Abelian and consta-Abelian polyadic codes over affine algebras with a finite commutative chain coefficient ring

In this paper, we define Abelian and consta-Abelian polyadic codes over rings defined as affine algebras over chain rings. For that aim, we use the classical construction via splittings and multipliers of the underlying Abelian group. We also derive some results on the structure of the associated polyadic codes and the number of codes under these conditions.


Introduction
Polyadic codes were first introduced in [1].There is a rich literature on this type of codes, see for example [2,6,8] and the references therein.Recently in [3,4] they extended some of these ideas to codes over rings.
In this paper we extend those results in a twofold way: we answer some questions settled in [8], namely the generalization of polyadic Abelian codes to the case of chain rings and, also to consider the more general case of considering a class of serial rings as ambient space that do not entirely split into linear factors over the base chain ring.Up to our knowledge, this type of base rings have been considered only for the case that the polynomials defining it split completely into linear factors and the underlying ring is a finite field, see [3,4].Note that general linear codes over this type of rings, namely affine algebras with a finite commutative chain coefficient ring was already studied in [12] and a concrete example of those ambient spaces was studied in [5] that is closely related to the construction in [3,4], but in [12] only general linear codes are studied and there is no advantage taken of the underlying group structure as in the present paper.
The outline of the paper will be as follows.In Section 2, we introduce some preliminaries on finite chain rings and serial polynomial rings over them as well as their idempotents.Section 3 is devoted to codes over those types of rings, we will take particular care of the structure of constacyclic codes and multiconstacyclic codes.In Section 4, we defined Abelian and consta-Abelian polyadic codes over chain rings via splittings and multipliers.Sections 5 and 6 are the core part of the paper where we study Abelian and consta-Abelian polyadic codes over affine alebra rings with a finite commutative chain coefficient ring.We finish with some conclusions in Section 7.

Preliminaries
In this section, we will fix our notation and recall some basic facts about finite chain rings (see for example [13] for a complete account) and serial polynomial rings over a chain ring (see [9]).In this paper, all rings will be associative, commutative, and with identity.A ring R is called a local ring if it has a unique maximal ideal.A local ring is a chain ring if its lattice of ideals is a chain under inclusion.In this case, since the ideals are linearly ordered by inclusion, the ring is also called uniserial.It can be shown [8,Proposition 2.1] that R is a finite commutative chain ring if and only if R is a local ring and its maximal ideal is principal.We will denote by a ∈ R a fixed generator of the maximal ideal m, and let t be its nilpotency index, thus the ideals of R are m i = a i for i = 0, . . ., t.Also, we will denote the residue field of R by F q = R/m, where q = p l , for a prime number p.We will denote the polynomial ring in the indeterminates X i by R[X 1 , . . ., X s ] for i = 1, . . .s with its coefficients in R. We can extend the natural ring homomorphism .from R to F q given by r → r = r + m to the polynomial rings R[X 1 , . . ., X s ] and F q [X 1 , . . ., X s ] just by applying .on each coefficient of the polynomial.Let t i (X i ) ∈ R[X i ] (i = 1, . . ., s) be monic polynomials such that each ti (X i ) ∈ F q [X i ] is a square-free polynomial.During this paper, we are interested in codes over the following alphabet Note that this setting includes as particular cases the alphabets considered in [3][4][5].In [9] the ideals of R have been described explicitly.This structure has been also studied in [18,19], in the case where the ring R is the finite field F q .
In the finite field case, the square-free condition on the polynomials t i (X i ) is known as the "semisimple condition" because of the structure of the ring R (it can be decomposed as a direct sum of simple ideals).In the general case, the square-free condition on the polynomials t i (X i ) leads to a decomposition of the ring R as a direct sum of finite chain rings, and therefore it is a serial ring [15].
In the remaining part of the preliminaries, we will follow [9] to explicitly show decomposition in terms of primitive idempotents.Let H i be the set of roots of ti (X i ) in a suitable extension of F q for i = 1, . . ., s.For each ν ∈ H = s i=1 H i we define the class of ν as C(ν) = {(ν q j 1 , . . ., ν q j s ) | j ∈ N}.We will denote the set of all the classes as C = C(t 1 , . . ., t s ), the elements of C form a partition of H and for any ideal I ⊳ R/m the set of the common zeros of the elements in I is a union of classes.Also the size of each class is given by where d i is the degree of the irreducible polynomial of ν i over F q .
For all i = 1, . . ., s and a class C, let p C,i (X i ) denote the polynomial Irr(ν i , F q ) and (ν 1 , . . ., ν s ) ∈ C .Also for all i = 2, . . ., s we consider the polyno- bC,i(Xi) .Note that it is clear that they are independent of the element ν chosen within C and that b C,i (X i ) and bC,i (X i ) are coprime polynomials.Then, define the multivariable polynomials w C,i (X 1 , . . ., X i ), and π C,i (X 1 , . . ., X i ) obtained from b C,i (X i ) and bC,i (X i ) respectively by substituting ν i by X i .One has that and we denote the Hensel lifts to R of the polynomials p C,i , w C,i and π C,i by q C,i , z C,i and σ C,i respectively.If we denote by σ qC,i(X2,...,Xi) , then I C +I = Ann( h C +I ), h C +I ≃ R[X 1 , . . ., X s ]/I C and R ≃ C∈C h C + I ( for a proof see [9, Proposition 3.7, Lemma 3.8 & Theorem 3.9]).This decomposition of the ring R is equivalent to the existence of primitive orthogonal idempotents elements e C ∈ R where C ∈ C such that 1 = C∈C e C and e C R ≃ h C + I , i.e. there exists a polynomial g C such that the idempotent e C is the element g C h C +I and g C h C +I C = 1+I C .Any ideal of R is principally generated by G + I where G = t−1 i=0 a i G i and G i is a sum of primitive idempotents e C described before (see [9,Corollary 3.12]).
Remark 1.Note that this decomposition includes the case in [3,5] where, in the first reference there are two variables where R = F q and t 1 , t 2 split completely in linear factors over F q [X] and, in the second one there is one polynomial in R = F q whose roots are all the elements in the field.

Structure of codes over R
We will define a linear code K of length n over the ring R as an R-submodule of R n .The Euclidean dual of K will be denoted by K ⊥ and it is given by the set {x ∈ R n | x • k = 0 for all k ∈ K}, where • is the Euclidean inner product in R n .Note that, since R = C∈C e C R, for each x ∈ R n we can define the projection of x by e C ′ as x C ′ = (x 1,C ′ , . . ., x n,C ′ ) ∈ R n where x i = C∈C x i,C e C and x i,C ∈ R for i = 1, . . ., n.Indeed, x C ′ = x • e C ′ for each C ∈ C and for a given linear code K of length n over the ring R we can define the following codes where C ranges in the set of classes in C. It is clear that K C is an R-linear code and that C e C (note that we slightly abuse the notation since first orthogonality is in R and the second one in R).

Constacyclic codes over
Then the following result follows and it can be proven in the same fashion as [4,Theorem 3].
The following result characterizes, in a polynomial way, the class of constacyclic codes over R. First we will introduce the following lemma that characterizes constacyclic codes over chain rings, it can be found in [14] or in a more general way that can be also use for the multivariable case in the language of Canonical Sets of Generators in [10].

Lemma 3 ([14]
).A non-zero λ C -constacyclic code K C over the chain ring R with maximal ideal m = a and nilpotency index s has a generating set in standard form S = {a b0 g b0 , a b1 g b1 , . . ., a bu g bu } (3) and the code is principal as ideal Proof.It is straightforward from the reasoning above.

Consta-abelian codes over R
We can state a similar result as Lemma 3 in the case of abelian codes over finite chain rings in terms of Canonical Set of Generators [9,10] but, as it was pointed in [11, Corollary 1], in the case of Abelian codes they are principal if the length of the code is coprime with the characteristic of the chain ring, thus we will restrict ourselves to that case stated in [9, Sections 4 and 5].In the case of consta-Abelian, that is not the general case (see [11,Example 1] where it is shown that negacyclic codes over Z 4 defined by a multiple root polynomial can be seen as a principal ideals).Anyway, since our purpose is defining polyadic codes using splittings of the roots we will restrict to simple root codes.Given an Abelian group A expressed as A = δ i=1 Z i where, for each i, Z i is a cyclic group and let r = δ i=1 r i where r i is the size of the cyclic component Z i and gcd(r i , q) = 1 for each i = 1, . . ., δ.

Definition 5. An Abelian code over the ring R with underlying group
where the element λ i is an invertible element in R for each i = 1, . . ., δ.An ideal in R A,λ is called a λ = (λ 1 , . . ., λ δ )-consta-abelian code with underlying group A.
As in Section 3.1 we can state the following results.The first one is a straightforward generalization of Proposition 2.
A particular version of Theorem 3.13 in [9] (and [9, Corollary 3.14] for devising the sizes of the ideals) suited to our setting provides us the following analogous results to Lemma 3 and Proposition 4.
) and the code is principal as ideal

Polyadic codes over chain ring 4.1 Splittings and multipliers
As in the section before, let A = δ i=1 Z i be a finite Abelian group, r = δ i=1 r i where r i = |Z i | and gcd(r i , q) = 1 for each i = 1, . . ., δ.We will associate to A the ideal We will denote as C A the set of the cyclotomic classes associated to I A as in Section 2. We will define a splitting of A following the notation in [8].For that, we will consider the commutative group A ⋆ = (A, ⋆) given by the component-wise multiplication ⋆ in A derived from the multiplication in the components Z i ≃ Z/Z ri and A * ⋆ its group of units.Any u = (u 1 , . . ., u δ ) ∈ A * ⋆ defines an action u ⋆ over A given by a = (a 1 , . . ., a δ ) → u ⋆ (a) = (u 1 a 1 , . . ., u δ a δ ) for all a in A .We extend this concept to a union C of cyclotomic classes in C A defining u ⋆ (C) as the union of the exponents of the images of the elements u ⋆ (a) where a ∈ A is associated to an element in C. We call this u ⋆ a multiplier.In the literature, usually, the splittings are defined over cyclotomic cosets of modular integers, note that for this case there is a one-to-one correspondence with our classes of roots.It is clear that in a splitting the class given by {0} is contained in the set S ∞ .
For a given set S which is a union of cyclotomic classes and for a chain ring R with quotient field F q we will denote the ideal on R[Y 1 , . . ., Y δ ]/I A by I S the ideal given by I S = C∈CA,C∈S I C .Note that in the case that R is a finite field, I S denotes the polynomial ideal in R[Y 1 , . . ., Y δ ]/I A whose elements vanish when evaluated in all the elements in S.

Polyadic abelian codes over chain rings
Definition 10 (Polyadic Abelian codes over a chain ring).Let R be a chain ring.Let A be a finite Abelian group, 0 ≤ i ≤ m − 1, S = (S ∞ , S 0 , S 1 , ..., S m−1 ) a m-splitting of the cyclotomic classes C A associated to I A and S ′ ∞ = S ∞ \ {0}.The ideals (codes) defined in R[Y 1 , . . ., Y δ ]/I A are called polyadic codes.K i and K i are called even-like codes and L i , L i are called odd-like codes.
The following result follows directly from the definition (see [8, Theorem 2.1]) for its counterpart of cyclic codes over finite fields).
Proposition 11.For i = j, i, j ∈ {0, 1, . . ., m − 1} • The following identities hold • For 0 ≤ i ≤ m − 1, all the codes K i are equivalent codes.The same is true for the other families of codes K i , L i , and L i .
Note that the last fact is a straightforward result of multiplier u * having the property u * (S i ) = S i+1 .Since each K i is uniquely determined by the m-splitting set S i it can be regarded as a permutation thus each code K i is equivalent to the other K i+1 .
Proposition 12.For an m-adic code K over a finite chain ring R, let K = K • I S∞ as ideals in R[Y 1 , . . ., Y δ ]/I A .Then, for all i ∈ {0, 1, . . ., m − 1} we have The code K = I S∞ in the proposition above is called the even-like subcode of K when S ∞ = {0} and the codewords in K \ K are called odd-like in that case.
For 0 ≤ i ≤ m − 1, let ēi and ē′ i be the even-like idempotent generators of even-like codes K i and K i respectively, di and d′ i be the odd-like idempotent generators of even-like codes L i and L i respectively in R[Y 1 , . . ., Y δ ]/I A , respectively given in Section 2.
If we take the element σ = −1 ∈ A ⋆ it is clear that it induces a permutation of the cyclotomic cosets but it could be the case that it does not induce a permutation on the sets S 0 , . . ., S m−1 of an m-splitting.The following result follows directly form the finite field case in [8, Proposition 2.2] and the characterization of the dual of an Abelian code over a chain ring in [9, Sections 4 & 5].
Proposition 13.Suppose that σ ⋆ (S ∞ ) = S ∞ and that σ ⋆ is a permutation of S 0 , . . ., S m−1 such that σ ⋆ (X i ) = X σ(i) , for i ∈ {1, . . ., m − 1}.Then Remark 14.Note that the existence of polyadic abelian codes over the finite chain ring R relies on the existence of polyadic abelian codes over F q = R/m since we used the same cyclotomic clases.We refer to [8, Section III] for that study.

Polyadic Consta-Abelian codes over chain rings
We will follow mainly the ideas in [7] for describing consta-Abelian serial codes over a chain ring R defined by the Abelian group A = δ i=1 Z i and λ = (λ 1 , . . ., λ δ ) ∈ (R * ) δ .The ambient space is given by R
As in the section above, we will consider the commutative group A ⋆ = (A, ⋆).For each set S ⊂ A one can define S = {1+r(s−1) | s ∈ S}.We say that S ⊂ A defines an orbit with respect to r if S is a cyclotomic coset of A, in other words, defines the exponents of a cyclotomic class of the associated Abelian code R[A] that we will denote as C S .We say that the splitting is non-trivial if S ∞ Θ, that is S i = ∅ for i = 0, . . ., n − 1.
Given a splitting of Θ w.r.t.r as above, the ideals (codes) given by its defining sets defined in R[Y 1 , . . ., Y δ ]/I A are called polyadic codes.K i and K i are called even-like codes and L i , L i are called odd-like codes.Note that in the case of A being a cyclic group and Θ = A we are in the case of splittings for constacyclic codes, moreover in that case if S ∞ = ∅ they are called Type I, otherwise they are called Type II (See for example [2]).Now we have a similar result to Proposition 11 that can be stated for chain rings also (see [7,Theorem 7.2] for the finite field case).
Proposition 16.For i = j, i, j ∈ {0, 1, . . ., m − 1} • The following identities hold • For 0 ≤ i ≤ m − 1, all the codes L i are equivalent codes.The same is true for the family of codes L i .

Polyadic abelian codes over serial rings
In this section we describe polyadic codes of length n over the ring R expressed as in Equation 1 by using its primitive idempotents e C in the Section 1 where C ∈ C is identified.Let us divide the classes of C from {1, . . ., |C|} into m disjoint sets and denote these disjoint sets with A i for i = 1, . . ., m. C can be written as follows: Consider the condition that each Otherwise, |C| sets in the partition in the Equation 8 are non-empty set has only one element and the remaining m − |C| are empty.So in this case, It can be easily seen that |C| = m i=1 t i .We define θ Ai = Cj ∈Ai e Cj for each i = 1, . . ., m. Assume that θ Ai = 0 when A i = ∅.It is easily seen that m i=1 θ Ai = e C = 1 R , θ 2 Ai = θ Ai and θ Ai .θAj = 0 for all i = j.Note also that this idempotents in R seen as generators correspond to a disjoint decomposition of R as a sum of serial codes.
Let E i , E ′ i , D i , D ′ i be the idempotent generators of polyadic codes over R[A] defined as in the Section 4.1 for i = 1, . . ., m. E i and E ′ i 's are even-like ones, while the others are odd-like ones.
From now on, we can define the idempotents to obtain polyadic codes over the serial ring R[A] = R[X 1 , . . ., X s , Y 1 , . . ., Y δ ]/ I, I A .These idempotents can be written as follows: Note that the following index number k ij we will use in order to enumerate all idempotents is the smallest positive integer which is equivalent to the number i − j + 1 (i.e k ij = i − j + 1 mod (m)) and the structure of the positive integer k ij forces the cyclicity of the mapping u ⋆ over the new idempotents.
Therefore, we can obtain odd-like (or even-like) polyadic codes over R by using the odd-like idempotents D j and D ′ j (or even-like idempotents E j and E ′ j ) for j = 1, . . ., m.Let the polyadic codes associated with the idempotents D j , D ′ j , E j and E ′ j over R be called as P j , P j , Q j and Q j , respectively.So, the desired polyadic codes are generated by the idempotents such that P j = D j , P j = D ′ j , Q j = E j and Q j = E ′ j .Remark 17.By choosing A i once, we now get a new idempotent to define a polyadic code (odd-like or even-like) and one can easily see that the polyadic codes obtained by taking image of these idempotents under the multiplier u * are all equivalent codes.But if we change our choice for A i we will get a new odd-like (or even-like) polyadic code that is inequivalent code to the other codes corresponding to other selections.So we can count the number of these inequivalent codes as in the follow theorem.In the same fashion, θ Am−1 can be chosen in |C|−(t1+t2+...+tm−2) tm−1 different ways and finally θ Am has to be taken in only one way.Thus, the total number of odd-like idempotent generators is equal to odd-like idempotent generators and similarly the number of idempotent generators generated by D ′ j is same.Therefore, we have just counted the number of odd-like idempotent generators (i.e. the number of odd-like inequivalent polyadic codes) as the number stated in the expression of the theorem.The same quantity is valid for the number of even-like inequivalent polyadic codes.Finally, consider the case |C| < m.Since all non-empty A i has only one element, the total number of choices A i is equal to m |C| and the number of their displacement is (|C|)!, the total number of odd-like idempotent generators is (|C|)!m |C| .The number of inequivalent ones is 1 m (|C|)!m |C| by using the same argument as the previous case.Finally by considering the equality of arrivals from both D j and D ′ j , the total number of inequivalent odd-like polyadic codes is 2 m (|C|)!m |C| .Similar operations in both cases can be done to compute the number of inequivalent even-like polyadic codes.
We will denote by Rep(n) the repetition code of length n, that is, the code generated by the polynomial i Y i ∈ R[A] (i.e. the polynomial with all ones as coefficients), and, as usual, the even weight code is just Rep(n) ⊥ .The following two theorems extend those in [4] to affine algebra rings and Abelian codes.
Theorem 19.Let B be a subset of {1, 2, . . ., m} with at least two elements.The following propositions are satisfied for the polyadic codes P i , Q i over R defined as above. 1.
If we consider P i 's and Q i 's instead of P i 's and Q i 's respectively, the previous statements also hold.
Proof.For the proof of the first statement of the theorem, recall that for any Abelian codes C and D, the defining set of the Abelian code C ∩ D is the union of the defining set of C and D. So the union of all defining sets of P i generates the repetition code.Recall also that for any two Abelian codes C and D whose idempotent generators of these codes are E 1 and E 2 respectively, the idempotent generators of C ∩ D and C + D are E 1 E 2 and E 1 + E 2 − E 1 E 2 respectively.By generalizing of these properties we can obtain the second statement of the theorem.By using that the idempotent generators of P i are D i as above and the fact that the basic properties of idempotents we get the following expression Theorem 20.Let B be a subset of {1, 2, . . ., m} with at least two elements.The following statements are satisfied by the polyadic codes P i and Q i over R defined as above.
3. P i + P i = R[A] and P i ∩ P i = Rep(n) Proof.Let E i and i Rep be the generator idempotents of Q i and Rep(n) respectively.Then the idempotent generator of the code Q i + Rep(n) can be written as idempotent and its pair, say D i and D i .Now we can define the polyadic consta-Abelian codes over R using p of polyadic consta-Abelian codes over a chain ring R Thus we get the following codes (in this case, we have also Type I codes since S ∞ can be the empty set) over R[A, λ] = R[X 1 , . . ., X s , Y 1 , . . ., Y δ ]/ I, I A,λ .
Definition 22.Let Θ = A and an S = (S ∞ , S 0 , S 1 , ..., S m−1 ) an m− splitting of Θ w.r.t.r.Let k ij be integers such that k ij = i − j + 1 mod (m).Let L i , L i , K i and K i be defined as in Section 4.3.
Let the polyadic consta-abelian codes associated with the idempotents D j , D ′ j , E j and E ′ j over R[A, λ] be called as P j , P j , Q j and Q j , respectively.So, the desired polyadic consta-abelian codes are generated by the idempotents such that P j = D j , P j = D ′ j , Q j = E j and Q j = E ′ j .It is a straightforward exercise to check that the following results can be proven in the same fashion as the Abelian case in Section 5 above since they only rely on the decomposition of the idempotents in the polynomial rings R[X 1 , . .
a λ-constacyclic of length n over R and suppose that the λ C -constacyclic codes K C are generated by G C (x) defined as in the previous Lemma for each C ∈ C. Then there exists a polynomial

Lemma 7 .
A non-zero λ C consta-Abelian code K C over the chain ring R with maximal ideal m = a and nilpotency index s has a generating set in standard form abelian code with underlying group A of length n over R and suppose that the λ Cconstacyclic codes K C are generated by G C (x) defined as in the previous Lemma for each C ∈ C. Then there exists a polynomial G(X) = C∈C G C e C such that K = G ⊳ R A,λ and |K| = C∈C |K λC |.

Definition 9 . 2 . 3 .
For a positive integer m ≥ 2 and a nonempty set S ∞ ⊂ A, an m− splitting of A is a m-tuple S = (S ∞ , S 0 , S 1 , ..., S m−1 ) which satisfies the follow conditions:1.Each set S ∞ , S 0 , S 1 , ..., S m−1 is a union of cyclotomic classes in C A , The sets S ∞ , S 0 , S 1 , ..., S m−1 are disjoint and form a partition of C A , There exists u ∈ A * ⋆ such that u * (S ∞ ) = S ∞ and u ⋆ (S i ) = S i+1 where u ⋆ is a multiplier.

Definition 15 .
Let Θ be a union of orbits in A w.r.t.r.For a positive integer m ≥ 2 and a set S ∞ ⊂ Θ, an m− splitting of Θ w.r.t.r is a m-tuple S = (S ∞ , S 0 , S 1 , ..., S m−1 ) which satisfies the follow conditions: 1.Each set S ∞ , S 0 , S 1 , ..., S m−1 is a union of orbits in A w.r.t.r, 2. The sets S ∞ , S 0 , S 1 , ..., S m−1 are disjoint and form a partition of Θ, 3.There exists u ∈ A * ⋆ such that u * (C S∞ ) = C S∞ and u * (C Si ) = C Si+1 where u * is a multiplier.

Theorem 18 ( 1 ,
Number of polyadic codes).The following statements are hold: 1.If |C| ≥ m, then the number of inequivalent odd-like (or even-like) polyadic codes over the ring R is equal to where T i = i j=1 t j .2. If |C| < m, then the number of inequivalent odd-like (or even-like) polyadic codes over the ring R is equal to 2 m (|C|)!m |C| Proof.First, we will prove the desired number in the case |C| ≥ m.To count the odd-like polyadic codes, we should calculate the number of idempotent generators by using the equalities D j = u * (D j−1 ) = m i=1 θ Ai D kij and D 1 = m i=1 θ Ai D i for j = 2, . . ., m.The total number of odd-like idempotent generators obtained by these equations is determined by the number of choices of θ Ai for i = 1, . . ., m. Recall that |θ Ai | = t i and the total number of idempotents of type e C is |C|.θ A1 can be chosen in |C| t1 different ways out of |C| idempotents.θ A2 can be chosen in |C|−t1 t2 different ways out of |C|−t 1 remained idempotents.

1 |C| − T 1 t 2
. . .|C| − T m−2 t m−1 .Since D 1 = m i=1 θ Ai D i , D j = u * (D j−1 ) = m i=1 θ Ai D kij for j = 2, .. ., m and the fact that u * only permutes the idempotents D i , the number of idempotent generators obtained D j is equal for j = 1, . . ., m.So, only one D j By applying similar operations we obtain the following equalityj∈B D j = m j=1 D jwe can get the 3) in the theorem.The expression in item 4) can easily be obtained by considering 1).Now, Let the idempotent generator of the Rep(n) be i Rep .By using that thei=1 θ Ai D i E i = 0 and D 1 + E 1 = m i=1 θ Ai (D i + E i ) = m i=1 θ Ai = 1, we get that P 1 ∩ Q 1 = D i E i = {0} and P 1 + Q 1 = D i + E i − D i E i = 1 .

θθθ
Aj E j + i Rep • Aj (E j + i Rep ) = m j=1 Aj D ′ j = D ′ j .
. , X s ]/ I and R[Y 1 , . . ., Y δ ]/ I A,λ .The first two theorems are related to type I codes and the remaining ones to type II codes.Theorem 23 (Number of polyadic consta abelian codes of Type I).Consider Type I codes in Definition 22.The following statements hold.