Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉

Xiying Zheng; Bo Kong

doi:10.1515/math-2018-0045

Article Open Access

Constacyclic codes over 𝔽_p^m[u₁, u₂,⋯,u_k]/〈 u_i² = u_i, u_iu_j = u_ju_i〉

Xiying Zheng and Bo Kong

Published/Copyright: May 10, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, we study linear codes over ring R_k = 𝔽_p^m[u₁, u₂,⋯,u_k]/〈ui2 = u_i, u_iu_j = u_ju_i〉 where k ≥ 1 and 1 ≤ i, j ≤ k. We define a Gray map from RkntoFpm2kn and give the generator polynomials of constacyclic codes over R_k. We also study the MacWilliams identities of linear codes over R_k.

Keywords: Constacyclic codes; Cyclic codes; Gray map; Self-orthogonal codes

MSC 2010: 94B15

1 Introduction

Constacyclic codes are an important class of linear codes and have good error-correcting properties as well as have practical applications since they can be encoded with shift registers. have practical applications as they can be encoded with shift registers. Constacyclic codes over finite rings are well-known as they have rich algebraic structures for efficient error detection and correction, which explain their preferred role in engineering. In recent years, due to their rich algebraic structure, constacyclic codes have been studied over finite fields [1, 2, 3, 4]. The class of finite chain rings has been studied, by many authors, [5, 6, 7, 8]. There is a lot of work on constacyclic codes over finite rings of the form 𝔽_p^m + u 𝔽_p^m + ⋯ + u^e−1𝔽_p^m by many authors, where u^e = 0. For example, Chen et al. in [9] gave the structures of all (a + bu)-constacyclic codes of length 2p^s over ring 𝔽_p^m + u 𝔽_p^m. Sobhani in [10] completely determined the structure of (δ + αu²)-constacyclic codes of length p^k over 𝔽_p^m + u 𝔽_p^m + u²𝔽_p^m. Liu and Xu in [11] gave the structure of cyclic and negacyclic codes of length 2p^s over 𝔽_p^m + u 𝔽_p^m. Abualrub and Siap in [12] gave the structure of (1 + u)-constacyclic codes of arbitrary length n over 𝔽₂ + u𝔽₂. Kai et al. in [13] studied the (1 + λu)-constacyclic codes of arbitrary length n over 𝔽_p[u]/〈u^m〉, where (1 + λu) is a unite of 𝔽_p[u]/〈u^m〉. Guenda and Gulliver in [14] gave the structure of repeated root constacyclic codes of length mp^s over 𝔽_p^r + u𝔽_p^r + ⋯ + u^e−1𝔽_p^r.

The class of finite commutative rings of the form R + uR has been studied by many authors, where u² = 1. For example, in [15] Cengellenmis gave the structure of cyclic codes over 𝔽₃ + v𝔽₃, where v² = 1. ¨Qzen et al. in [16] gave the structure of cyclic and some constacyclic codes over the ring ℤ₄[u]/〈u² − 1〉. The class of finite commutative rings of the form 𝔽_p^m + u 𝔽_p^m has been studied by many authors, where u² = u. For example, in [17], Kong and Chang described the structure of cyclic codes and self dual cyclic codes over 𝔽_p + u𝔽_p, where u² = u. Cengellenmis et al. in [18] gave the structure of codes over 𝔽₂[u₁, u₂,⋯, u_k]/〈ui2 = u_i, u_iu_j = u_ju_i〉 with a Gray map. Li et al. in [19] gave the structure of linear codes over ℤ₄[u, v]/〈u² = u, v² = v, uv = vu〉. In [20], the generators of cyclic codes and (λ₁ + λ₂u + λ₃v + λ₄uv)-constacyclic codes over 𝔽_p + u𝔽_p + v𝔽_p + uv𝔽_p were given. The purpose of this paper is to continue this line of research. We determine the algebraic structures of all λ-constacyclic codes of 𝔽_p^m[u₁, u₂,⋯, u_k]/〈ui2 = u_i, u_iu_j = u_ju_i〉, where λ is an arbitrary unit of the ring 𝔽_p^m[u₁, u₂,⋯, u_k]/〈ui2 = u_i, u_iu_j = u_ju_i〉.

The remainder of this paper is organized as follows. In section 2, we provide the preliminaries that we need and define a Gray map from RkntoFpm2kn. In section 3, we study the Gray image of linear codes over R_k. In section 4, we give the structure of constacyclic codes of arbitrary length over R_k.

2 Preliminaries

An ideal I of a finite commutative ring R is called principal if it is generated by one element. R is a principal ideal ring if its ideals are principal. R is called a local ring if R has a unique maximal ideal. R is called a chain ring if its ideals are linearly ordered by inclusion.

As defined in [18], let

Rk=Fpm[u1,u2,⋯,uk]/〈ui2=ui,uiuj=ujui〉.

For any subset A ⊆ {1, 2, ⋯, k}, let

uA=∏i∈Aui

with the convention that u_∅ = 1. Then any element of R_k can be represented as

∑A⊆{1,2,⋯,k}cAuA,cA∈Fpm.

We can easily observe that

uAuB=uA⋃B.

Let P_k be the power set of the set {1, 2, ⋯, k}.

It follows that

(∑A∈PkcAuA)(∑B∈PkcBuB)=∑D∈Pk(∑A∪B=DcAcB)uD.

By the same method of Theorem 2.3 and Lemma 2.4 in [18] we have the following theorem:

Theorem 2.1

The ideal 〈w₁, w₂,⋯, w_k〉, wherew_i ∈ {u_i, 1 − u_i}, is an ideal of cardinalityp^{m(2^k−1)}and there are 2^ksuch ideals.

Let ω_i = 〈w_i1, w_i2,⋯, w_ik〉 be an ideal as described in Theorem 2.1, where w_ij ∈ {u_j, 1 − u_j}, 1 ≤ i ≤ 2^k. An element e is called an idempotent element if e² = e. For x, y ∈ R_k, x, y are called orthogonal if xy = 0. Let e_i = w_i1w_i2 ⋯ w_ik, where i = 1, 2, ⋯, 2^k. We know that u_i² = u_i, (1 − u_i)² = 1 − u_i, u_i(1 − u_i) = 0, so e₁, e₂, ⋯, e_2^k are pairwise orthogonal non-zero idempotent elements over R_k. By the induction method over R_k, we have 1 = e₁ + e₂ + ⋯ + e_2^k. By the Chinese Remainder Theorem, we have that R_k = e₁R_k + e₂R_k + ⋯ + e_2^kR_k, and for any element r ∈ R_k, r can be expressed uniquely as r = r₁e₁ + r₂e₂ + ⋯ + r_2^ke_2^k, where r_i ∈ 𝔽_p^m, i = 1, 2,⋯, 2^k.

Theorem 2.2

R_k ≅ R_k/ω₁ × … × R_k/ω_2^k.

Proof

First, we prove that ⋂i=12kωi={0}.

We use mathematical induction over R_k.

Base case: Setting over R₁, we get

⋂i=12ωi=〈u1〉⋂〈1−u1〉=〈u1−u12〉={0}.

Induction step: Over R_k−1, suppose that

⋂i=12k−1ωi′={0},

where ωi′ = 〈w_i1, w_i2, ⋯, w_ik−1〉, w_ij ∈ {u_j, 1 − u_j}, 1 ≤ i ≤ 2^k−1, 1 ≤ j ≤ k − 1.

Then over R_k

⋂i=12kωi=⋂i=12k〈wi1,wi2,⋯,wik〉=(⋂i=12k−1〈wi1,wi2,⋯,wik−1,uk〉)⋂(⋂i=12k−1〈wi1,wi2,⋯,wik−1,1−uk〉)=(⋂i=12k−1(ωi′+〈uk〉))⋂(⋂i=12k−1(ωi′+〈1−uk〉))=(⋂i=12k−1ωi′+〈uk〉)⋂(⋂i=12k−1ωi′+〈1−uk〉)=〈uk〉⋂〈1−uk〉=〈uk−uk2〉={0},

where ω_i = 〈w_i1, w_i2,⋯, w_ik〉, w_ij ∈ {u_j, 1 − u_j}, 1 ≤ i ≤ 2^k, 1 ≤ j ≤ k.

Secondly, we prove that ω₁, ω₂, ⋯, ω_2^k are pairwise coprime. For any two different ideals ω_i, ω_j, there exist u_t ∈ ω_i, (1 − u_t) ∈ ω_j, such that 1 ∈ ω_i + ω_j, then ω_i + ω_j = R_k. So ω₁, ω₂, ⋯, ω_2^k are pairwise coprime.

By the Chinese Remainder Theorem, we can get that R_k ≅ R_k/ω₁ × … × R_k/ω_2^k. □

Theorem 2.3

The ringR_khas cardinalityp^{m2^k}. The idealω_iis a maximal ideal ofR_k, wherei = 1, 2,⋯, 2^k. Consequently, R_k ≅ Fpm2k.

Proof

By Theorem 2.2, we have that |Rk|=|Rk||ω1|×⋯×|Rk||ω2k|. By Theorem 2.1 |ω_i| = p^{m(2^k−1)}, where i = 1, 2,⋯, 2^k.

We have that |R_k| = p^{m2^k}. Thus |Rk||ωi| = p^m, where i = 1, 2,⋯, 2^k. So ω_i is a maximal ideal of R_k, we can get that R_k/ω_i≅ 𝔽_p^m, where i = 1, 2,⋯, 2^k. So R_k ≅ Fpm2k. □

Corollary 2.4

There are (p^m − 1)^{2^k}units in the ringR_k.

Proof

There are (p^m − 1) units in 𝔽_p^m. By Theorem 2.3, we know there are (p^m − 1)^{2^k} units in the ring R_k. □

Theorem 2.5

(cf. [21, Theorem 2]). The ringR_kis a principal ideal ring, not a chain ring.

We define the Gray map as follows:

For r = r₁e₁ + r₂e₂ + ⋯ + r_2^ke_2^k ∈ R_k, we define ϕ : r ↦ (r₁, r₂, ⋯, r_2^k). We expand ϕ as:

Φ:Rkn→Fpm2kn(c0,c1,⋯,cn−1)↦(r1,0,⋯,r1,n−1,r2,0,⋯,r2,n−1,⋯,r2k,0,⋯,r2k,n−1),

where c_i = r_1,ie₁ + r_2,ie₂ + ⋯ + r_2^k,ie_2^k ∈ R_k.

A linear code C of length n over R_k is an R_k-submodule of Rkn. Every codeword c in such a code C is just an n-tuple of the form c = (c₀, c₁, ⋯, c_n−1) ∈ Rkn, and can be represented by a polynomial in R_k[x] as follows:

c=(c0,c1,⋯,cn−1)⟷c(x)=∑i=0n−1cixi∈Rk[x].

We define a constacyclic shift operator as:

σλ(c0,c1,⋯,cn−1)=(λcn−1,c0,⋯,cn−2).

If for any c ∈ C, we have σ_λ(c) ∈ C, then C is called λ-constacyclic code over R_k. Let a = (a₀, a₁, ⋯, a_n−1) and b = (b₀, b₁, ⋯, b_n−1) be two elements of Rkn. Then the usual inner product of a and b is defined as a⋅b=∑i=0n−1aibi. If a ⋅ b = 0, then a and b are said to be orthogonal.

The dual code of C is C^⊥ = {a| ∀ b ∈ C, a ⋅ b = 0}, which is also a linear code. A code C is self-orthogonal if C ⊆ C^⊥ and self dual if C = C^⊥.

For all r ∈ R_k, define the Lee weight of r as follows: w_L(r) = w_H(ϕ(r)), where let w_H(ϕ(r)) denote the Hamming weight of the image of r under ϕ.

For all x = (x₁, x₂,⋯, x_n) ∈ Rkn, define the Lee weight of x as follows w_L(x) = ∑i=1nw_L(x_i), the Lee distance of codewords x, y over Rkn is defined as d_L(x, y) = w_L(x − y). The Lee distance of C is defined by

dL(C)=min{dL(x−y),x,y∈C,x≠y}.

By the definition of the Gray map and the Lee weight of R_k, we can get that Φ is one-to-one and a distance preserving linear map from RkntoFpm2kn.

3 Linear codes over R_k

Using the polynomial representation of codewords in Rkn, we easily have the following.

Lemma 3.1

A subsetCofRknisa λ-constacyclic code of lengthnoverR_kif and only if its polynomial representation is an ideal of the ringR_k[x]/〈xⁿ - λ〉.

For any r = (r⁽⁰⁾, r⁽¹⁾, ⋯, r⁽ⁿ⁻¹⁾) ∈ Rkn, where r⁽ⁱ⁾ = ∑j=12kr_ije_j, i = 0, 1, ⋯, n − 1. Then r can be uniquely express as r = ∑j=12kr_je_j, where r_j = (r_0j, r_1j, ⋯, r_n−1,j) ∈ Fpmn,j = 1, 2,⋯, 2^k.

For any r, s ∈ Rkn, where s = ∑j=12ks_je_j, s_j = (s_0j, s_1j, ⋯, s_n−1,j) ∈ Fpmn, we can get that

r⋅s=∑j=12k(rj⋅sj)ej,

where rj⋅sj=∑i=0n−1(rijsij).

Let C be a linear code over R_k. For j = 1, 2,⋯, 2^k, we denote C_j as follows:

Cj={rj∈Fpmn|∑i=12kriei∈C,ri∈Fpmn,},j=1,2,⋯,2k.

Clearly, C_j is a linear code of length n over 𝔽_p^m.

By the definition above we have the following theorems easily.

Theorem 3.2

LetCbe a linear code overR_k, thenC=∑j=12kejCj,|C|=∏j=12k|Cj|,whereC₁, C₂, ⋯, C_2^kare linear codes of lengthnover 𝔽_p^m, and the direct sum decomposition is unique.

Theorem 3.3

LetCbe a linear code overR_k, thenC⊥=∑j=12kejCj⊥,whereCj⊥is the dual code ofC_j, wherej = 1, 2,⋯, 2^k.

Proof

Let C~=∑j=12kejCj⊥. For anyc∈C,c~∈C~,c⋅c~=∑j=12k(cjcj~)ej, wherec=∑j=12kejcj,c~=∑j=12kejcj~,cj∈Cj,cj~∈Cj⊥. Then c ⋅ c̃ = 0, and thus C̃ ⊆ C^⊥. The ring R_k is a principal ideal ring and thus a Frobenius ring, we have |C||C^⊥| = |R_k|ⁿ. Thus

|C~|=∏j=12k|Cj⊥|=∏j=12kpn|Cj|=|Rk|n|C|=|C⊥|.

So C^⊥ = C̃. □

Theorem 3.4

LetCbe a linear code overR_k, thenCis a self-orthogonal code if and only ifC_jis a self-orthogonal over 𝔽_p^m, wherec=∑j=12kejcj.Cis a self-dual code if and only ifC_jis a self-dual code over 𝔽_p^m, wherej = 1, 2,⋯, 2^k.

Proof

By Theorems 3.2 and 3.3, C ⊆ C^⊥ if and only if C_j ⊆ Cj⊥, so if C is a self-orthogonal code then C_j is a self-orthogonal code over 𝔽_p^m, where j = 1, 2,⋯, 2^k. Similarly, C is a self-dual code then C_j is a self-dual code over 𝔽_p^m, where j = 1, 2,⋯, 2^k. □

Let C be a linear code of length n over R_k, for any c = c₁e₁ + c₂e₂ + ⋯ + c_2^ke_2^k ∈ C, Φ(c) = (c₁, c₂, ⋯, c_2^k) ∈ Fpm2kn. Let C₁,C₂, ⋯, C_2^k be linear codes of length n over 𝔽_p^m, we define

C1×C2×⋯×C2k={(c1,c2,⋯,c2k),ci∈Ci,i=1,2,⋯,2k}.

Theorem 3.5

LetC = e₁C₁ + e₂C₂ + ⋯ + e_2^kC_2^kbe a linear code of lengthn over R_kwith |C| = p^mland the minimum Lee distanced_L(C) = d. ThenΦ(C) = C₁× C₂× ⋯ × C_2^kis a linear code with parameter [2^kn, l, d] andΦ(C)^⊥ = Φ(C^⊥). IfCis a self-dual code overR_k, thenΦ(C) is a self-dual code over 𝔽_p^m.

Proof

By the definition above, we can know that

C1×C2×⋯×C2k⊆Φ(C)

and

|C1×C2×⋯×C2k|=|C1||C2|⋯|C2k|=|C|.

This gives that

Φ(C)=C1×C2×⋯×C2k.

Let c=∑j=12kejcj∈C,d=∑j=12kejdj∈C⊥,wherecj∈Cj,dj∈Cj⊥,thenc⋅d=∑j=12kejcjdj=0, which implies c_jd_j = 0, so

Φ(c)⋅Φ(d)=∑j=12kcjdj=0,

which implies

Φ(C)⊥⊇Φ(C⊥).

By Theorem 3.3, we have

Φ(C⊥)=C1⊥×C2⊥×⋯×C2k⊥.

Since Φ is one-to-one, we have

|Φ(C⊥)|=pm2kn|C|=pm2kn|Φ(C)|=|Φ(C)⊥|.

Φ(C)⊥=Φ(C⊥).

□

Let τ be a cyclic shift operator on Fpmn.Leta=(a(1)|a(2)|⋯|a(2k))∈Fpm2kn, wherea(j)∈Fpmn for j = 1, 2,⋯, 2^k. Let τ_2^k be the quasi-shift given by

τ2k(a(1)|a(2)|⋯|a(2k))=(τ(a(1))|τ(a(2))|⋯|τ(a(2k))).

Proposition 3.6

Letσbe a cyclic shift onRkn, letΦbe the Gray map fromRkntoFpm2kn, and letτ_2^kbe as above. ThenΦσ = τ_2^kΦ.

Proof

Let r = (r₀, r₁, ⋯, r_n−1) ∈ Rkn, where ri=∑j=12kr_ije_j, i = 0, 1, ⋯, n − 1. We have σ(r) = (r_n−1, r₀, ⋯, r_n−2). If we apply Φ, we have

Φ(σ(r))=Φ(rn−1,r0,⋯,rn−2)=(r1,n−1,r1,0,⋯,r1,n−2,r2,n−1,r2,0,⋯,r2,n−2,⋯,r2k,n−1,r2k,0,⋯,r2k,n−2).

On the other hand,

τ2k(Φ(r))=τ2k(Φ(r0,r1,⋯,rn−1))=τ2k(r1,0,r1,1,⋯,r1,n−1,r2,0,r2,1,⋯,r2,n−1,⋯,r2k,0,r2k,1,⋯,r2k,n−1)=(r1,n−1,r1,0,⋯,r1,n−2,r2,n−1,r2,0,⋯,r2,n−2,⋯,r2k,n−1,r2k,0,⋯,r2k,n−2).

Therefore, we have

Φσ=τ2kΦ.

□

Theorem 3.7

LetCbe a cyclic code of lengthnoverR_k. ThenΦ(C) is a quasi-cyclic code of index 2^kover 𝔽_p^mwith length 2^kn.

Proof

Since C is a cyclic code, then σ(C) = C. If we apply Φ, we have Φσ(C) = Φ(C). By the Proposition 3.6, Φ(σ(C)) = Φ(C) = τ_2^k(Φ(C)), so Φ(C) is a quasi-cyclic code of index 2^k over 𝔽_p^m with length 2^kn. □

Let C be a linear code of length n over R_k, let A₀, A₁, ⋯, A_2^kn denote the number of codewords in C of the Lee weight, and the Lee weight distribution of C is simply the tuple of numbers {A₀, A₁, ⋯, A_2^kn}.

Let Lee_C(x, y) = ∑i=02knAix2kn−iyi denote the Lee weight enumerator of C, we get that

LeeC(x,y)=∑c∈Cx2kn−wL(c)ywL(c).

Let W_C(x, y) = ∑c∈Cx2kn−wH(c)ywH(c) denote the Hamming weight enumerator of C.

By the results of [22], we have

WC⊥(x,y)=1|C|WC(x+(|Rk|−1)y,x−y).

By a proof similar to (cf. [23, Lemma 1]), we obtain the following lemma.

Lemma 3.8

Letxandybe two vectors inRkn, and letd_H(Φ(x), Φ(y)) denote the Hamming distance ofΦ(x),Φ(y), whereΦ(x), Φ(y) are codewords inFpm2kn. Letw_H(Φ(x)) denote the Hamming weight ofΦ, then

w_L(x) = w_H(Φ(x)),
d_L(x, y) = d_H(Φ(x), Φ(y)).

Theorem 3.9

LetCbe a linear code of lengthnoverR_k, then Lee_C^⊥(x, y) = 1|Φ(C)|W_Φ(C)(x + (p^{m2^k} − 1)y, x − y).

Proof

By Theorem 3.5, we have that

LeeC⊥(x,y)=WΦ(C⊥)(x,y)=WΦ(C)⊥(x,y).

LeeC(x,y)=∑c∈Cx2kn−wL(c)ywL(c)=∑Φ(c)⊆Φ(C)x2kn−wH(Φ(c))ywH(Φ(c))=WΦ(C)(x,y).

As Φ is one-to-one, we have that |Φ(C)| = |C|, hence

LeeC⊥(x,y)=WΦ(C⊥)(x,y)=1|Φ(C)|WΦ(C)(x+(pm2k−1)y,x−y).

□

4 λ-Constacyclic codes over R_k

Theorem 4.1

LetC = e₁C₁ + e₂C₂ + ⋯ + e_2^kC_2^kbe a linear code overR_k, thenCis a (λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k)-constacyclic code overR_kif and only ifC₁, C₂, ⋯, C_2^kare λ_i-constacyclic codes over 𝔽_p^m, where λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^kis a unit overR_k.

Proof

For any c_i = (c_i,0, c_i,1, ⋯, c_i,n−1) ∈ C_i, where i = 1, 2,⋯, 2^k. Then

c=e1c1+e2c2+⋯+e2kc2k=(∑i=12keici,0,∑i=12keici,1,⋯,∑i=12keici,n−1)∈C.

If λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k is a unit over R_k, it is easy to know that for any element r = r₁e₁ + r₂e₂ + ⋯ + r_2^ke_2^k ∈ R_k, r is a unit if and only if r_i ≠ 0, where i = 1, 2,⋯, 2^k.

For i = 1, 2,⋯, 2^k, if C_i is a λ_i-constacyclic code over 𝔽_p^m, then

σλi(ci)=σλi(ci,0,ci,1,⋯,ci,n−1)=(λici,n−1,ci,0,⋯,ci,n−2)∈Ci.

Then we have

σλ1e1+λ2e2+⋯+λ2ke2k(c)=((λ1e1+λ2e2+⋯+λ2ke2k)∑i=12keici,n−1,∑i=12keici,0,⋯,∑i=12keici,n−2)=e1σλ1(c1)+e2σλ2(c2)+⋯+e2kσλ2k(c2k)∈C.

This proves that C is a (λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k)-constacyclic code over R_k.

Conversely, if C is a (λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k)-constacyclic code over R_k, then

σλ1e1+λ2e2+⋯+λ2ke2k(c)=e1σλ1(c1)+e2σλ2(c2)+⋯+e2kσλ2k(c2k)∈C.

Thus σ_{λ_i}(c_i) ∈ C_i, where i = 1, 2,⋯, 2^k.

So C_i is a λ_i-constacyclic code over 𝔽_p^m, where i = 1, 2,⋯, 2^k. □

Theorem 4.2

LetC = e₁C₁ + e₂C₂ + ⋯ + e_2^kC_2^kbe a (λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k)-constacyclic code of lengthnoverR_k, then there exists a polynomiale₁g₁(x) + e₂g₂(x) + ⋯ + e_2^kg_2^k(x) inR_k[x] that dividesxⁿ-(λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k) generates the code, whereg_iis the generator polynomial ofC_i, i = 1, 2,⋯, 2^k.

Proof

If C = e₁C₁ + e₂C₂ + ⋯ + e_2^kC_2^k be a (λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k)-constacyclic n over R_k, by Theorem 4.1 we know that C_i is λ_i-constacyclic code over 𝔽_p^m, where i = 1, 2,⋯, 2^k. Let g_i be the generator polynomial of C_i, where i = 1, 2,⋯, 2^k. It follows that C has the form

C=〈e1g1(x),e2g2(x),⋯,e2kg2k(x)〉.

Let C′ = 〈e₁g₁(x) + e₂g₂(x) + ⋯ + e_2^kg_2^k(x)〉. We have that C′ ⊆ C.

Note that

ei[(e1g1(x)+e2g2(x)+⋯+e2kg2k(x)]=eigi(x),

where i = 1, 2,⋯, 2^k.

We get that C ⊆ C′. So C = C′, and C is generated by a single element g(x) = e₁g₁(x) + e₂g₂(x) + ⋯ + e_2^kg_2^k(x).

We know that g_i divides xⁿ-λ_i, since g_i is the generator polynomial of C_i, where i = 1, 2,⋯, 2^k. Let f_i(x) be the polynomial such that g_i(x)f_i(x) = xⁿ-λ_i, where i = 1, 2,⋯, 2^k.

Then we have

[e1g1(x)+e2g2(x)+⋯+e2kg2k(x)][e1f1(x)+e2f2(x)+⋯+e2kf2k(x)]=λ1e1+λ2e2+⋯+λ2ke2k.

So we have e₁g₁(x) + e₂g₂(x) + ⋯ + e_2^kg_2^k(x) in R_k[x] that divides xⁿ-(λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k). □

By Theorem 4.2 we have the following theorem easily:

Theorem 4.3

LetC = e₁C₁ + e₂C₂ + ⋯ + e_2^kC_2^kbe a (λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k)-constacyclic code of lengthnoverR_k. ThenC⊥=〈e1f1∗(x)+e2f2∗(x)+⋯+e2kf2k∗(x)〉,|C⊥|=pm(∑i=12kdeg(gi)),wherefi∗ (x) is the reciprocal polynomial off_i(x),i.e., f_i(x) = (xⁿ-λ_i)/g_i(x), fi∗ (x) = x^deg(f_i)f(x⁻¹), fori = 1, 2,⋯, 2^k.

Example 4.4

Letn = 10 andR₂ = 𝔽₃ + u₁𝔽₃ + u₂𝔽₃ + u₁u₂𝔽₃, λ = − 1, x¹⁰ + 1 = (x² + 1)(x⁴ + x³ + 2x + 1)(x⁴ + 2x³ + x + 1) in 𝔽₃(x). Letf₁(x) = f₂(x) = (x⁴ + x³ + 2x + 1), f₃(x) = f₄(x) = (x⁴ + 2x³ + x + 1), C = 〈(1 + u₁ + u₂ + u₁u₂)f₁(x),(u₁ + u₁u₂)f₂(x),(u₂ + u₁u₂)f₃(x),(u₁u₂)f₄(x)〉. C₁, C₂, C₃, C₄are [10, 6,4] linear codes of length 10 with the minimum Lee weightd_L = 4. SoΦ(C) is a [40, 24,4] linear code.

Example 4.5

Letn = 15 andR₃ = 𝔽₂[u₁, u₂, u₃]/〈ui2 = u_i, u_iu_j = u_ju_i〉, x¹⁵ − 1 = (x + 1)(x² + x + 1)(x⁴ + x + 1)(x⁴ + x³ + 1)(x⁴ + x³ + x² + x + 1) in 𝔽₂(x). Letf₁(x) = f₂(x) = f₃(x) = f₄(x) = (x⁴ + x + 1), f₅(x) = f₆(x) = f₇(x) = f₈(x) = (x⁴ + x³ + 1), C = 〈 ∏i=13 (1 + u_i)f₁(x), u₁(1 + u₂)(1 + u₃)f₂(x), u₂(1 + u₁)(1 + u₃)f₃(x), u₃(1 + u₁)(1 + u₂)f₄(x), u₁u₂(1 + u₃)f₅(x), u₁u₃(1 + u₂)f₆(x), u₂u₃(1 + u₁)f₇(x), u₁u₂u₃f₈(x)〉. C_iis a [15, 11,3] linear code of length 15 with the minimum Lee weightd_L = 3, i = 1, 2, ⋯, 8. SoΦ(C) is a [120, 88,3] linear code.

5 Conclusion

In this paper, we studied the constacyclic codes over R_k = 𝔽_p^m[u₁, u₂,⋯, u_k]/〈ui2 = u_i, u_iu_j = u_ju_i〉. We proved that the (λ₁e₁ + λ₂e₂ + ⋯ + λ_2^ke_2^k)-constacyclic codes of arbitrary length over R_k can be generated by one polynomial.

Acknowledgement

This work was supported by the Basic and Advanced Technology Research Project of Henan Province (No.162300410083) and the Science and Technology Developing Project of Henan Province(No.172102210243).

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Received: 2017-08-30

Accepted: 2018-03-22

Published Online: 2018-05-10

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0045

Keywords for this article

Constacyclic codes; Cyclic codes; Gray map; Self-orthogonal codes

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Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉

Article

Abstract

1 Introduction

2 Preliminaries

Theorem 2.1

Theorem 2.2

Proof

Theorem 2.3

Proof

Corollary 2.4

Proof

Theorem 2.5

3 Linear codes over Rk

Lemma 3.1

Theorem 3.2

Theorem 3.3

Proof

Theorem 3.4

Proof

Theorem 3.5

Proof

Proposition 3.6

Proof

Theorem 3.7

Proof

Lemma 3.8

Theorem 3.9

Proof

4 λ-Constacyclic codes over Rk

Theorem 4.1

Proof

Theorem 4.2

Proof

Theorem 4.3

Example 4.4

Example 4.5

5 Conclusion

Acknowledgement

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

Constacyclic codes over 𝔽_p^m[u₁, u₂,⋯,u_k]/〈 u_i² = u_i, u_iu_j = u_ju_i〉

3 Linear codes over R_k

4 λ-Constacyclic codes over R_k