Codes over an inﬁnite family of algebras

: In this paper, we will show some properties of codes over the ring B k = F p [ v 1 , . . . , v k ] / ( v 2 i = v i , ∀ i = 1 , . . . , k ) . These rings, form a family of commutative algebras over ﬁnite ﬁeld F p . We ﬁrst discuss about the form of maximal ideals and characterization of automorphisms for the ring B k . Then, we deﬁne certain Gray map which can be used to give a connection between codes over B k and codes over F p . Using the previous connection, we give a characterization for equivalence of codes over B k and Euclidean self-dual codes. Furthermore, we give generators for invariant ring of Euclidean self-dual codes over B k through MacWilliams relation of Hamming weight enumerator for such codes.


Introduction
Codes over finite rings has been an interesting topic in algebraic coding theory since the discovery of codes over Z 4 , see [4].An example of finite rings which has interesting properties is the ring because it has two Gray maps which relate codes over such ring and binary codes, see [2].This ring also has non-trivial automorphisms which can be used to define skew-cyclic codes, for example in [1], skew-cyclic codes over the ring A 1 = F 2 + vF 2 , where v 2 = v, which give some optimal Euclidean and Hermitian self-dual codes.Furthermore, Abualrub et al. show that skew-cyclic codes over A 1 have a connection to left submodules over a skew-polynomial ring and give skew-polynomial generators for these codes.In [6], skew-cyclic codes over the ring A 1 have been characterized using a Gray map.This characterization gives a way to construct skew-cyclic codes over the ring A 1 from binary cyclic or quasi-cyclic codes, and also gives decoding algorithm for some codes over such ring.Meanwhile, Gao [3] consider skew-cyclic codes over the ring B 1 = F p + vF p , where v 2 = v, and found that these codes are equivalent to either cyclic codes or quasi-cyclic codes.Using this connection, Gao is able to give an enumeration for skew-cyclic codes which are constructed using an automorphism with order relatively prime to the length of the codes.
In this paper, we consider codes over the ring [2] and B 1 in [3].We study its maximal ideals, automorphisms, equivalence codes, and Euclidean self-dual codes over these rings, including the generators for its invariant ring.This paper is organized as follows: Section 2 describes some properties of the ring B k such as maximal ideals and automorphisms.Meanwhile, in Section 3, we describe a Gray map for the ring B k , and we characterize linear codes and equivalent codes over the ring B k .Finally, in Section 4, we characterize Euclidean self-dual codes, give the shape of MacWilliams relation and generators of invariant rings for Euclidean self-dual codes.

The ring B k
As we readily see, the ring B k forms a commutative algebra over prime field F p .Let Ω = {1, 2, . . ., k} and 2 Ω is the collection of all subsets of Ω.Also, let w i be an element in the set Then, we will prove the following observation.
(=⇒) Consider the equation, Therefore, if a + bv k = 1, then γ = 1 and = −β(α(β + α)) −1 .Which implies, α + βv k is a unit if and only if α and α + β are also units.Considering this observation for elements in B k−1 , B k−2 , . . ., B 1 , we have α + βv ∈ B 1 is a unit if and only if α, α + β ∈ F p are non zero elements.Since, every element in finite commutative ring is either a unit or a zero divisor, we can see that the only zero divisors in B 1 are the elements in the ideals generated by βv or α(1 − v).By generalizing this result recursively, we have the intended conclusion.Also, we can easily show that The following lemma is needed to prove Proposition 2.4.
since F p has characteristic p and β p−1 = 1 for all β ∈ F p .If we continue this procedure, then we have α p = α.
The following result shows that the ring B k is a principal ideal ring.
Otherwise, if i ∈ A, then there is a unique A = A ∪ {i} ⊆ {1, . . ., m} such that So, every term will be vanish except It is clear that The following proposition shows that the ideal in Lemma 2.2 is the only maximal ideal in B k .(=⇒) Let J be a maximal ideal in B k .By Proposition 2.4, B k is a principal ideal ring.Then, let J = ω , for some ω ∈ B k .Note that, ω is not a unit in B k , so it is a zero divisor.By Lemma 2.1, ω is an element of some m i = w 1 , w 2 , . . ., w k , which means J ⊆ m i .Consequently, J = m i , because J is a maximal ideal.
Using the above result, we have the following lemmas.
Lemma 2.6.The ring B k can be viewed as an F p -vector space with dimension 2 k whose basis consists of elements of the form w S = i∈S w i , where S ∈ 2 Ω .
Proof.As we can see, every element a ∈ B k can be written as a = S∈2 Ω α S v S , for some α S ∈ F p , where v S = i∈S v i and v ∅ = 1.So, B k is a vector space over F p whose basis consists of elements of the form v S = i∈S v i , where v ∅ = 1 and there are k j=0 k j = 2 k elements of basis.Now, we will show that the set {1, w S2 , . . ., w S 2 k } is also a basis.Consider, for some α i ∈ F p , for all i = 1, . . ., 2 k , which gives, We have to note that, the set with elements of the w S , where S ∈ 2 Ω , is also linearly independent over F p , because S k is a vector space over F p with element of basis are of the form v S , where S ⊆ Ω.Therefore, Proof.It is immediate since characteristic of F p is p, and B k can be viewed as a F p -vector space with dimension The following theorem characterizes the shape of automorphisms in the ring B k .
Theorem 2.8.Let θ be an endomorphism in B k .Then, θ is an automorphism if and only if θ(v i ) = w j , for every i ∈ Ω, and θ, when restricted to F p , is an identity map.

Proof. (=⇒) Let
We can see that the map λ is a ring homomorphism.For any a, b ∈ B k /J where λ Consequently, a − b = 0 + J, which means a = b, in other words, λ is a monomorphism.Moreover, for any a ∈ B k /J θ , let a = a 2 + J θ for some a 2 ∈ B k , then there exists a = θ −1 (a 2 ) + J such that λ(a) = a .Therefore, F p B k /J B k /J θ , which implies J θ is also a maximal ideal.By Proposition 2.5, which means, A⊆Ω,A =∅ (−1) |A|+1 ( j∈A w j ) p−1 and A⊆Ω,A =∅ (−1) |A|+1 ( j∈A θ(v j )) p−1 are associate.Therefore, θ(v i ) = βw j for some unit β which satisfies β |A| p−1 = β, for all A = ∅.Consequently, we have β p−1 = β, but by Lemma 2.3, β p = β.Since β is a unit, we have that β p−1 = 1.Therefore, β must be equal to 1.Moreover, since θ is an automorphism, θ(v i ) = θ(v j ) whenever i = j.Also, since the only automorphism in F p is identity map, we have the conclusion.
(⇐=) Suppose that θ(v i ) = w j , and θ(v i ) = θ(v j ) whenever i = j.By Lemma 2.6, we can see that θ is also an automorphism.Now, we have to note that every element a in B k can be written as for some α S ∈ F p , where w S = i∈S w i .Define a map ϕ as follows.
We can show that this map ϕ is a bijection map.Furthermore, this map can be extended n tuples of B k as follows.
Since ϕ is a bijection map, we also have ϕ is a bijection map.We have to note that, the map ϕ is a permutation, based on the choice of subsets S i ∈ 2 Ω , of Gray maps in [2].

Codes over the ring B k
The following proposition gives a characterization of B k -linear codes using the map ϕ.Proposition 3.1.C is a linear code over B k if and only if there exist linear codes C 1 , . . ., C Proof.(=⇒) Since ϕ is a bijection, there exist C 1 , . . ., C 2 k such that C = ϕ −1 (C 1 , . . ., C 2 k ).Now, we only need to show that C i is a linear code over F p for all i = 1, . . ., 2 k .For any C i , let c 1 and c 2 be two codewords in C i .For l = 1, 2, let c l = (α for any λ l in F × p for all l = 1, 2. The last equality holds since t , 0, . . ., 0) (⇐=) Take any two codewords c 3 and c 4 in C. Let S w S , . . ., S w S , . . ., for some α i , β i in F p , where i = 1, . . ., 2 k .For any λ 3 and λ 4 in F × p we have is also in (C 1 , . . ., C 2 k ) , since C i is a linear code for every i = 1, . . ., 2 k .Therefore, λ 3 c 3 + λ 4 c 4 is also in C. Now, following [5], we define permutation equivalence of codes as follows.
Definition 3.2.Two codes are permutation equivalent if one can be obtained from the other by permuting the coordinates.
Using Definition 3.2, we can define the following notion of equivalence between two codes.Note that, the above definition is similar to the one in [5].Now, let Π θ be a permutation on 2 k tuples of F p induced by automorphism θ.Then we have for any c ∈ B n k .Then, we have the following characterization.Theorem 3.4.Let C and C be two codes over B k .Then, C and C are equivalent if and only if there exists a permutation which sends (C 1 , . . ., C 2 k ) to (C 1 , . . ., C 2 k ) or to (Π θ (C 1 ), . . ., Π θ (C 2 k )).
Note that the matrix T is derived from the identity in Lemma 4.2 and the matrix D is derived from the condition that n is always even.Also, it is easy to see that G = {I, D, T, −T }.Formally, we have the following result.Let R G be a set of all polynomials in two variables which are invariant under the action • of G.We can easily prove that R G is a ring, and by the above Lemma we can see that every Hamming weight enumerator of Euclidean self-dual codes must be inside R G .This ring R G called invariant ring for Euclidean self-dual codes over B k .The following theorem gives generators for R G .

Proposition 2 . 5 .
An ideal I in B k is maximal if and only if I = w 1 , w 2 , . . ., w k .Proof.(⇐=) It is clear by Lemma 2.2.
which means they are linearly independent over F p .Lemma 2.7.The ring B k has characteristic p and cardinality p 2 k .

Definition 3 . 3 .
Two codes C and C over B k are equivalent if either they are permutation-equivalent or C is permutation equivalent to the code θ(C ) for some automorphism θ in B k , i.e. the code θ(C ) obtained from C by changing α with θ(α) in all coordinates.

Lemma 4 . 3 .
If W C (X, Y ) is a Hamming weight enumerator for an Euclidean self-dual code C over B k , then W C (X, Y ) is invariant under the action of G.