Finite Fields and Their Applications 43 (2017) 1–21

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Finite Fields and Their Applications

www.elsevier.com/locate/ffa

Constacyclic codes over finite local Frobenius

non-chain rings with nilpotency index 3

C.A. Castillo-Guillén a,∗, C. Rentería-Márquez b, H. Tapia-Recillas a

a Departamento de Matemáticas, Universidad Autónoma Metropolitana,Unidad Iztapalapa, 09340, México City, D.F., Mexicob Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, 07300 México City, D.F., Mexico

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 January 2016Received in revised form 15 June 2016Accepted 25 August 2016Available online xxxxCommunicated by W. Cary Huffman

MSC:13H9994B15

Keywords:Length of a moduleGalois extension of local commutative finite ringChain ringsFrobenius ringsConstacyclic codes

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4(hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with p4 elements, (p a prime) are given. Second, the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined.

© 2016 Elsevier Inc. All rights reserved.

* Corresponding author.E-mail addresses: carlos_53@hotmail.com (C.A. Castillo-Guillén), renteri@esfm.ipn.mx

(C. Rentería-Márquez), htr@xanum.uam.mx (H. Tapia-Recillas).

http://dx.doi.org/10.1016/j.ffa.2016.08.0041071-5797/© 2016 Elsevier Inc. All rights reserved.

2 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

1. Introduction

After the work of R. Hammons et al. (see [4]) the study of linear codes over finite rings has been an interesting research topic. Results in several directions including the description of structural properties of codes over several families of rings, particularly finite fields and finite chain rings, are available in the literature. The γ-constacyclic codes where γ is a unit of the finite ring A taken as the alphabet, i.e., those codes invariant under the mapping σγ : An −→ An given by σγ(a0, a1, . . . , an−1) = (γan−1, a0, . . . , an−2)are a generalization of cyclic codes. The 1-constacyclic codes are the usual cyclic codes and (−1)-constacyclic codes are the negacyclic codes. Finite Frobenius rings represent an interesting family of rings in Coding theory due to the fact that MacWilliams identities on the weight enumerator polynomial of a linear code are satisfied (see [13]). A finite Frobenius ring can be expressed as a direct sum of finite Frobenius local rings, (see [13]), and this induces a decomposition of the linear codes over Frobenius local rings. Finite chain rings are a subfamily of the family of finite Frobenius local rings and γ-constacyclic codes over finite chain rings has been considered by several researchers (see [2,3,11]), so it would be interesting to study γ-constacyclic codes over finite local Frobenius non-chain rings (γ a unit of the ring).

If p is a prime number, it is well-known that up to isomorphism there is only one local ring with p elements, namely the Galois field: 1) GF(p). The local rings with p2 elements are: 2) GF(p2), 3) Zp2 and 4) GF(p)[X]/〈X2〉. If p is odd, the local rings with p3 elements are: 5) GF(p3), 6) Zp3 , 7) GF(p)[X]/〈X3〉, 8) Zp2 [X]/〈X2 − p, pX〉, 9) Zp2 [X]/〈X2−ζp, pX〉, where ζ is a primitive element of GF(p), 10) GF(p)[X, Y]/〈X, Y〉2and 11) Zp2 [X]/〈X2, pX〉. If p = 2, the local rings with 23 elements are: 12) GF(23), 13) Z23 , 14) GF(2)[X]/〈X3〉, 15) Z22 [X]/〈X2 − 2, 2X〉, 16) GF(2)[X, Y]/〈X, Y〉2 and 17) Z22 [X]/〈X2, 2X〉, (see [9]). The rings 10), 11), 16) and 17) are not Frobenius lo-cal rings, because the annihilator of their maximal ideal is not a simple ideal, and the other rings are chain rings. In [5] the ring GF(2)[X, Y]/〈X2, Y2〉 with 24 elements, whichis a local Frobenius non-chain ring was introduced and linear cyclic codes over this al-phabet were considered. Recently, finite local Frobenius non-chain rings with 24 elements were determined and linear codes over these rings were studied [7]. Now if p > 2 is a prime it would be interesting to determine the family of finite local Frobenius non-chain rings with p4 elements.

The relation |M| = |GF(pd)|�A(M), (see section 2), where M is an A-module over a finite local ring A with residue field GF(pd) and �A(M) is the length of M, and the fact that a local ring of length one and two are chain rings, (see section 2), implies that if Ais a finite local Frobenius non-chain ring with p4 elements then A has length 4. On the other hand the family of finite local Frobenius non-chain rings of nilpotency index 3 is large, including finite local Frobenius non-chain rings of length 4 and the rings A(l,pd) =GF(pd)[X1, . . . , Xl]/〈XiXj−X1X2, X2

1, . . . , X2l : (i, j), 1 ≤ i < j ≤ l, (i, j) �= (1, 2)〉, where

p is a prime and l ≥ 3 an integer such that (l − 1, p) = 1. Since the length of the ring A(l,pd) is l + 2, this family contains rings of all lengths.

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 3

The purpose of this paper is twofold. First, to determine the family of finite local Frobenius non-chain rings of length four (hence with nilpotency index three). As a by-product all finite local Frobenius non-chain rings with p4 elements, p a prime, are given, in particular those with 16 elements appearing in [7]. Second, determine the number and structure of γ-constacyclic codes whose alphabets are finite local Frobenius non-chain rings whose maximal ideal has nilpotency index 3, and the length of the code is rela-tively prime to the characteristic of the residue field of the ring. It would be interesting to describe γ-constacyclic codes over finite local Frobenius non-chain rings of arbitrary nilpotency index.

The paper is organized as follows: in Section 2 basic facts on finite local rings and modules over these rings are recalled. In Section 3 the family of finite local Frobenius non-chain rings of length 4 is determined, and as a consequence the finite local Frobenius non-chain rings with p4 elements, p a prime, are given, in particular those with 16 elements (see [7]). In Section 4 the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings whose maximal ideal has nilpotency index 3, and the length of the code is relatively prime to the characteristic of the residue field of the ring are determined. Some examples are included to illustrate the main results. In the last section some conclusions are given.

2. Preliminaries

Throughout this work all rings are assumed to be finite, commutative with unit el-ement and all modules are finitely generated. As usual, GF(q) is the Galois field with q = pd elements, p a prime. For details we refer the reader to [1,8,9].

Let A and B be rings and M an A-module. We say that B is an extension of A if A is a subring of B. If B is an extension of A and I is an ideal of A, the ideal IB is called the expansion of I to B. The annihilator ideal of M in A is annA(M) := {a ∈ A : am = 0,∀ m ∈ M}. Let LA(M) denote the set of all A-submodules of M. If M = A, LA(M) is the set of ideals of A and will be denoted by L(A). Recall that a lattice is a partially ordered set in which any two elements have a supremum and an infimum. Then the set LA(M) is a lattice with respect to “⊇” (set-theoretic inclusion).

Let A be a ring and M an A-module. For each ideal I of A contained in the annihilator of M there is a natural structure of (A/I)-module on M. The scalar multiplication is given by (a +I, m) �→ am, a ∈ A, m ∈ M and the lattice of A-submodules and (A/I)-submodules of M are the same, i.e., LA(M) = LA/I(M).

A chain of A-submodules of an A-module M is a sequence of submodules with strict inclusion: M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 〈0〉 and such a chain is said to have length l (the number of links). The chain is a composition series if each A-module Mi/Mi+1 is a nonzero simple module, that is, has no nonzero proper submodules. The length of M, denoted by �A(M), is the length of a composition series for M, or ∞ if Mhas no finite composition series, which by the Jordan–Holder–Schreier Theorem (see [9])

4 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

is independent of the composition series. Observe that if K is a field, the length of a K-module is the same as the dimension of the K-vector space.

Recall that a ring A with only one maximal ideal m is called local, k = A/m its residue field (isomorphic to a GF(q)) and it will be denoted by the triple (A, m, GF(q)). If (A, m, GF(q)) is a finite local ring there is an integer t ≥ 1 such that mt = 〈0〉 and mt−1 �= 〈0〉, called the nilpotency index of m. By Nakayamas’ Lemma (see [8]) A has the chain of ideals A ⊃ m ⊃ m2 ⊃ . . . ⊃ mt−1 ⊃ mt = 〈0〉, which implies �A(A) ≥ t. Also, if M is an A-module, l = �A(M) and M = M0 ⊃ M1 ⊃ . . . ⊃ Ml−1 ⊃ Ml = 〈0〉is a composition series of submodules of M, then Mi/Mi+1 ∼= GF(q). Consequently |M| = |M/M1||M1/M2| · · · |Ml−2/Ml−1||Ml−1| = |GF(q)|�A(M).

Let (A, m, GF(q)) be a finite local ring and M an A-module. A subset G of M generates M if and only if its image G in M/mM generates M/mM. A set of generators of M obtained from lifting a basis of the A/m-vector space M/mM is called a minimal A-generating set for M and vA(M) denotes the number of elements in a minimal A-generating set for the A-module M, (see [8], Theorem 2.3). Note that vA(M) = dimGF(q)(M/mM) =�A(M/mM).

Recall that a finite ring A is called chain ring if the lattice of its ideals is a chain under set-theoretic inclusion. The ring A is a finite chain ring, if and only if A is local and its maximal ideal is principal, if and only if A is local and �A(A) = t, where t is the nilpotency index of the maximal ideal of A, (see [3]). A finite local ring (A, m, GF(q)) is Frobeniusif annA(m) is the unique minimal ideal of A, if and only if annA(m) is a simple ideal of A, (see [13]). If (A, m, GF(q)) is a finite Frobenius local ring, then annA(m) = mt−1, where t is the nilpotency index of m. The family of Frobenius rings is large including finite chain rings and the following rings: GF(q)[X1, . . . , Xl]/〈X2

1, . . . , X2l 〉, where l ≥ 2;

GF(q)[X1, . . . , Xl]/〈X21, . . . , X2

l , XiXj −X1X2 : (i, j) : 1 ≤ i < j ≤ l, (i, j) �= (1, 2)〉, where p is a prime and l ≥ 3 an integer such that (l−1, p) = 1, Zp2 [X]〈X2〉, Zp3 [X]〈X2−p2, pX〉, (see [13] for other examples).

Let (A, m, GF(q)) be a finite local ring. Two polynomials f1, f2 ∈ A[T] are called coprime if 〈f1〉 + 〈f2〉 = A[T]. Let ¯ : A[T] → GF(q)[T] denote the natural ring homo-morphism that maps a �→ a +m and the variable T to T. The element f ∈ A[T] is called basic irreducible if f is irreducible in GF(q)[T]. Hensel’s Lemma (see [9], Theorem XIII.4) guarantees that factorization into a product of pairwise coprime polynomials in GF(q)[T]lifts to such a factorization over A. Hence if (q, n) = 1 and γ is a unit of A, the poly-nomial Tn − γ factors into pairwise coprime basic irreducible polynomials in A[T], i.e., Tn − γ = f1 · · · fr.

Let (A, m, GF(q)) be a finite local ring, f ∈ A[T] be a basic irreducible polynomial and s = deg(f), then there is a monic polynomial g in A[T] and a unit v in A[T] such that f = g and g = vf (see [9], Theorem XIII.6). Let B = A[T]/〈f〉 = A[T]/〈g〉 ={a0 + a1T + · · · + as−1Ts−1 : ai ∈ A}. This ring is a separable extension of A, is local with maximal ideal mB and residue field GF(qs), (see [9]). If T ⊂ A is a set of representatives of GF(q) the set Ts := {a0+a1T +· · ·+as−1Ts−1 : ai ∈ T} ⊂ B is a set of representatives of GF(qs). If I is an ideal of A and B is as above, then �A(I) = �B(IB), (see

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 5

[8], Exercise 22.1), and (annA(I))B = annB(IB), (see [8], Theorem 7.4(iii)). This implies that �A(annA(m)) = �B(annB(mB)), vA(m) = �A(m/m2) = �B(mB/m2B) = vB(mB), A is a Frobenius ring if and only if �A(annA(m)) = �B(annB(mB)) = 1 if and only if B is a Frobenius ring, and if {α1, . . . , αl} is a minimal A-generating set for m, then it is also a minimal B-generating set for mB.

Observe that, if A = Zpk the ring B is just the Galois ring GR(pk, s). And in general the ring B is not a Galois extension of A (see [12]).

Let V be a n-dimensional vector space over GF(q), {α1, . . . , αn} be a basis of V and k

an integer with 0 < k < n. Recall that the number of k-dimensional vector subspaces of V is given by the Gaussian binomial coefficient:(

n

k

)q

= (qn − 1)(qn − q)(qn − q2)(qn − q3) · · · (qn − qk−1)(qk − 1)(qk − q)(qk − q2)(qk − q3) · · · (qk − qk−1) ,

and the Gaussian number, Gn(q) := 2 +∑n−1

k=1(nk

)q, gives the total number of vector

subspaces of V, (see [10]).A (k× n) matrix over GF(q) is said to be in reduced row echelon form, (rre)-form, if

in each row, i = 1, . . . , k, the first nonzero entry is equal to 1, the index of the column in which the 1 occurs, called a pivotal column, strictly increases with i, and the k pivotal columns are, in order, the columns of the (k × k) identity matrix. Since each matrix is row equivalent to a unique reduced row echelon form matrix, the k-dimensional vector subspaces of V are in one-to-one correspondence with the (k × n) matrices over GF(q)in (rre)-form. The (k × n) matrix (aij) over the field in (rre)-form corresponds to the subspace 〈

∑ni=1 a1iαi, . . . ,

∑ni=1 akiαi〉, (see [10]).

The following result is the well-known Correspondence Theorem, (see [1]), and will be used later.

Theorem 1. Let M and N be A-modules, LA(M) and LA(N), be the lattice of submodules of M and N , respectively, and ψ : N → M be a surjective homomorphism of A-modules. Let J = {N1 ∈ LA(N) : ker(ψ) ⊆ N1}. Then there is a bijection

ψ∗ : LA(M) −→ J, ψ∗(M1) = ψ−1(M1).

The inverse of ψ∗ is given by (ψ∗)−1(N1) = ψ(N1).

Lemma 1. Let (A, m, GF(q)) be a finite local ring, T ⊂ A a set of representatives of GF(q), M an A-module and {α1, . . . , αl} be a minimal A-generating set for M. Then the A-submodules of M between M and mM of length k + �A(mM), where 0 <k < l = dimGF(q)(M/mM), are in one to one correspondence with the (k × l) ma-trices over GF(q) in (rre)-form. The matrix H = (aij) corresponds to the submodule 〈∑n

i=1 a1iαi, . . . , ∑n

i=1 akiαi〉 + mM. In particular, the number of A-submodules of Mbetween M and mM is the Gaussian number Gl(q).

6 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

Proof. Recall that if α ∈ A then α ∈ GF(q) is the image of α under the natural mapping from A to GF(q). The matrix H = (aij) corresponds to the GF(q)-subspace of M/mM, 〈∑n

i=1 a1iαi, . . . , ∑n

i=1 akiαi〉. This GF(q)-subspace of M/mM is the A-submodule of M/mM, 〈

∑ni=1 a1iαi, . . . ,

∑ni=1 akiαi〉, by change of ring. And, by the Correspon-

dence Theorem, this A-submodule of M/mM corresponds to the A-submodule of M, 〈∑n

i=1 a1iαi, . . . , ∑n

i=1 akiαi〉 + mM. �3. Finite local Frobenius non-chain rings of length 4

In the following we focus on describing the family of finite local Frobenius non-chain rings of length 4. As a corollary the finite local Frobenius non-chain rings with p4 el-ements, p a prime, are given, particularly those with 24 elements appearing in [7]. It would be interesting to determine such rings of arbitrary length.

Let L4 be the family of finite local Frobenius non-chain rings of length 4 and F3 be the family of finite local Frobenius non-chain rings of nilpotency index 3. Observe that if (A, m, GF(q)) is a finite local Frobenius ring and t the nilpotency index of m, then t = 1implies A is a finite field; t = 2 implies annA(m) = m is a simple ideal, m is principal and hence A is a chain ring. Consequently, L4 ⊂ F3, because for local non-chain rings its length is greater than its nilpotency index, (see section 2).

The following result is a well-known fact on finite local rings.

Theorem 2 (Structure theorem for finite local rings, [9], Theorem XVII.1). Let (A, m, k)be a finite local ring of characteristic pk, {α1, . . . , αl} a minimal A-generating set of mand d = [k : Fp]. Then a subring S of A exists such that

(a) S ∼= GR(pk, d), S is unique and is the largest Galois ring extension of Zpk in A.(b) A is a homomorphic image of S[X1, . . . , Xl], i.e., A = S[α1, . . . , αl].

Lemma 2. Let I be an ideal of the ring GR(pk, d)[X1, . . . , Xl] such that for all i ∈{1, . . . , l}, Xki

i ∈ I, for some ki ∈ N. Then the ring GR(pk, d)[X1, . . . , Xl]/I is local with maximal ideal 〈p, X1, . . . , Xl〉/I and residue field GF(pd).

Proof. Let J be a maximal ideal of GR(pk, d)[X1, . . . , Xl] such that I ⊆ J. Since Xkii ∈ J,

pk = 0 and J is a prime ideal, then 〈p, X1, . . . , Xl〉 ⊆ J and since 〈p, X1, . . . , Xl〉 is maximal, 〈p, X1, . . . , Xl〉 = J. The first claim follows from the Correspondence Theorem. For the second part, just note that

(GR(pk, d)[X1, . . . ,Xl]/I

)/(〈p,X1, . . . ,Xl〉/I

) ∼= GF(pd). �The following result is central in proving the main result of this section.

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 7

Lemma 3. Let (A, m, GF(q)) ∈ L4, T ⊂ A a set of representatives of GF(q) and x ∈m \ m2. Then y ∈ m \ m2 exists such that {x, y} is a minimal A-generating set of m, x3 = y3 = x2y = xy2 = 0 and one of the following two relations is satisfied:

(i) xy = 0 and x2 = fy2 �= 0 for some f ∈ T \ {0}.(ii) xy �= 0, x2 +uxy = 0 and y2 +vxy = 0 for some u, v ∈ T such that uv �= 1 in GF(q).

Proof. Since m has nilpotency index equal to 3, annA(m) = m2 is the unique minimal ideal of A and x ∈ m \ m2, then v(m) = �A(m/m2) = �A(m) − �A(m2) = 2, there exists y ∈ m \ m2 such that {x, y} is a minimal A-generating set of m and x3 = y3 = x2y =xy2 = 0. If annA(x) = m2, then 〈x〉 = annA(annA(x)) = annA(m2) = m, a contradiction. If annA(x) = m, then 〈x〉 = annA(annA(x)) = annA(m) = m2, which is not possible. Then m2 ⊂ annA(x), annA(y) ⊂ m and annA(x) �= annA(y).

By Lemma 1 and since (0, 1) and (1, λ), where λ ∈ GF(q), are all the 1 ×2 matrices in (rre)-form over GF(q), we have annA(x) ∈ {〈y〉, 〈x +uy〉} and annA(y) ∈ {〈y〉, 〈x + vy〉}, where u, v ∈ T. In the following lines assertions i) and ii) are proved.

i) Observe that, annA(x) = 〈y〉 ⇔ annA(y) = annA(annA(x)) = 〈x〉. In this case, xy = 0 and since x /∈ 〈y〉 = annA(x), y /∈ 〈x〉 = annA(y), then x2 �= 0 and y2 �= 0. Since m2 is generated by any of its nonzero elements because it is a simple ideal of A, m2 = 〈x2, xy, y2〉 = 〈x2〉 = 〈y2〉 implies x2 = fy2, for some f ∈ T \ {0}.

ii) Observe also that annA(x) = 〈x + uy〉, where u ∈ T ⇔ annA(y) ∈ {〈y〉, 〈x + v1y〉}, where v1 ∈ T \ {0} ⇔ annA(y) = 〈y + vx〉, where v ∈ T. In this case x2 + uxy = 0 and y2 + vxy = 0 for u, v ∈ T. Also xy �= 0 and uv �= 1 in GF(q). Indeed, if xy = 0 then x2 = y2 = 0 and m2 = 〈x2, xy, y2〉 = 〈0〉, a contradiction. If u = 0 or v = 0 we have the assertion. If uv = 1 + m, where m ∈ m, then uvx2 = x2 and x2 + uxy = uvx2 + uxy =ux(y + vx) = 0 implies annA(y) = 〈y + vx〉 ⊆ annA(x), consequently 〈x〉 ⊆ 〈y〉, which is not possible. �

The following results on finite fields will be used later on. Some of them may be found in the literature but we include them all here for completeness.

Proposition 1. Let F = GF(pd) be a finite field and ζ be a primitive element of this field. Then,

(1 ) γ1, γ2 ∈ F exist such that ζ = γ21 + γ2

2 .(2 )

√−1 ∈ F if and only if p = 2 or pd ≡ 1 mod 4.

(3 ) The equation X2ζ + Y2 = 0 has nontrivial solutions in F if and only if p = 2 or pd ≡ 3 mod 4, if and only p = 2 or

√−1 /∈ F.

(4 ) If u, v ∈ F are such that √u,

√v /∈ F, then

√uv ∈ F.

(5 ) Let u, v, w ∈ F with uw �= 0. Then the system of equations X2+Y2 = uw . . . (a), Z2+W2 = vw . . . (b), XZ + YW = w . . . (c) has solution in F if and only if

√uv − 1 ∈ F.

If p is odd and w = ζ2k a solution is ζk(u, 0, 1, √uv − 1).

u

8 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

If p is odd and wu = ζ2k+1 a solution is ζk(uγ1, uγ2, γ1 +γ2√uv − 1, γ2−γ1

√uv − 1),

where γ1, γ2 are as in (1).If p = 2 a solution is (

√uw, 0,

√wu ,

√wu

√uv − 1).

(6 ) Let u ∈ F with u �= 0. The system of equations X2 + Y2 = uζ . . . (a), Z2 + W2 =u . . . (b), XZ + YW = 0 . . . (c) has solution in F if and only if p = 2.

Proof. (1): The case p = 2 is trivial since (F∗)2 = F∗. Let p be an odd prime and let Γ = {1 − ζ2i+1 : 0 ≤ i ≤ pd−1

2 − 1}. Since |Γ| = pd−12 and 0, 1 /∈ Γ, Γ ⊂ {ζ2i : 1 ≤ i ≤

pd−12 − 1} ∪ {ζ2i+1 : 0 ≤ i ≤ pd−1

2 − 1} and Γ ∩ {ζ2i+1 : 0 ≤ i ≤ pd−12 − 1} �= ∅. The

assertion follows.(2): ⇒) If p is an odd prime,

√−1 ∈ F∗ has order 4 and the assertion follows from

Lagrange’s Theorem. Conversely: if p = 2, (F∗)2 = F∗ and the result follows. If pd =4h + 1, then

√−1 = ζh ∈ F.

(3): ⇒) If p is an odd prime and (a, b) is a nontrivial solution of the equation. Then

ζk = ba , for some k, and ζ+( b

a )2 = ζ+ζ2k = ζ(1 +ζ2k−1) = 0 implies ζ2k−1 = −1 = ζpd−1

2

from which pd − 1 | pd−12 − (2k− 1) and hence pd ≡ 3 mod 4. Conversely: the case p = 2

follows from the fact that (F∗)2 = F∗. If pd = 4h + 3, then (1, ζh+1) is a solution of X2ζ + Y2 = 0.

(4): Since GF(p2d) ∼= GF(pd)(√u) = GF(pd)(

√v),

√u = a +b

√v, where a, b ∈ GF(pd)

and b �= 0. Then 2ab√v = u − a2 − b2v, consequently a = 0 and b2 = u

v ∈ GF(pd).(5): ⇒) The case p = 2 is trivial because (F∗)2 = F∗ implies

√uv − 1 ∈ F and a

solution of the system is X =√uw, Y = 0, Z =

√wu−1, W =

√wu−1

√uv − 1.

Assume that p is an odd prime. ⇒) Z and W will be determined in terms of the values for X, Y, u and v. If Y �= 0, then W = −XZ+w

Y and Z2 + W2 = Z2 +(−XZ+w

Y )2 = Z2Y2+X2Z2−2wXZ+w2

Y2 = vw, consequently uZ2 − 2XZ + w − vY2 = 0, Z = 2X±

√4X2−4u(w−vY2)

2u = X±Y√uv−1

u and W = −XZ+wY = Y∓X

√uv−1

u . If Y = 0then X2 = uw, Z2 + W2 = vw, XZ = w; consequently Z2 = w2

X2 = u−1w and

W2 = vw−Z2 = vw−u−1w = u−1w(uv−1) = w2

X2 (uv−1). Conversely: if u−1w = ζ2k+1, then uw = u2u−1w = u2ζ2kζ = (uζk)2(γ2

1 + γ22) = (uζkγ1)2 + (uζkγ2)2, where γ1, γ2 are

as in (1). We take X = uζkγ1 and Y = uζkγ2 and according to the previous discussion we obtain, Z = ζkγ1 ± ζkγ2

√uv − 1 and W = ζkγ2 ∓ ζkγ1

√uv − 1. If u−1w = ζ2k, then

uw = u2u−1w = (uζk)2. We take X = uζk and Y = 0 and based on the above arguments, we get Z = ζk and W = ζk

√uv − 1.

(6): ⇒) If Y �= 0, then W = −XZY and Z2 + W2 = Z2Y2+X2Z2

Y2 = uZ2ζY2 = u implies

ζ ∈ (F∗)2, F∗ = (F∗)2 and p = 2; if Y = 0, then X2 = uζ, XZ = 0 and Z2 + W2 = u

implies Z = 0, W2 = u, X2 = uζ = W2ζ, ζ ∈ (F∗)2, F∗ = (F∗)2 and p = 2. Conversely: take X =

√uζ, Y = 0, Z = 0, W =

√u. �

Now we have the main result of this Section. Before proceeding with the proof we observe the following.

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 9

Recall that if A is a local ring with residue field GF(pd) and M is an A-module of finite length �A(M), then |M| = pd�A(M), (see section 2). In particular, if (A, m, GF(pd)) ∈ L4, |A| = p4d and since the nilpotency index of m is equal to 3, p3 ∈ m3 = (0). Hence char(A) ∈ {p, p2, p3}. Thus concerning the characteristic of the ring A, the only possi-bilities are the following:

(a): char(A) = p2 and p ∈ m2.(b): char(A) = p2 and p /∈ m2.(c): char(A) = p3 and p ∈ m2.(d): char(A) = p3 and p /∈ m2.(e): char(A) = p.Each one of these cases is treated in the following propositions. Observe that the

case (c): char(A) = p3 and p ∈ m2, is not possible since if p ∈ m2 then p2 ∈ m4 = 〈0〉.

Proposition 2. Let (A, m, GF(pd)) ∈ L4 be such that char(A) = p2 and p ∈ m2. Then,

1. If p is odd the ring A is isomorphic toGR(p2, d)[X, Y]/〈X2 − Y2, Y2 − p, XY, Y3, pX, pY〉, orGR(p2, d)[X, Y]/〈X2 − ζY2, Y2 − p, XY, Y3, pX, pY〉, {0, 1, . . . , ζpd−2} is the Teich-müller set of the Galois ring GR(p2, d).

2. If p = 2 the ring A is isomorphic toGR(4, d)[X, Y]/〈X2 − Y2, Y2 − 2, XY, Y3, 2X, 2Y〉, orGR(4, d)[X, Y]/〈X2, Y2, XY − 2, 2X, 2Y〉.

Proof. By Theorem 2 we may assume that the Galois ring GR(p2, d) ⊂ A and let T ={0, 1, ζ, . . . , ζpd−2} be the Teichmüller set of this Galois ring. Let {x, y} be a minimal A-generating set for the maximal ideal m satisfying statements (i) or (ii) of Lemma 3 for the values 0 �= f1, u1, v1 ∈ T. Since p ∈ m2 then px, py ∈ m3 = (0). Since m2 is a simple ideal of A it is generated by any one of its nonzero elements. In the case of statement (i) of Lemma 3, m2 = 〈y2〉 = 〈p〉 which implies y2 = g1p for some g1 ∈ T \{0}. In case (ii), m2 =〈xy〉 = 〈p〉 implying xy = w1p for some w1 ∈ T \{0}. Again by Theorem 2, in case (i) there is an epimorphism from A(f1,g1) := GR(p2, d)[X, Y]/〈X2−f1Y2, Y2−g1p, XY, Y3, pX, pY〉onto A, and in the case (ii), from B(u1,v1,w1) := GR(p2, d)[X, Y]/〈X2−u1XY,Y2−v1XY,

XY − w1p, pX, pY〉 onto A, with u1, v1, w1, f1, g1 ∈ T such that w1, f1 and g1 are not zero and u1v1 �= 1 in GF(pd). From Lemma 2 it follows that A(f1,g1) and B(u1,v1,w1) are local with maximal ideal 〈x, y〉 and residue field GF(pd). Observe that every element of the rings A(f1,g1) or B(u1,v1,w1) can be uniquely written as a + bx + cy, where a ∈GR(p2, d) and b, c ∈ T, and the elements of their maximal ideal can be uniquely written as ap +bx +cy, where a, b, c ∈ T. Consequently |A(f1,g1)| = |B(u1,v1,w1)| = p4d and in each case the epimorphism mentioned above is an isomorphism. It should be observed that although the elements of the rings A(f1,g1) and B(u1,v1,w1) are the same, its arithmetic is different.

10 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

Recall that the rings A(f1,g1) and B(u1,v1,w1) introduced above are determined by particular elements u1, v1, w1, f1, g1 ∈ T such that w1, f1 and g1 are not zero and u1v1 �= 1 in GF(pd). The aim now is to prove that for all elements u, v, w, f, g ∈ T

such that w, f and g are not zero and uv �= 1 in GF(pd), the rings A(f,g) and B(u,v,w) defined in the same way as above replacing u, v, w, f, g by u1, v1, w1, f1, g1

respectively, are as stated in the Proposition. Let x and y be the elements in these rings corresponding to X and Y modulo the respective ideals. From Lemma 2 it fol-lows that these rings are local with maximal ideal m = 〈x, y〉 satisfying the relations x2 = fy2, y2 = gp, xy = px = py = 0 in the case of the ring A(f,g); and x2 = uxy, y2 = vxy, xy = wp, px = py = 0 in the case of the ring B(u,v,w). Their ele-ments are written uniquely as a + bx + cy where a ∈ GR(p2, d) and b, c ∈ T; and the elements of the maximal ideal can be written uniquely as ap + bx + cy, where a, b, c ∈ T.

Observe that for u, v, w ∈ T such that u and w are not zero and uv �= 1 in GF(pd), B(u,v,w) ∼= A(1,1) if and only if there exist a, a1, b, b1, c, c1 ∈ T such that {α = ap +bx + cy, β = a1p + b1x + c1y} is a minimal A-generating set for the maximal ideal of A(1,1), and these elements must satisfy the relations satisfied by x and y in B(u,v,w), i.e., α2 = uαβ, β2 = vαβ, αβ = wp, pα = pβ = 0. From these relations and the expression for α and β we have: (b2 + c2)p = uwp, (b21 + c21)p = vwp and (bb1 + cc1)p = wp. These last relations hold if and only if b, b1, c, c1 ∈ T exist such that b2 + c2 = uw, b21 + c21 = vw and bb1 + cc1 = w in GF(pd) if and only if

√uv − 1 ∈ GF(pd), from (5) of

Proposition 1.From the above argument, when u, v, w ∈ T are such that u and w are not zero,

uv �= 1 in GF(pd) and √uv − 1 ∈ GF(pd), it is easy to see that an isomorphism between

B(u,v,w) and A(1,1) is given by: x �→ bx + cy and y �→ b1x + c1y, where b, b1, c, c1 ∈ T are such that their image in GF(pd) is a solution of the equations in (5) of Proposition 1. That is:

(1) If p is odd, u and w are not zero, uv �= 1 in GF(pd), √

uv − 1 ∈ GF(pd) and wu = ζ2k, an isomorphism between B(u,v,w) and A(1,1) is given by, x �→ uζkx and

y �→ ζkx + ζkξy, where ξ ∈ T is such that ξ =√

uv − 1.(2) If p is odd, u and w are not zero, uv �= 1 in GF(pd),

√uv − 1 ∈ GF(pd) and wu =

ζ2k+1, an isomorphism between B(u,v,w) and A(1,1) is given by, x �→ uζk[γ1x + γ2y]and y �→ ζk[(γ1 +γ2ξ)x +(γ2−γ1ξ)y], where ξ, γ1, γ2 ∈ T are such that ξ =

√uv − 1

and γ21 + γ2

2 = ζ.(3) If p = 2 and u and w are not zero an isomorphism between B(u,v,w) and A(1,1) is

given by, x �→ ax and y �→ a1x + b1y, where a, a1, b1 ∈ T are such that a =√uw,

a1 =√

wu and b1 =

√wu

√uv − 1.

Similarly, it can be seen that the following maps are isomorphism between the respec-tive rings.

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 11

(4) If p is odd, u and w are not zero, uv �= 1 in GF(pd), √

uv − 1 /∈ GF(pd) and wu = ζ2k, an isomorphism between B(u,v,w) and A(ζ,1) is given by, x �→ uζky and

y �→ ζk[y + ξx], where ξ ∈ T is such that ξ =√

uv−1ζ

.

(5) If p is odd, u and w are not zero, uv �= 1 in GF(pd), √

uv − 1 /∈ GF(pd) and wu = ζ2k+1, an isomorphism between B(u,v,w) and A(ζ,1) is given by, x �→ uζkx and

y �→ ζk[x + ξζy], where ξ ∈ T is such that ξ =√

uv−1ζ

.

(6) If p is odd and √

−1 ∈ GF(pd) an isomorphism between B(0,0,w) and A(1,1)is given by, x �→ w(x + ξy) and y �→ 1

2x − ξ2y, where ξ ∈ T is such that

ξ =√−1.

(7) If p is odd and √−1 /∈ GF(pd) an isomorphism between B(0,0,w) and A(ζ,1) is given

by, x �→ w(x + ζh+1y) and y �→ 12ζ [x + y

ζh ], where pd = 4h + 3 and (1, ±ζh+1) is a

solution of X2ζ + Y2 = 0 in GF(pd).(8) B(u,0,w) → B(0,u,w), x �→ y and y �→ x.(9) B(0,0,w) → B(0,0,1), x �→ x and y �→ wy.

(10) A(ζ2i+1,ζ2j) → A(ζ,1), x �→ ζi+jx and y �→ ζjy.(11) A(ζ2i,ζ2j) → A(1,1), x �→ ζi+jx and y �→ ζjy.(12) A(ζ2i+1,ζ2j+1) → A(ζ,1), x �→ ζi+j+1y and y �→ ζjx.(13) A(ζ2i,ζ2j+1) → A(1,1), x �→ ζi+j(γ1x +γ2y) and y �→ ζj(−γ2x +γ1y), where γ1, γ2 ∈ T

are such that γ21 + γ2

2 = ζ.

On the other hand, A(1,1) ∼= A(ζ,1) if and only if a, a1, b, b1, c, c1 ∈ T exist such that {α = ap + bx + cy, β = a1p + b1x + c1y} is a minimal A-generating set for the maximal ideal of A(1,1), and these elements must fulfill the same relations satisfied by x and yin A(ζ,1), i.e., α2 = ζβ2, β2 = p, αβ = 0 and pα = pβ = 0. From these relations and the expression for α and β we have: (b2 + c2)p = ζ(b21 + c21)p, (b21 + c21)p = p and (bb1 + cc1)p = 0. These last relations hold if and only if b, b1, c, c1 ∈ T exist such that b2 + c2 = ζ, b21 + c21 = 1 and bb1 + cc1 = 0 if and only if p = 2, by (6) of Proposition 1.

Finally, for p = 2, since the elements of the maximal ideal of B(0,0,1) have square zero and the element x ∈ A(1,1) is such that x2 = y2 �= 0, B(0,0,1) � A(1,1). �Proposition 3. Let (A, m, GF(pd)) ∈ L4 be such that char(A) = p2 and p /∈ m2. Then,

1. If p is odd the ring A is isomorphic toGR(p2, d)[X]/〈X2〉.

2. If p = 2 the ring A is isomorphic toGR(4, s)[X]/〈X2〉, orGR(4, s)[X]/〈X2 − 2X〉.

Proof. The same arguments as in Proposition 2 can be followed with the ring Au =GR(p2, d)[X]/〈X2 −upX〉, where u is in the Teichmüller set of the Galois ring GR(p2, d), give the following:

12 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

(1) If p is odd an isomorphism between A0 and Au is given by, x �→ 2x − up.(2) If p = 2 and u �= 0 an isomorphism between Au and A1 is given by, x �→ ux.

If p = 2, since all elements of the maximal ideal of A0 have square zero and the element x ∈ A1 is such that x2 = 2x �= 0, then A1 � A0. �Proposition 4. Let (A, m, GF(pd)) ∈ L4 be such that char(A) = p3 and p /∈ m2. Then,

1. If p is odd the ring A is isomorphic toGR(p3, d)[X]/〈X2 − p2, pX〉, orGR(p3, d)[X]/〈X2 − ζp2, pX〉, {0, 1, . . . , ζpd−2} is the Teichmüller set of the Galois ring GR(p3, d).

2. If p = 2 the ring A is isomorphic toGR(8, s)[X]/〈X2 − 4, 2X〉.

Proof. The same arguments as in Proposition 2 can be followed with the rings A(u,v) =GR(p3, d)[X]/〈X2 − upX, p2 − vpX, X3〉 and Bw = GR(p3, d)[X]/〈X2 − wp2, pX〉, where u, v, w are in the Teichmüller set of the Galois ring GR(p3, d), v �= 0, w �= 0 such that uv �= 1 in GF(pd); and we have:

(1) If √

uv − 1 ∈ GF(pd) an isomorphism between B1 and A(u,v) is given by x �→−ξp + vξx, where ξ ∈ T is such that ξ = 1√

uv−1.

(2) If √

uv − 1 /∈ GF(pd) an isomorphism between Bζ and A(u,v) is given by x �→−ξp + vξx, where ξ ∈ T is such that ξ =

√ζ

uv−1 .(3) Bζ2k → B1, x �→ ζkx.(4) Bζ2k+1 → Bζ , x �→ ζkx.

Finally, B1 ∼= Bζ if and only if a, b, c ∈ T exist such that {α = ap + bp2 + cx, p} is a minimal A-generating set for the maximal ideal of B1, and these elements must satisfy the relations satisfied by p and x in Bζ , i.e., α2 = ζp2 and pα = 0. From these relations and the expression for α we have: (a2 + c2)p2 = ζp2, ap2 = 0. These last relations hold if and only if c2 = ζ if and only if p = 2. �Proposition 5. Let (A, m, GF(pd)) ∈ L4 be such that char(A) = p. Then,

1. If p is odd the ring A is isomorphic toGF(pd)[X, Y]/〈X2 − Y2, XY, Y3〉, orGF(pd)[X, Y]/〈X2 − ζY2, XY, Y3〉, ζ is a primitive element of GF(pd).

2. If p = 2 the ring A is isomorphic toGF(2s)[X, Y]/〈X2, Y2〉, orGF(2s)[X, Y]/〈X2 − Y2, XY, Y3〉.

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 13

Proof. With the same arguments as in Proposition 2 for the rings A(u,v) = GF(pd)[X, Y]/〈X2 − uXY, Y2 − vXY, Y3, X2Y〉 and Bw = GF(pd)[X, Y]/〈X2 − wY2, XY, Y3〉, where u, v, w ∈ GF(pd) are such that uv �= 1 and w �= 0, we have:

(1) If √uv − 1 ∈ GF(pd) and u �= 0 an isomorphism between A(u,v) and B1 is given by,

x �→ ux and y �→ x −√uv − 1y.

(2) If √uv − 1 /∈ GF(pd) and u �= 0, then

√ζ(uv − 1) ∈ GF(pd), by (4) of Proposition 1.

An isomorphism between A(u,v) and Bζ is given by, x �→ ux and y �→ x +√

ζ(uv − 1)y.(3) If p is odd and

√−1 ∈ GF(pd) an isomorphism between A(0,0) and B1 is given by,

x �→ x +√−1y and y �→ x −

√−1y.

(4) If √−1 /∈ GF(pd) then pd = 4h + 3 and (1, ±ζh+1) is a solution of X2ζ + Y2 = 0, by

(3) of Proposition 1. An isomorphism between A(0,0) and Bζ is given by x �→ ζh+1xand y �→ x − ζh+1y.

(5) Bζ2k → B1, x �→ ζkx and y �→ y.(6) Bζ2k+1 → Bζ , x �→ ζkx and y �→ y.(7) A(u,v) → A(v,u), x �→ y and y �→ x.

Finally, B1 ∼= Bζ if and only if a, a1, b, b1 ∈ T exist such that {α = ax +by, β = a1x +b1y}is a minimal A-generating set for the maximal ideal of B1, and these elements must satisfy the relations satisfied by x and y in Bζ , i.e., α2 = ζβ2 and αβ = 0. From these relations and the expression for α and β we have: a2+b2 = ζ(a2

1+b21), a21+b21 �= 0 and aa1+bb1 = 0

if and only if p = 2, by (6) of Proposition 1.For p = 2, since all elements of the maximal ideal of A(0,0) have square zero and the

element x ∈ B1 is such that x2 = y2 �= 0, then A(0,0) � B1. �In the following Theorem we summarize the previously proven claims as the main

result of this section.

Theorem 3. Let (A, m, GF(pd)) be a finite local Frobenius non-chain ring of length 4. Then A is isomorphic to one of the following rings:

(a) If p is odd:GR(p2, d)[X, Y]/〈X2 − Y2, Y2 − p, XY, Y3, pX, pY〉 orGR(p2, d)[X, Y]/〈X2 − ζY2, Y2 − p, XY, Y3, pX, pY〉, {0, 1, . . . , ζpd−2} is the Teich-müller set of the Galois ring GR(p2, d).If p = 2:GR(4, s)[X, Y]/〈X2 − Y2, Y2 − 2, XY, Y3, 2X, 2Y〉 orGR(4, s)[X, Y]/〈X2, Y2, XY − 2, 2X, 2Y〉.In this case, char(A) = p2 and p ∈ m2.

(b) If p is odd:GR(p2, d)[X]/〈X2〉.

14 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

If p = 2:GR(4, s)[X]/〈X2〉 orGR(4, s)[X]/〈X2 − 2X〉.In this case, char(A) = p2 and p /∈ m2.

(c) If p is odd:GR(p3, d)[X]/〈X2 − p2, pX〉 orGR(p3, d)[X]/〈X2 − ζp2, pX〉, {0, 1, . . . , ζpd−2} is the Teichmüller set of the Galois ring GR(p3, d).If p = 2:GR(8, s)[X]/〈X2 − 4, 2X〉.In this case, char(A) = p3, p /∈ m2.

(d) If p is odd:GF(pd)[X, Y]/〈X2 − Y2, XY, Y3〉 orGF(pd)[X, Y]/〈X2 − ζY2, XY, Y3〉, ζ is a primitive element of GF(pd).If p = 2:GF(2s)[X, Y]/〈X2, Y2〉 orGF(2s)[X, Y]/〈X2 − Y2, XY, Y3〉.In this case, char(A) = p.

Let (A, m, GF(pd)) ∈ F3 be such that A has p4 elements. Since �A(A) ≥ 4 and |A| =p4 = (pd)�A(A), then d = 1, �A(A) = 4 and we have the following:

Corollary 1. Let (A, m, GF(p)) be a finite local Frobenius non-chain ring with p4 elements. Then A is isomorphic to one of the following rings:

(a) If p is odd:Zp2 [X, Y]/〈X2 − Y2, Y2 − p, XY, Y3, pX, pY〉 orZp2 [X, Y]/〈X2 − ζY2, Y2 − p, XY, Y3, pX, pY〉, where ζ is a primitive element of GF(p).If p = 2:Z4[X, Y]/〈X2 − Y2, Y2 − 2, XY, Y3, 2X, 2Y〉 orZ4[X, Y]/〈X2, Y2, XY − 2, 2X, 2Y〉.In this case, char(A) = p2 and p ∈ m2.

(b) If p is odd:Zp2 [X]/〈X2〉.If p = 2:Z4[X]/〈X2〉 orZ4[X]/〈X2 − 2X〉.In this case, char(A) = p2 and p /∈ m2.

(c) If p is odd:Zp3 [X]/〈X2 − p2, pX〉 orZp3 [X]/〈X2 − ζp2, pX〉, ζ is a primitive element of GF(p).

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 15

If p = 2:Z8[X]/〈X2 − 4, 2X〉.In this case, char(A) = p3, p /∈ m2.

(d) If p is odd:GF(p)[X, Y]/〈X2 − Y2, XY, Y3〉 orGF(p)[X, Y]/〈X2 − ζY2, XY, Y3〉, ζ is a primitive element of GF(p).If p = 2:GF(2)[X, Y]/〈X2, Y2〉 orGF(2)[X, Y]/〈X2 − Y2, XY, Y3〉.In this case, char(A) = p.

Remark. For the case p = 2 the above rings were given in [7].

4. Constacyclic codes over finite local Frobenius non-chain ring with nilpotency index 3

Let (A, m, GF(q)) be a finite local ring and γ a unit of A. Assume that the integer n > 1 is not divisible by p, so that by Hensel’s Lemma, Tn − γ is the product of basic irreducible pairwise coprime polynomials in A[T]. Recall that a linear code of length n over A is γ-constacyclic if it is invariant under the permutation (a0, a1, . . . , an−1) �→(γan−1, a0, . . . , an−2). If γ = 1, the code is cyclic, and if γ = −1 the code is negacyclic. As usual, γ-constacyclic codes of length n over A can be identified as ideals in the quotient ring A[T]/〈Tn − γ〉 via the isomorphism from An to A[T]/〈Tn − γ〉 defined by (a0, . . . , an−1) �→ a0 + a1T + . . . + an−1Tn−1, (the polynomial representation of An), (see [6]).

Recall that F3 is the family of finite local Frobenius non-chain rings with nilpotency index 3. In this Section the structure and the number of constacyclic codes over rings in F3 of length relatively prime to the characteristic of the residue field of the ring are determined.

The following results are a direct consequence of the Chinese Remainder Theorem. Re-call that if R is a ring, L(R) is the lattice of ideals of R, and the fact that if (A, m, GF(q))is a local ring, f ∈ A[T] is a basic irreducible polynomial and M is an A-module, then Aand A[T]/〈f〉 have the same length and |M| = |GF(q)|�A(M).

Lemma 4. Let (A, m, GF(q)) be a finite local ring, l = �A(A), γ a unit of A and n an integer relatively prime to q. Let Tn − γ = f1 · · · fr be a representation of Tn − γ as a product of basic irreducible pairwise coprime polynomials in A[T], Ai = A[T]/〈fi〉 and si = deg(fi). Then

(a) A[T]/〈Tn − γ〉 ∼= ⊕ri=1Ai.

(b) Any ideal I of A[T]/〈Tn − γ〉 is a direct sum of ideals of Ai and there is a partition of [1, . . . , r], U0, U1, . . . , Ul, such that:

16 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

I =⊕u∈U0

Iu ⊕⊕u∈U1

Iu ⊕ . . .⊕⊕

u∈Ul−1

Iu ⊕⊕u∈Ul

Iu

where Ui = {u : �Au(Iu) = i}.

(c) Let I and U0, U1, . . . , Ul be as above, then:

|I| = q∑

u∈U1su+2

∑u∈U2

su+...+(l−1)∑

u∈Ul−1su+l

∑u∈Ul

su .

(d) The number of γ-constacyclic codes of length n over A is:

|L(A1)| · · · |L(Ar)|.

For ease of notation, if f(T) is a factor of Tn−γ, let f(T) = Tn−γf(T) and we will just write

a0+a1T +. . .+an−1Tn−1 for the corresponding coset a0+a1T +. . .+an−1Tn−1+〈Tn−γ〉in A[T]/〈Tn − γ〉.

Let (A, 〈π〉, GF(q)) be a finite chain ring, t the nilpotency index of π, f ∈ A[T] be a basic irreducible polynomial and B = A[T]/〈f〉, then B ⊃ πB ⊃ . . . ⊃ πt−1B ⊃ 〈0〉 is the chain of ideals of B, (see [3]), which implies that �B(B) = t and �B(πiB) = t − i. Consequently, if in Lemma 4 the local ring is a chain ring, then for any ideal I of ⊕r

i=1Ai∼= A[T]/〈Tn − γ〉 there is a partition of [1, . . . , r], U0, U1, . . . , Ut, such that:

I = πt−1⊕u∈U1

Au ⊕ πt−2⊕u∈U2

Au ⊕ . . .⊕⊕u∈Ut

Au,

I corresponds in A[T]/〈Tn−γ〉 to 〈πt−1F1, πt−2F2, . . . , πFt−1, Ft〉, where Fi =∏

u∈Uifu,

and |I| = q∑

u∈U1su+...+(t−1)

∑u∈Ut−1

su+t∑

u∈Utsu = q

∑ti=1 ideg(Fi), as in [3].

For the remainder of the manuscript the following notation will be used. Given (A, m, GF(q)) a finite local ring, f ∈ A[T] a basic irreducible polynomial, s = deg(f), B = A[T]/〈f〉, T ⊂ A a set of representatives of GF(q), Ts = {a0 +a1T + · · ·+as−1Ts−1 :ai ∈ T} ⊂ B the set of representatives of B/mB = GF(qs), M an B-module, α = {α1, . . . , αl} a sequence of elements of M and H = (aij) a (k × l) matrix over GF(qs). Furthermore,

(1) For a ∈ GF(qs), a(Ts) will denote the only representative of a in Ts.(2) The following B-submodule of M will be denoted by HTs

(α):

〈l∑

i=1a1i(Ts)αi, . . . ,

l∑i=1

aki(Ts)αi〉.

Example 1. Let A = Z4[X]/〈X2〉, B = A[T]/〈T3 + T2 + T + 1〉, M a B-module, α =

{α1, α2, α3, α4} a sequence in M and H =

⎛⎜⎝T2 + T 0 T + 1 1T2 T 1 01 0 0 1

⎞⎟⎠ over GF(23). Since

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 17

T = {0, 1} ⊂ Z4 ⊂ A is a set of representatives for the residue field of A, T3 = {a0 +a1T + a2T2 : ai ∈ {0, 1}} ⊂ B is a set of representatives of the residue field of B. Then HT3(α) = 〈(T2 + T)α1 + (T + 1)α3 +α4, T2α1 + Tα2 +α3, α1 +α4〉. Observe that the 1in the matrix H has additive order 2 and the 1 in HT3(α) has additive order 4.

For our purposes the following result on the ideals of a ring in the family F3 will be use-ful. Let (A, m, GF(q)) ∈ F3, f ∈ A[T] a basic irreducible polynomial, B = A[T]/〈f〉, recall that a minimal A-generating set for m is also a minimal B-generating set for mB; A and Bhave the same length, m2B is the unique minimal ideal of B, v(m) = dimGF(q)(m/m2) =�A(A) − 2 ≥ 2 and �A(A) ≥ 4.

Lemma 5. Let (A, m, GF(q)) ∈ F3, l = �A(A), α = {α1, . . . , αl−2} be a minimal A-generating set for m, f ∈ A[T] be a basic irreducible polynomial, deg(f) = s and B = A[T]/〈f〉, then the ideals of length k ∈ {2, . . . , l − 2} of B are in one to one corre-spondence with the (k − 1) × (l − 2) matrices over GF(qs) in (rre)-form. Such a matrix corresponds to the ideal HTs

(α). In particular, the number of ideals of B is Gl−2(qs) +2.

Proof. The ideals of length k ∈ {2, . . . , l−2} of B are between mB and m2B. The assertion follows from Lemma 1, if we take the ring as B and the module as M = mB. �

The following examples illustrate the above Lemma.

Example 2. Let A = GF(2)[X, Y, Z, W]/〈XZ −XY, XW−XY, YZ −XY, YW−XY, ZW−XY, X2, Y2, Z2, W2〉, f ∈ A[T] be a basic irreducible polynomial, deg(f) = s and B =A[T]/〈f〉. It is easy to see that A ∈ F3, �A(A) = 6, {x, y, z, w} is a minimal A-generating set for the maximal ideal of A, T = GF(2) is a set of representatives for the residue field of A, Ts = {a0 + a1T + . . . + as−1Ts−1 : ai ∈ GF(2)} is a set of representatives of the residue field of B.

(a) Since (1, a1, a2, a3), (0, 1, b1, b2), (0, 0, 1, c1) and (0, 0, 0, 1), where ai, bi, c1 ∈ GF(ps), are all the 1 × 4 matrices in (rre)-form over GF(ps).(

0 0 1 00 0 0 1

), (

0 1 h1 00 0 0 1

), (

1 f1 f2 00 0 0 1

),(

0 1 0 g10 0 1 g2

), (

1 e1 0 e20 0 1 e3

)and

(1 0 d1 d20 1 d3 d4

), where di, ei, fi, gi, h1 ∈

GF(ps), are all the 2 × 4 matrices in (rre)-form over GF(ps).⎛⎜⎝ 1 0 0 m10 1 0 m20 0 1 m

⎞⎟⎠,

⎛⎜⎝ 1 0 n1 00 1 n2 00 0 0 1

⎞⎟⎠,

⎛⎜⎝ 1 o1 0 00 0 1 00 0 0 1

⎞⎟⎠,

3

18 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

⎛⎜⎝ 0 1 0 00 0 1 00 0 0 1

⎞⎟⎠, where mi, ni, o1 ∈ GF(ps), are all the 3 × 4 matrices in (rre)-form

over GF(ps). Then the ideals of A are:〈0〉, 〈xy〉, 〈x + a1y + a2z + a3w〉, 〈y + b1z + b2w〉, 〈z + c1w〉, 〈w〉, 〈x + d1z + d2w, y +d3z +d4w〉, 〈x +e1y +e2w, z +e3w〉, 〈x +f1y +f2z, w〉, 〈y +g1w, z +g2w〉, 〈y +h1z, w〉, 〈z, w〉, 〈x + m1w, y + m2w, z + m3w〉, 〈x + n1z, y + n2z, w〉, 〈x + o1y, z, w〉, 〈y, z, w〉, 〈x, y, z, w〉, 〈1〉, where a1, a2, a3, b1, b2, c1, di, ei, fi, gi, h1, mi, ni, o1 ∈ GF(2).

(b) The ideals of B have the same expression as the ideals of A but with the coefficients a1, a2, a3, b1, b2, c1, di, ei, fi, gi, h1, mi, ni, o1 in Ts.

Example 3. Let A = Z8[X]/〈X2−4, 2X〉, (cf. Corollary 1, (c)), f ∈ A[T] a basic irreducible polynomial, deg(f) = s and B = A[T]/〈f〉, T = {0, 1} ⊂ Z8 ⊂ A is a set of representatives for the residue field of A, Ts = {a0 + a1T + . . . + as−1Ts−1 : ai ∈ {0, 1}} ⊂ B is a set of representatives of the residue field for B.

(a) Since {x, 2} is a minimal A-generating set for the maximal ideal of A, (1, a) and (0, 1) where a ∈ GF(2s), are all the 1 × 2 matrices in (rre)-form over GF(2s). Then the ideals of A are: 〈0〉 ⊂ 〈4〉, 〈x + 2a〉, 〈2〉, 〈x, 2〉, A, where a ∈ {0, 1}.

(b) The ideals of B have the same expression as the ideals of A but with the coefficient ain Ts.

Corollary 2. Let (A, m, GF(q)) ∈ F3, l = �A(A), γ a unit of A and (n, q) = 1. Let Tn − γ = f1 · · · fr, where the f ′is are basic irreducible pairwise coprime polynomials in A[T] and si = deg(fi), for i = 1, . . . , r. Then the number of γ-constacyclic codes of length n over A is:

[Gl−2(qs1) + 2][Gl−2(qs2) + 2] · · · [Gl−2(qsr) + 2].

Proof. Recall that Gl(q) denotes the total number of vector subspaces of an l-dimensional GF(q)-vector space (see Section 2). The assertion follows from Lemma 4(d) and Lemma 5. �

The main result of this section, on the structure of γ-constacyclic codes over a ring of the family F3 can now be established.

Theorem 4. Let (A, m, GF(q)) ∈ F3, γ be a unit of A, l = �A(A), T and Ts as above, α = {α1, . . . , αl−2} a minimal A-generating set for m, and C a γ-constacyclic code of length n over A, (n, q) = 1. Let Tn − γ = f1 · · · fr be a representation of Tn − γ as a product of basic irreducible pairwise coprime polynomials in A[T] and si = deg(fi). Then

(1 ) There exists a partition of [1, . . . , r], U0, U1, . . . , Ul.

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 19

(2 ) For each i ∈ {2, . . . , l − 2} and each u ∈ Ui, there exists a (i − 1) × (l − 2) matrix in (rre)-form over GF(qsu), Hu, such that:

(3 ) C = 〈m2 ∏u/∈U1

fu, m ∏

u/∈Ul−1fu,

∏u/∈Ul

fu, (Hu)Tsu(α)fu : u ∈ ∪l−2

i=2〉.(4 ) |C| = q

∑u∈U1

su+2∑

u∈U2su+...+(l−1)

∑u∈Ul−1

su+l∑

u∈Ulsu .

Proof. Let Ai = A[T]/〈fi〉. From Lemma 4(b) and since m2Ai is the unique minimal ideal of Ai and mAi is the maximal ideal of Ai, there is a partition of [1, . . . , r], U0, U1, . . . , Ul, such that C has the form:⊕

u∈U1

m2Au ⊕⊕u∈U2

Iu ⊕ . . .⊕⊕

u∈Ul−2

Iu ⊕⊕

u∈Ul−1

mAu ⊕⊕u∈Ul

Au

where Ui = {u : �Au(Iu) = i}.

Let i ∈ {2, . . . , l− 2} and u ∈ Ui, by Lemma 5, Iu is of the form (Hu)Tsu(α) and it is

identified in A[T]/〈Tn − γ〉 with (Hu)Tsu(α)fu, where Hu is a (i − 1) × (l− 2) matrix in

(rre)-form over GF(qsu). Finally, ⊕

u∈UiAu is identified in A[T]/〈Tn−γ〉 with

∏u/∈Ui

fu. The last assertion follows from Lemma 4(c). �

Observe that the conditions (1) and (2) in Theorem 4 are uniquely determined by the code C and any code can be obtained in this way.

The following examples are given illustrating the previous results.

Example 4. Let A = GF(2)[X, Y, Z, W]/〈XZ −XY, XW−XY, YZ −XY, YW−XY, ZW−XY, X2, Y2, Z2, W2〉, be the ring of Example 2 and γ be a unit of A. Since A ∈ F3, �A(A) = 6, {x, y, z, w} is a minimal A-generating set for m, the maximal ideal of A, m2 = 〈xy〉, T = GF(2) is a set of representatives for the residue field of A and, by Hensel’s Lemma, T7−γ = f1f2f3, where f1 = λT +1, f2 = λT3+T2+1 and f3 = T3+T +λ ∈ A[T]. Then A[T]/〈T7 − γ〉 ∼= A ⊕A[T]/〈f2〉 ⊕A[T]/〈f3〉, T3 = {a1 + a2T + a3T2 : ai ∈ GF(2)}is a set of representatives for the residue field of the rings A[T]/〈f2〉 or A[T]/〈f3〉 and:

(a) If Ui = ∅, i ∈ {0, 1, 2, 4, 5, 6}, U3 = {1, 2, 3}, let H1 =(

1 a 0 b

0 0 1 c

)

over GF(2), H2 =(

1 α0 + α1T + α2T2 β0 + β1T + β2T2 00 0 0 1

)over GF(8) and

H3 =(

0 1 0 η0 + η1T + η2T2

0 0 1 σ0 + σ1T + σ2T2

)over GF(8).

The associated ideal, in A ⊕ A[T]/〈f2〉 ⊕ A[T]/〈f3〉, is:

〈x + ay + bw, z + cw〉 ⊕ 〈x + (α0 + α1T + α2T2)y + (β0 + β1T + β2T2)z,w〉 ⊕〈y + (η0 + η1T + η2T2)w, z + (σ0 + σ1T + σ2T2)w〉,

and corresponds, in A[T]/〈T7 − γ〉, to the ideal:

20 C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21

〈(x + ay + bw)f2f3, (z + cw)f2f3, (x + (α0 + α1T + α2T2)y + (β0 + β1T + β2T2)z)

f1f3,wf1f3, (y + (η0 + η1T + η2T2)w)f1f2, (z + (σ0 + σ1T + σ2T2)w)f1f2〉.

(b) If Ui = ∅, for i ∈ {0, 2, 5, 6}, U1 = {2}, U3 = {1}, U4 = {3} and let H1 =(1 e1 0 e20 0 1 e3

)over GF(2), H3 =

⎛⎜⎝ 1 0 0 α0 + α1T + α2T2

0 1 0 β0 + β1T + β2T2

0 0 1 η0 + η1T + η2T2

⎞⎟⎠ over GF(8).

The associated ideal, in A ⊕ A[T]/〈f2〉 ⊕ A[T]/〈f3〉, is:

〈x + e1y + e2w, z + e3w〉 ⊕ 〈xy〉 ⊕ 〈x + (α0 + α1T + α2T2)w,

y + (β0 + β1T + β2T2)w, z + (η0 + η1T + η2T2)w〉,

and corresponds, in A[T]/〈T7 − γ〉, to the ideal:

〈(x + e1y + e2w)f2f3, (z + e3w)f2f3, xyf1f3, (x + (α0 + α1T + α2T2)w)f1f2,

(y + (β0 + β1T + β2T2)w)f1f2, (z + (η0 + η1T + η2T2)w)f1f2〉.

Example 5. Let A = Z8[X]/〈X2 − 4, 2X〉 = {u + vx : u ∈ Z8, v ∈ {0, 1} ⊂ Z8}, (see Corollary 1(c)) and γ be a unit of A (hence γ = 1 + n, where n = 2a + 4b + cx, a, b, c ∈ {0, 1} ⊂ Z8). Since A ∈ F3, �A(A) = 4, {x, 2} is a minimal A-generating set for m, the maximal ideal of A, m2 = 〈4〉, T = {0, 1} ⊂ Z8 is a set of representatives for the residue field of A and, by Hensel’s Lemma, T7 −γ = f1f2f3, where f1 = n2T2 +T +7 +n, f2 = n2T4 + T3 + (3 + 4a + 3n)T2 + (2 + n2)T + 7 + n2 + 3n and f3 = T3 + (6 +4a)T2 + (5 + 4c)T + 7 + 7n ∈ A[T]. Then A[T]/〈T7 − γ〉 ∼= A ⊕ A[T]/〈f2〉 ⊕ A[T]/〈f3〉and: T3 = {a1 + a2T + a3T2 : ai ∈ {0, 1}} is a set of representatives of the residue field of the rings A[T]/〈f2〉 or A[T]/〈f3〉 and:

(a) If U0 = U3 = U4 = ∅, U1 = {3}, U2 = {1, 2} and let H1 = (1, 1) over GF(2), H2 = (1, a0 + a1T + a2T2) over GF(8), (i.e., 1, ai ∈ GF(2)).The associated ideal, in A ⊕ A[T]/〈f2〉 ⊕ A[T]/〈f3〉, is:

〈x + 2〉 ⊕ 〈x + 2(a0 + a1T + a2T2)〉 ⊕ 〈4〉,

and corresponds, in A[T]/〈T7 − γ〉, to the ideal:

〈(x + 2)f2f3, (x + 2(a0 + a1T + a2T2))f1f3, 4f1f2〉 (here 1, ai ∈ {0, 1} ⊂ Z8).

(b) If U0 = U1 = ∅, U2 = {1}, U3 = {2}, U4 = {3} and take H1 = (1, a) over GF(2).The associated ideal, in A ⊕ A[T]/〈f2〉 ⊕ A[T]/〈f3〉, is

〈x + 2a〉 ⊕ 〈x, 2〉 ⊕ 〈1〉,

C.A. Castillo-Guillén et al. / Finite Fields and Their Applications 43 (2017) 1–21 21

and correspond, in A[T]/〈T7 − γ〉, to the ideal:

〈(x + 2a)f2f3, xf1f3, 2f1f3, f1f2〉.

5. Conclusion

In this paper the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined, particularly those rings with p4 elements (p a prime), which include those appearing in the literature with 24 elements. Furthermore, the number and structure of γ-constacyclic codes having finite local Frobenius non-chain rings with nilpotency index 3 as alphabet, of length relatively prime to the characteristic of the residue field of the ring, are determined. Several examples are included illustrating the main results.

Acknowledgments

The authors would like to thank the referees for their valuable observations and comments. The first author was partially supported by CONACYT fellowship 104564, México.

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