R i such that ξ = γ(y + w(x)) + pδ R i (y + w(x)) + R i p. Hence C R i (y + w(x)) + R i p C and so C = R i (y + w(x)) + R i p. Moreover, by Lemma 3.7(ii) and Equation (7) it follows that C = Γ i (w(x) + y) Γ i = p 3md i and w (A i) H (C) = w(k i) H ((C : p)) = w(k i) H (Γ i) = 1. (iii-2) (C : p) = Γ i ((w(x) + y). Then by the proof of (i-2), we know that p(w(x) + y) = p(y + w(x)) C. As α R i, there exist a(x), b(x) A i such that α = a(x) + b(x)y. Denote τ(x) a(x) b(x)w(x) (mod p), where τ(x) A i = K i. Then pτ(x) = p(a(x) b(x)w(x)) and hence y + w(x) + pτ(x) = y + w(x) + p(a(x) + b(x)y) p(b(x)y + b(x)w(x)) = (y + w(x) + pα) b(x) p(y + w(x)) C. Therefore, R i (y + w(x) + pτ(x)) C. Conversely, let ξ C. By C = Γ i (w(x) + y) = R i (y + w(x) + pτ(x)), there exist γ, δ R i such that ξ = γ(y + w(x) + pτ(x)) + pδ, which implies pδ = ξ γ(y + w(x) + pτ(x)) C, i.e., δ (C : p). From this and by (C : p) = Γ i ((w(x) + y), we deduce that δ = ρ(y + w(x)) + pη for some ρ, η R i. Therefore, ξ = γ(y + w(x) + pτ(x)) + p(ρ(y + w(x)) + pη) = γ(y + w(x) + pτ(x)) + ρ p((y + w(x) + pτ(x)) = (γ + pρ)(y + w(x) + pτ(x)) R i (y + w(x) + pτ(x)). Hence C = R i (y + w(x) + pτ(x)). Suppose that C = R i (y + w(x) + pν(x)) for some ν(x) K i as well. Then p(τ(x) ν(x)) = (y + w(x) + pτ(x)) (y + w(x) + pν(x)) C, which implies τ(x) ν(x) (C : p) = Γ i (w(x) + y) with ϑ(x) ν(x) K i. Since (Γ i(w(x) + y)) = 2 by Lemma 3.7(ii), we have τ(x) ν(x) = 0, i.e., τ(x) = ν(x) in K i. By Lemma 3.6(iii), we see that w(x) θ i (w(x)) = 1, i.e., θ i (w(x)) = w(x) 1. From this and by p 2 = 0, we deduce that (θ i (w(x)) + pθ i (τ(x))) 1 = w (K i) H 22

23 w(x) 2 (θ i (w(x)) pθ i (τ(x))). As C is a left ideal of R i and y 2 = 1, we have y + w(x) + p( w(x) 2 θ i (τ(x))) = y + w(x) 2 (θ i (w(x)) pθ i (τ(x))) = (θ i (w(x)) + pθ i (τ(x))) 1 ((θ i (w(x)) + pθ i τ(x))) y + 1) = (θ i (w(x)) + pθ i (τ(x))) 1 y (w(x) + pϑ(x) + y) C, which implies w(x) 2 θ i (τ(x)) = τ(x) in K i, i.e., θ i ( θi (w(x))τ(x) ) + θ i (w(x))τ(x) = w(x)θ i (τ(x)) + θ i (w(x))τ(x) = 0 since θ i (w(x)) = w(x) 1. Denote ϑ(x) = θ i (w(x))τ(x) K i. Then τ(x) = w(x)ϑ(x) and pτ(x) = pw(x)ϑ(x) where ϑ(x) satisfies the following equation ϑ(x) p md i 2 + ϑ(x) = θ i (ϑ(x)) + ϑ(x) = 0, (8) as θ i (ξ) = σ d i 2 i (ξ) = ξ p mdi 2 for all ξ K i. Let p = 2. Then Equation (8) is equivalent to (ϑ(x) 2 md i 2 1 1)ϑ(x) = 0. Since ζ i (x) is a primitive element of the finite field K i with multiplicative order 2 md i 1, in this case Equation (8) has exactly 2 md i 2 solutions in K i : ϑ(x) V i where V i = {0} {ζ i (x) (2 md i 2 +1)l l = 0, 1,..., 2 md i 2 2}. Let p be an odd prime. Since ζ i (x) is a primitive element of the finite field K i with multiplicative order p md i 1, we have ζ i (x) pmd i 1 2 = 1. Then Equation (8) is equivalent to ϑ(x) = 0 or ϑ(x) p md i 2 1 ζ i (x) pmd i 1 2 = ϑ(x) p md i = 0. (9) Let ϑ(x) = ζ i (x) k where 0 k p md i 2. Then Equation (9) is equivalent to k(p md i 2 1) pmd i 1 (mod p md i 1), i.e., 2 k 1 2 (p md i 2 + 1) (mod p md i 2 + 1). Therefore, Equation (8) has exactly p md i 2 solutions in K i : ϑ(x) V i where V i = {0} {ζ i (x) 1 2 (p md i 2 +1)+(p md i 2 +1)l l = 0, 1,..., p md i 2 2}. 23

24 As stated above, we conclude that C = R i (y + w(x) + pτ(x)) = R i (y + w(x)(1 + pϑ(x))) is uniquely determined by the pair (w(x), ϑ(x)) of polynomials where w(x) W i and ϑ(x) V i. Hence the number of left ideals is equal to (p md i 2 + 1)p md i 2 = p md i + p md i 2 in this case. Then by Lemma 3.7(ii) and Equation (7), we have C = Γ i (ζ i (x) k + y) 2 = p 2md i and w (A i) H (C) = w(k i) H ((C : p)) = w(k i) H (Γ i(ζ i (x) k + y)) = 2. (II) r + 1 i t. In this case, f i (x) = ρ i (x)ρ i (x) where ρ i (x) is a monic basic irreducible polynomial such that ρ i (x) and ρ i (x) are coprime in R[x]. Using the notations in Section 2, we have A i = R[x]/ f i (x) which is an extension commutative ring of R and A i = R 2d i = (p 2m ) 2d i = p 4md i. As is a ring homomorphism from R onto F p m defined in Section 2, we see that ρ i (x) is a mionic irreducible polynomial in F p m[x] of degree d i, the reciprocal polynomial of ρ i (x) in F p m[x] is given by ρ i (x) = ρ i (x). Then ρ i (x) is also a mionic irreducible polynomial in F p m[x] of degree d i satisfying gcd(ρ i (x), ρ i (x)) = 1. Hence f i (x) = ρ i (x)ρ i (x) which is a monic self-reciprocal polynomial in F p m[x] of degree 2d i. In the following, let K i = F p m[x]/ f i (x) = F p m[x]/ ρ i (x)ρ i (x). Then we extend the surjective homomorphism of rings : R F p m to a surjective homomorphism of rings from A i onto K i, which is also denoted by, in the natural way: : 2d i 1 j=0 β j x j 2d i 1 j=0 β j x j ( β j A i, j = 0, 1,..., 2d i 1). By Equation (2) in Section 2 and the definitions of ϵ i,1 (x) and ϵ i,2 (x) in Theorem 2.5(II), from classical ring theory we deduce the following lemma. Lemma 3.10 (cf. [37] Theorem 2.7 and its proof)using the notations above, we have the following conclusions: (i) ϵ i,1 (x) 2 = ϵ i,1 (x), ϵ i,2 (x) 2 = ϵ i,2 (x), ϵ i,1 (x)ϵ i,2 (x) = 0 and ϵ i,1 (x) + ϵ i,2 (x) = 1 in A i. (ii) A i = A i,1 A i,2, where A i,j = A i ϵ i,j (x) for j = 1, 2. (iii) The map χ : Υ i,1 Υ i,2 A i defined by χ(g 1 (x), g 2 (x)) = ϵ i,1 (x)g 1 (x) + ϵ i,2 (x)g 2 (x) (mod f i (x)), g j (x) Υ i,j 24

25 is a ring isomorphism where Υ i,1 = R[x]/ ρ i (x) and Υ i,2 = R[x]/ ρ i (x). (iv) θ i (ϵ i,1 (x)) = ϵ i,2 (x) and θ i (ϵ i,2 (x)) = ϵ i,1 (x). Then θ i induces a ring isomorphism between A i,1 and A i,2. Hence θ i is a ring automorphism of A i with multiplicative order 2. By Lemma 2.2(ii), Υ i,1 is a Galois ring of characteristic p 2 and cardinality p 2md i and there exists ζ i (x) = d i 1 k=0 ζ ik x k Υ i,1 with ζ ik R for all k such that ord(ζ i (x)) = p md i 1. Then every element of Υ i,1 has a u- nique p-expansion: a 0 (x) + pa 1 (x) where a 0 (x), a 1 (x) {0} {ζ i (x) l l = 0, 1,..., p md i 1} (mod ρ i (x)). Let θ i : K i K i be defined by θ i (c(x)) = c(x n 1 ) (mod f i (x)), c(x) = d i 1 j=0 c j x j K i with c j F p m. As f i (x) (x n 1) in F p m[x], θ i is a ring automorphism of K i such that the following diagram commutes: A i K i θ i Ai θ i Ki, i.e., θ i (β) = θ i (β), β A i. (10) Lemma 3.11 Using the notations above, denote π i,j = ϵ i,j (x) K i and K i,j = K i π i,j for j = 1, 2. Then we have the following conclusions: (i) π 2 i,1 = π i,1, π 2 i,2 = π i,2, π i,1 π i,2 = 0 and π i,1 π i,2 = 1 in K i. (ii) K i = K i,1 K i,2 where K i,j = A i,j = K i π i,j for j = 1, 2. (iii) K i,1 = {0} {ϵ i,1 (x)ζ i (x) k k = 0, 1,..., p md i 2} and K i,2 = {0} {ϵ i,2 (x)ζ i (x 1 ) k k = 0, 1,..., p md i 2}, which are both finite fields of cardinality p md i. (iv) θ i (π i,1 ) = π i,2 and θ i (π i,2 ) = π i,1. Then θ i induces a field isomorphism between K i,1 and K i,2. Hence θ i is a ring automorphism of K i with multiplicative order 2. 25

26 Proof. (i), (ii) and (iv) follow from Lemma 3.10 (i), (ii) and (iv), respectively. (iii) As ζ i (x) Υ i,1 and ord(ζ i(x)) = p md i 1 by Lemma 2.2(ii), every element of Υ i,1 has a unique p-expansion: b 0 (x) + pb 1 (x), b 0 (x), b 1 (x) T i,1 where T i,1 = {0} {ζ i (x) k k = 0, 1,..., p md i 2}. From this, by Lemma 3.10 (ii) and (iii), we deduce that A i,1 = {ϵ i,1 (x)b 0 (x) + pϵ i,1 (x)b 1 (x) b 0 (x), b 1 (x) T i,1 } (mod f i (x)), which implies K i,1 = A i,1 = {ϵ i,1 (x)b 0 (x) b 0 (x) T i,1 } = {0} {ϵ i,1 (x)ζ i (x) k 0 k p md i 2}. Hence K i,2 = θ i (K i,1 ) = θ i (A i,1 ) = {0} {ϵ i,2 (x)ζ i (x 1 ) k k = 0, 1,..., p md i 2} by Lemma 3.10(iv). Now, let K i [y; θ i ] be the skew polynomial ring over the finite commutative ring K i determined by θ i and denote Γ i = K i [y; θ i ]/ y 2 1 which is the residue class ring of K i [y; θ i ] modulo its two-sided ideal y 2 1 generated by y 2 1. Then we extend the surjective ring homomorphism : A i K i to the a map from R i onto Γ i, denoted by as well, by the natural way: β 0 + β 1 y = β 0 + β 1 y, β 0, β 1 A i. For any β A i, by Equation (10) it follows that yβ = θ i (β)y = θ i (β)y = θ i (β)y = yβ. From this, it can be verify easily that is a surjective ring homomorphism from R i onto Γ i. As U i,1 = {0} {ϵ i,1 (x)ζ i (x) k k = 0, 1,..., p md i 2} and W i = {u(x) + 1 u(x 1 ) 0 u(x) U i,1} (see Theorem 2.5(II)), we have that W i = {u(x) + 1 u(x 1 ) u(x) = ϵ i,1(x)ζ i (x) k, 0 k p md i 2} K i. Obviously, W i = W i = p md i 1. For ideals of Γ i = K i [y; θ i ]/ y 2 1, we have the following conclusions. Lemma 3.12 Let r + 1 i t. We have the following conclusions: (i) (cf. [18] Theorem 4.2) all distinct left ideals of Γ i = K i [y; θ i ]/ y 2 1 are given by: {0}, Γ i, Γ i ϵ i,1, Γ i ϵ i,2, Γ i (w(x) + y) where w(x) W i. 26

27 Moreover, we have the following properties for nonzero left ideals of Γ i : (i-1) Γ i ϵ i,j (x) = p 2md i and wt (K i) H (Γ iϵ i,j (x)) = 1 for j = 1, 2; Γ i = p 4md i and wt (K i) H (Γ i) = 1. (i-2) J = p 2md i and wt (K i) H (J) = 2 for any J = Γ i(w(x) + y) with w(x) W i. (ii) Let L = {Γ i ϵ i,j (x) j = 1, 2} {Γ i (w(x) + y) w(x) W i }. Then for any J 1, J 2 L satisfying J 1 j 2, we have Γ i = J 1 J 2. Hence every left ideal of Γ i contained in L is a minimal left ideal of Γ i. Proof. (ii) Let J 1, J 2 L satisfying J 1 J 2. Then J 1 = J 2 = p 2md i by (i). If J 1 J 2 = {0}, we have J 1 + J 2 = J 1 J 2 = p 4md i = Γ i, which implies J 1 + J 2 = Γ i, and hence Γ i = J 1 J 2. Suppose that J 1 J 2 {0}. Since J 1 J 2 is also a left ideal of Γ i satisfying J 1 J 2 J s for all s = 1, 2, by (i) it follows that J 1 J 2 = p 2md i, which implies J 1 = J 2, and we get a contradiction. Therefore, Γ i = J 1 J 2. Now, we list all left ideals of the ring R i = A i [y; θ i ]/ y 2 1 as follows. Theorem 3.13 Let r + 1 i t. Using the notations above, all skew θ i -cyclic code C i over A i of length 2, i.e., all left ideals of the ring R i = A i [y; θ i ]/ y 2 1, are given by the following table: case N i C i (left ideals of R i ) C i d (1) 1 {0} 0 0 (2) 2 R i p j (j = 0, 1) p 4(2 j)md i 1 (3) 2 R i pϵ i,j (x) (j = 1, 2) p 2md i 1 (4) p md i 1 R i p(w(x) + y) (w(x) W i ) p 2md i 2 (5) 2 R i ϵ i,j (x) + R i p (j = 1, 2) p 6md i 1 (6) 2p md i R i (ϵ i,j (x) + pb i,j (x)y) p 4md i 1 (b i,j K i,j, j = 1, 2) (7) p md i 1 R i (w(x) + y) + R i p (w(x) W i ) p 6md i 1 (8) p 2md i p md i R i (w(x) + pϑ(x) + y) p 4md i 2 (ϑ(x) V (w(x)) i, w(x) W i ) where N i is the number of left ideals in the same line of the table, d = w (A i) H (C i) is the minimum Hamming weight of C i as a linear code over A i. Therefore, the number of left ideals in R i is equal to p 2md i + 3p md i

28 Proof. For any nonzero left ideal C of R i, let C = {α α C} and (C : p) = {α R i pα C}. Similar to the proof of Theorem 3.9, it can be verified that (C : p) is a left ideal of R i satisfying C (C : p), which implies that both C and (C : p) are left ideals of Γ i satisfying C (C : p) and C = Im(τ i ) Ker(τ i ) = C (C : p). (11) Then by Lemma 3.12(i), we have one of the following cases. (i) C = {0}. As C {0}, by (11) it follows that (C : p) is a nonzero left ideal of Γ i. Then by Lemma 3.12(ii) we have one of the following subcases: (i-1) (C : p) = Γ i. By Lemma 3.12 and an argument similar to (i-1) in the proof of Theorem 3.9, it follows that C = R i p, C = Γ i = p 2md i and w (A i) H (C) = 1. (i-2) (C : p) = Γ i ϵ i,j (x), where j = 1, 2. Then (C : p) = R i ϵ i,j (x), which implies that ϵ i,j (x) + pα (C : p) for some α R i, and hence pϵ i,j (x) = p(ϵ i,j (x) + pα) C. From this we deduce that R i pϵ i,j (x) C. Conversely, let ξ C. By C = {0} there exists β R i such that ξ = pβ C, which implies β (C : p), and so β (C : p) = R i ϵ i,j (x). Then there exist γ, δ R i such that β = γϵ i,j (x) + pδ, and hence ξ = p(γϵ i,j (x) + pδ) = γ(pϵ i,j (x)) R i pϵ i,j (x). Therefore, C R i pϵ i,j (x) and so C = R i pϵ i,j (x). Moreover, by Lemma 3.12(i) and Equation (11) it following that C = Γ i ϵ i,j = p 2md i and w (A i) H (C) = w(k i) H ((C : p)) = w(k i) H (Γ iϵ i,j (x)) = 1. (i-3) (C : p) = Γ i (w(x) + y) where w(x) W i. By Lemma 3.12 and an argument similar to (i-2) in the proof of Theorem 3.9, it follows that C = R i p(w(x) + y), C = Γ i (w(x) + y) = p 2md i and w (A i) H (C) = 2. (ii) C = Γ i. By an argument similar to (ii) in the proof of Theorem 3.9, it follows that C = R i = R i p 0, C = R i = p 8md i and w (A i) H (C) = 1. (iii) C = Γ i ϵ i,j (x) where j = 1, 2. By ϵ i,j (x) C, there exists α R i such that ϵ i,j (x) + pα C. As C (C : p), by Lemma 3.7(ii) we have one of the following two subcases: (iii-1) (C : p) = Γ i. In this case, by the proof of (i-1), we know that p C. Hence ϵ i,j (x) = (ϵ i,j (x) + pα) α p C. Therefore, R i ϵ i,j (x) + R i p C. Conversely, let ξ C. By ξ C = Γ i ϵ i,j (x) = R i ϵ i,j (x), there exists γ, δ R i such that ξ = γϵ i,j (x) + pδ R i ϵ i,j (x) + R i p. Hence C R i ϵ i,j (x) + R i p C and so C = R i ϵ i,j (x) + R i p. Moreover, by Lemma 3.12(i) and Equation (11) it follows that C = Γ i ϵ i,j Γ i = p 6md i and w (A i) H (C) = w(k i) H ((C : p)) = w(k i) H (Γ i) = 1. 28