Self-Dual Codes over a Family of Local Rings Steven T. Dougherty∗, Cristina Fern´andez-C´ordoba† and Roger Ten-Valls‡§¶ July 22, 2016

Abstract We construct an infinite family of commutative rings Rq,∆ and we study self-dual codes over these rings. We define a cannonical Gray map from Rq,∆ to vectors over the residue finite field of q elements and use it to relate self-dual codes over Rq,∆ to codes over the finite field Fq . We determine the parameters for when self-dual codes exist and give various constructions for self-dual codes over Rq,∆ .

Key Words: Self-dual codes, codes over rings, local rings.

1

Introduction

Self-dual codes are one of the most widely studied families of codes both over fields and over rings. They have wide applications in mathematics including connections to unimodular lattices, modular forms and combinatorial designs. Codes over rings have received a great deal of attention because of their use in finding new binary self-dual and formally self-dual codes, as well for their connection to lattices and number theory. In particular, codes over finite algebraic extension of finite fields have been widely studied. In the case when the finite field is F2 , there have been studies of extensions with just one indeterminant of different orders (see [1], [15] or [13]), with ∗ S. T. Dougherty is with the Department of Mathematics, University of Scranton, Scranton, PA 18510, USA (e-mail:[email protected]). † C. Fern´ andez-C´ ordoba is with the Department of Information and Communications Engineering, Universitat Aut` onoma de Barcelona, 08193-Bellaterra, Spain (e-mail: [email protected]). ‡ R. Ten-Valls is with the Department of Information and Communications Engineering, Universitat Aut` onoma de Barcelona, 08193-Bellaterra, Spain (e-mail: [email protected]). § Manuscript received Month day, year; revised Month day, year. ¶ This work has been partially supported by the Spanish MINECO grants TIN201340524-P and MTM2015-69138-REDT, and by the Catalan AGAUR grant 2014SGR-691.

1

two commutative indeterminants ([17], [19]) or an arbitrary number of commutative indeterminatns ([5], [6]). In all the cited cases with more than one indeterminant, all of them have order 2. A generalization was given in [3], with an arbitrary number of indeterminants whose order is a prime number. In the case the finite field is not F2 , then there are codes over some algebraic extensions of F4 , F22m , or F2m with one indeterminant of order 2 ([14], [11] and [12], resp.) or in general over Fq with one indeterminant of order s ≥ 0 ([16]). Finally, the self-duality of codes over finite algebraic extensions of some of the previous finite fields has also been studied in [4], [7], [9], [20] and [18]. These references are only a small portion of the papers that have been written about self-dual codes over this type of ring. In this paper, we shall make a unified approach to study these families of rings by making a generalization of the family of rings that we first studied in [3]. In that paper, this family of rings was first defined using only F2 rather than Fq as is done in this paper as a finite field. However, self-dual codes were not studied, but rather cyclic codes over these rings was the focus. The rings in this paper are, therefore, a large family of rings which contains the various families of rings described in the references as subfamilies. In this way, we make a unified approach to the study self-dual codes over these rings; we determine a linear Gray map from this family of rings to codes over finite fields and use it to relate these self-dual codes over these rings to codes over a finite field. We define a code C of length n over a ring R as any subset of Rn . If the code is also a submodule then we say that the code is linear. Let π act on the elements of Rn by π(c0 , c1 , . . . , cn−1 ) = (cn−1 , c0 , c1 , . . . , cn−2 ). Then a code C is said to be cyclic if π(C) = C. If π s (C) = C then the code is said to be quasi-cyclic of index s. A code is self-dual when it is equal to its own orthogonal under the usual Euclidean inner-product. The family of rings was first defined in [3] so that the image under the Gray map of a cyclic code was a quasi-cyclic code of arbitrary index, thus giving an algebraic description of a large family of binary quasi-cyclic codes. Here we change the focus of the ring to self-dual codes. In this paper, we determine the parameters for when self-dual codes exist, both free and non-free, and give various constructions for self-dual codes over these rings.

2

The Family of Rings Rq,∆

In this section, we shall describe a family of rings which is a generalization of the family of rings described in [3]. Let q = pk ≥ 2, p prime number and k ≥ 1, let p1 , p2 , . . . , pt be prime numbers with t ≥ 0 and pi 6= pj if i 6= j, and let ∆ = pk11 pk22 · · · pkt t , for some ki ≥ 1, i = 1, . . . , t. Define the following ring Rq,∆ = Fq [up1 ,1 , . . . , up1 ,k1 , up2 ,1 . . . , up2 ,k2 , . . . , upt ,kt ]/huppii ,j = 0i, 2

where the indeterminants {upi ,j }(1≤i≤t,1≤j≤ki ) commute. Let i ∈ {1, · · · , t}, j ∈ {1, · · · , ki }. Consider the set of exponents Ji = {0, 1, . . . , pi − 1} for the indeterminant upi ,j . For αi ∈ Jiki denote ,ki uαpii,1,1 · · · uαpii,k by uαi i , and for a monomial uα1 1 · · · uαt t in Rq,∆ we write uα , i where α = (α1 , . . . , αt ) ∈ J1k1 × · · · × Jtkt . Let J = J1k1 × · · · × Jtkt . Then any element c in Rq,∆ can be written as X X ,kt ,k1 , (1) · · · uαptt,1,1 · · · uαptt,k c= cα uα = cα uαp11,1,1 · · · uαp11,k t 1 α∈J

α∈J

with cα ∈ Fq . It is easy to check that the ring Rq,∆ is a commutative ring k1 k2

kt

with |Rq,∆ | = q p1 p2 ···pt = q ∆ . The following lemma characterizes the subrings of the ring Rq,∆ . Lemma 2.1. The ring Rq,∆0 is a subring of Rq,∆ if ∆0 divides ∆. Proof. Follows easily from the fact that the set of indeterminants of Rq,∆0 is a subset of the set of indeterminants of Rq,∆ and the addition and multiplication is the same. Let ∆0 divide ∆, we define the projection Π∆,∆0 : Rq,∆ → Rq,∆0 ,

(2)

by Π∆,∆0 (ujpk ,i ) = 0 if pik does not divide ∆0 . It is easy to see that this map is a ring homomorphism. Example 1. For Rq,12 the indeterminants are u2,1 , u2,2 , u3,1 , u23,1 . Then Π12,6 sends u2,2 to 0 and acts as the identity on the other indeterminants. The ring Rq,∆ is Frobenius. In order to prove that, we will determine its Jacobson radical and its socle. Recall that, for a ring R, the Jacobson radical can be defined as the intersection of all its maximal right ideals, and the socle is the sum of all its minimal one sided ideals. For a complete description of codes over Frobenius rings, see [2]. First, consider the ideal m = hupi ,j i(1≤i≤t,1≤j≤k P i ) , and define the map µ : Rq,∆ → Fq , as µ(c) = c mod m. If c = α∈J cα uα , then µ(c) = c0 , where 0 is the all zero vector. We have that ker(µ) = m and we can write every element in Rq,∆ as Rq,∆ = {a + m | a ∈ Fq , m ∈ m}. Note that |R | |m| = q,∆ q . Since Rq,∆ is a commutative local ring, we have that its Jacobson radical is the unique maximal ideal which is Rad(Rq,∆ ) = hupi ,j i(1≤i≤t,1≤j≤ki ) = m, and the socle of the ring Rq,∆ is the unique minimal ideal of the ring p1 −1 pt −1 pt −1 Soc(Rq,∆ ) = {λupp11 −1 ,1 · · · up1 ,k1 · · · upt ,1 · · · upt ,kt | λ ∈ Fq }(1≤i≤t,1≤j≤ki ) .

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Theorem 2.2. The local ring Rq,∆ is a Frobenius ring. Proof. With the definition of Rad(Rq,∆ ) and Soc(Rq,∆ ), we have that Rq,∆ /Rad(Rq,∆ ) = Rq,∆ /m ∼ = Soc(Rq,∆ ). = Fq ∼ Therefore, Rq,∆ is a Frobenius ring. Since m = Rad(Rq,∆ ), and the Jacobson radical is necessarily nilpotent, we know that there exists e ≥ 0 such that me = 0. The following theorem gives the value of the nilpotency index e. Theorem 2.3. Let ∆ = pk11 · · · pkt t . Then the index of nilpotency of the Jacobson radical of Rq,∆ is e = max{p1 , . . . , pt }. Proof. The index of nilpotency of a monomial is at most the mini {pi } where upi ,j divides the monomial, for some j ∈ {1, . . . , ki }. The nilpotency of the monomial upi ,j is pi . Then the result follows. It is well known that for local rings, the Jacobson radical contains all non-units. As a result, the following lemma determines the units of the ring Rq,∆ . Lemma 2.4. An element r ∈ Rq,∆ is a unit if and only if µ(r) 6= 0. Proof. Let r ∈ Rq,∆ . We have that r = a + m with a ∈ Fq and m ∈ m. Hence r is a unit if and only if a is a unit in Fq ; that is, µ(r) 6= 0.

2.1

Linear, orthogonal and Cyclic Codes over Rq,∆

A linear code, C, of length n over Rq,∆ is a submodule of Rq,∆ n . The following theorim gives properties on the cardinalities of such codes. Theorem 2.5. Any code over the ring Rq,∆ has cardinality q k for some integer k. Proof. The ring Rq,∆ in an Fq -algebra. Therefore any code must have cardinality q k for some integer k. Since Rq,∆ is a Frobenius ring, we can construct generator matrices for the code and its orthogonal, and also detemine certain properties that relates the code and its orthogonal. We will start by defining when some vectors are independent, modular independent and a basis of a code. All these definitions can be found in [2]. Let v1 , . . . , vs be vectors in Rq,∆ n and m the maximal ideal of Rq,∆ defined before. Then we say that v1 , . . . , vs are modular independent if P λ v j j j P= 0 impies that λj ∈ m, for j ∈ {1, . . . , s}, and they are independent if j λj vj = 0 impies that λj = 0, for j ∈ {1, . . . , s}. 4

Let C be a linear code over Rq,∆ . The codewords c1 , . . . , cs are a basis of C if they are modular independent and generate C. The matrix whose rows are a basis of a code C is a generator matrix of C. It is well known that a minimal generating set exists for any code, namely when the rows are modular indpendent, see [2] for example. Note that when a ring is not local we also need independence to have a basis but for local rings all we need is modular independent. Let C be a linear code over Rq,∆ . The code C is said to be free if it has a generator matrix of the form (Ids | A) where Ids is the s by s identity matrix. We define the usual inner-product, namely X [w, v] = wi vi where w, v ∈ Rq,∆ n . Then, the orthogonal of a code C is C ⊥ = {w ∈ Rq,∆ n | [w, v] = 0, ∀v ∈ C}. Since Rq,∆ is Frobenious, we have that |C||C ⊥ | = |Rq,∆ n | = q ∆n for all linear codes C.

2.2

Ideals of Rq,∆

In this subsection, we shall study some ideals in the ring Rq,∆ . b∆ be the subset of A∆ Let A∆ be the set of all monomials of Rq,∆ and A of all monomials with one indeterminant. Clearly |A∆ | = pk11 pk22 · · · pkt t = ∆ b∆ | = pk1 +pk2 +· · ·+pkt . View each element a ∈ A∆ , a = uα for some and |A t 1 2 α b α ∈ J, as the subset {upii,j |α ,j i,j 6= 0}(1≤i≤t,1≤j≤ki ) ⊆ A∆ . We will denote by b∆ . For example, the element a = u2,1 u2 u3 b a the corresponding subset of A 3,4 5,2 is identified with the set b a = {u2,1 , u23,4 , u35,2 }. Note that 1 ∈ A∆ and b 1 = ∅, the empty set. Consider the vector of exponents α = (α1,1 , . . . , α1,k1 , . . . , αt,1 , . . . , αt,kt ) ∈ J and denote by α ¯ the vector (p1 − α1,1 , · · · , p1 − α1,k1 , · · · , pt − αt,kt ), note ¯ = α. that α Let Iα be the ideal Iα = huα i, for α ∈ J. Note that I0 = h1i = Rq,∆ . We also define I(p1 ,··· ,p1 ,p2 ··· ,pt ,··· ,pt ) = {0}. Now we define the ideal cα i = huαp i,j Ibα = hu | αi,j 6= 0i(1≤i≤t,1≤j≤ki ) . i ,j Whith these definitions, it was proved in [3] that Ibα⊥ = Iα¯ , with Ib0⊥ = Rq,∆ ⊥ = {0} = I(p1 ,··· ,p1 ,p2 ··· ,pt ,··· ,pt ) = I0¯ .

5

(3)

Proposition 2.6. The number of elements of Iα is q Q of elements of Ibα is q ∆− i∈α i .

Q

i∈α ¯

i

and the number

Proof. By considering the proof of [3, Proposition 2.3], |Iα | = q Q ∆− i∈α i b Equation 3, clearly we have that |Iα | = q .

2.3

Q

i∈α ¯

i

. By

Cyclic codes over Rq,∆

P Let consider the polynomial ring Rq,∆ [x]. A monicP polynomial f = ai xi ∈ Rq,∆ [x] is a basic irreducible polynomial if µ(f ) = µ(ai )xi is an irreducible polynomial in Fq [x], where µ(a) is the reduction modulo m. Any irreducible polynomial in Fq [x] is a basic irreducible polynomial in Rq,∆ [x] due to the fact that Fq is a subring of Rq,∆ . If gcd(n, q) = 1, then xn − 1 factors into a product of r pairwise coprime basic irreducible polynomials over Rq,∆ , xn − 1 = f1 f2 . . . fr , that are uniquely determined up to a rearrangement. Consider the bijective map between Rq,∆ n and Rq,∆ [x]/hxn − 1i given by (v0 , . . . , vn−1 ) → v0 +v1 x+· · ·+vn−1 xn−1 . By using this operation, as in any commutative ring we can identify cyclic codes with ideals in Rq,∆ [x]/hxn −1i. Notice that if gcd(q, n) = 1 then the factorization of xn − 1 is unique. From the cannonical decomposition of rings we obtain the following theorem. Theorem 2.7. Let n be an integer coprime with q. Let xn − 1 = f1 f2 . . . fr be the factorization of xn − 1 into basic irreducible polynomials over Rq,∆ . Then, the ideals in Rq,∆ [x]/hxn − 1i can be written as I ∼ = I1 ⊕ I2 ⊕ · · · ⊕ Ir where Ii is an ideal of the ring Rq,∆ [x]/hfi i, for i = 1, . . . , r. Let f be a basic monic irreducible polynomial over Rq,∆ . Using an analogous argument described in [3], it is easy to show that there is a one to one correspondence between ideals of Rq,∆ [x]/hf i and ideals of Rq,∆ . This result shows the importance of understanding the ideal structure of Rq,∆ . Corollary 2.8. Let n be an integer coprime with q. Let xn − 1 = f1 f2 . . . fr be the factorization of xn − 1 into basic irreducible polynomials over Rq,∆ and let I∆ be the number of ideals in Rq,∆ . Then, the number of linear cyclic codes of length n over Rq,∆ is (I∆ )r .

3

Gray map to the Hamming Space

We will consider the elements in Rq,∆ as q-ary vectors of ∆ coordinates. First, we order the elements of A∆ lexicographically and use this ordering ∆ to label the coordinate positions of F∆ q . Define the Gray map Ψ : A∆ → Fq as follows:  1 if bb ⊆ {b a ∪ 1}, Ψ(a)b = 0 otherwise, where Ψ(a)b indicates the coordinate of Ψ(a) corresponding to the position of the element b ∈ A∆ with the defined ordering. We have that Ψ(a)b is 1 if 6

each indeterminant upi ,j in the monomial b with non-zero exponent is also in the monomial a with the same exponent; that is, bb is a subset of b a. In order to consider all the subsets of b a, we also add the empty subset that is given when b = 1; that is we compare bb to b a ∪ 1. Then extend Ψ linearly for all elements of Rq,∆ . Example 2. Let ∆ = 6 = 2 · 3, then we have the following ordering of the monomials [1, u2,1 , u2,1 u3,1 , u2,1 u23,1 , u3,1 , u23,1 ]. Then, for example, for λ, µ ∈ Fq , Ψ(λu23,1 + µu2,1 u23,1 ) = λΨ(u23,1 ) + µΨ(u2,1 u23,1 ) = (λ + µ, µ, 0, µ, 0, λ + µ). As it was proved in [3], for a ∈ A∆ such that a 6= 1, we have that wtH (Ψ(a)) is even. Let C be a linear code of length n over Rq,∆ . The code Ψ(C) is a q-ary code of length ∆n. In [5], it was proved that for q = 2 and ∆ = 2k , the orthogonality is preserved over the map Ψ. Later, it was proved that the orthogonality is not preserved when ∆ is not a power of two ([3]). Therefore, the orthogonality is not preserved in general for codes over Rq,∆ . By using Ψ, we can consider the image of linear cyclic and quasi-cyclic codes over Rq,∆ as q-ary quasi-cyclic codes. Proposition 3.1. Let C be a linear code over Rq,∆ of length n. • If C is cyclic, then Ψ(C) is a linear q-ary quasi-cyclic code of length ∆n and index ∆. • If C is quasi-cyclic of index k, then Ψ(C) is a linear q-ary quasi-cyclic code of length ∆n and index ∆k. Proof. We can consider the same arguments as in [3], but considering the map Ψ : Rq,∆ −→ F∆ q , and therefore the image is a q-ary code. Let ∆0 divide ∆ and let A∆,∆0 be the subset of monomials of A∆ such b∆0 . In the same way that in their product does not appear any element of A ∆ 0

∆ that we have defined Ψ, we define Ψ∆,∆0 : A∆,∆0 → Rq,∆ 0 to consider the ∆ elements of Rq,∆ as vectors of ∆0 coordinates over Rq,∆0 . First, order the elements of A∆,∆0 , then define Ψ∆,∆0 analogous to Ψ, and finally extend Ψ∆,∆0 linearly for all elements of Rq,∆ .

Example 3. Let ∆ = 24 = 23 3 and ∆0 = 4 = 22 , then consider the following ordering of the monomials of A∆,∆0 : [1, u2,3 , u2,3 u3,1 , u2,3 u23,1 , u3,1 , u23,1 ]. As an example, let λ, µ, η ∈ Fq then Ψ∆,∆0 (λu23,1 + µu2,1 u23,1 + ηu2,2 u2,3 u3,1 ) = λΨ∆,∆0 (u23,1 ) + µu2,1 Ψ∆,∆0 (u23,1 ) + ηu2,2 Ψ∆,∆0 (u2,3 u3,1 ) = (λ + µu2,1 + ηu2,2 , ηu2,2 , ηu2,2 , 0, ηu2,2 , λ + µu2,1 ). At this point, it is natural to ask if orthogonality is preserved over the map Ψ∆,∆0 . 7

Example 4. Let ∆ = 32 and ∆0 = 3, then consider the following ordering of the monomials of A∆,∆0 : [1, u3,2 , u23,2 ]. Clearly, [u3,2 , u3,1 u23,2 ] = 0 ∈ Rq,∆ but we have that [Ψ∆,∆0 (u3,2 ), Ψ∆,∆0 (u3,1 u23,2 )] = [(1, 1, 0), (1, 0, u3,1 )] 6= 0 ∈ Rq,∆0 . The previous example shows that, in general, orthogonality will not be preserved. The following lemma presents the cases when it is preserved. Lemma 3.2. Let ∆ = 2k1 ∆0 , such that k1 ≥ 1 and 2 6 |∆0 , and let q = 2. Let v and w be vectors of Rq,∆ n , then [v, w] = 0 =⇒ [Ψ∆,∆0 (v), Ψ∆,∆0 (w)] = 0. P Proof. By (1), note that we can rewrite any element c = α cα uα of Rq,∆ P P 1 ,k1 , where c0α1 ∈ Rq,∆0 . Let as c = α1 c0α1 uα2 1 = α1 c0α1 uα2,11 ,1 · · · uα2,k 1 X X 0 0 v=( vα,1 uα2 1 , · · · , vα,n uα2 1 ) α1

α1

and let X X 0 0 w=( wα,1 uα2 1 , · · · , wα,n uα2 1 ), α1

α1

0 , w0 v, w ∈ Rq,∆ n , where vα,i α,i ∈ Rq,∆0 , for i ∈ {1, . . . , n}. Then by an analogous proof of [5, Lemma 6.2], the result is straightforward.

4

MacWilliams Relations

In this section, we shall establish MacWilliams relations for codes over Rq,∆ [ [ by determining a generating character for R q,∆ , where Rq,∆ is the character module of Rq,∆ . In order to establish MacWilliams relations, we begin by defining the complete weight enumerator. Let C be a code over Rq,∆ of length n. The complete weight enumerator of C is definded in the usual way, namely: cweC (X) =

n XY

xci .

(4)

c∈C i=1

We shall denote by X the variables (xci ), where the ci are the elements of Rq,∆ in some order. The Hamming weight enumerator is defined as X WC (x, y) = xn−wtH (c) y wtH (c) , (5) c∈C

8

where wtH (c) denotes the Hamming weight of the vector c, that is the number of non-zero coordinates of c. Let χFq : Fq → C∗ be defined as follows. Let a0 +a1 x+a2 x2 +· · ·+ae xe−1 be an element of Fq where q = pe . Then P

χFq (a0 + a1 x + a2 x2 + · · · + ae xe−1 ) = ξp

ai

,

where ξp is a p-th root of unity. It is well known that this is a generating character for the field Fq . We can know define the character for the ring Rq,∆ . Let χ : Rq,∆ −→ C? as X Y χ( cα uα ) = χFq (cα ). α∈J

α∈J

It is easy to see that this map is additive and hence a character of the ring Rq,∆ . In order to prove that this character is a generating character, it is sufficient to show that ker(χ) contains no non-trivial ideals of Rq,∆ . Since Rq,∆ is a local ring, every non-trivial ideal contains Soc(Rq,∆ ). Therefore all we need to show is that the map χ is non-trivial on the ideal Soc(R). Recall that the socle of the ring is p1 −1 pt −1 pt −1 Soc(Rq,∆ ) = {λupp11 −1 ,1 · · · up1 ,k1 · · · upt ,1 · · · upt ,kt | λ ∈ Fq }.

Therefore χ evaluated on this ideal is χFq evaluated on Fq which has as its image {1, ξp , ξp2 , . . . , ξpp−1 }. Hence χ is non-trivial on this ideal and is [ therefore the generating character of R q,∆ . We can now explicitly describe the MacWilliams relations for this ring. For any matrix M , using the elements of Rq,∆ as coordinates for the rows and columns of the matrix R, define T to be the |Rq,∆ | × |Rq,∆ | matrix given by Ta,b = χ(ab), for a, b  ∈ Rq,∆ . Let T · X be (T X t )t . This matrix  ∆ 1 q −1 then collapses to to produce the MacWilliams relations for 1 −1 the Hamming weight enumerator. Then we have the following MacWilliams relations. Theorem 4.1. Let C be a linear code over Rq,∆ . Then cweC ⊥ (X) = and WC (x, y) =

1 cweC (T · X), |C|

1 WC (x + (q ∆ − 1)y, x − y). |C|

9

5

Self-dual codes

Self-dual codes are an important family of codes over rings. A self-dual code is defined as a code that satisfies C = C ⊥ . These codes have numerous applications to desings, lattices and to self-dual codes over fields via Gray maps. Also, a great number of the best codes known are self-dual. It follows then that if C is a self-dual code over Rq,∆ the complete weight enumerator and the Hamming weight enumerator are held invariant by the actions of their respective MacWilliams relations. A code is formally self-dual with respect to a weight enumerator if the weight enumerator of C and the weight enumerator of C ⊥ are equal. Of course, a code can be formally self-dual with respect to one weight enumerator and not to another. We begin with a construction of self-dual codes of length one for any q and for ∆ even. Proposition 5.1. Let ∆ = 2k1 pk22 · · · pkt t . Then C = hu2,i i is a self-dual code of length one over Rq,∆ , for all i ∈ {1, · · · , k1 }. Proof. First note that u2,i u2,i = 0 so C ⊆ C ⊥ . The number of monomials ∆ ∆ ∆ in A∆ containing u2,i is |A2∆ | , so |C| = q 2 and then |C ⊥ | = q∆ = q 2 . q

2

Therefore C = C ⊥ . Notice that the previous proof works for any q, but we need that 2 divides ∆. The codes obtained in Proposition 5.1 are non-free self-dual codes. Example 5. Consider the ring R3,2 which has cardinality 9. Then hu2,1 i = {0, u2,1 , 2u2,1 } is a self-dual code of over F3 . The image of this code under the Gray map is {(0, 0), (1, 1), (2, 2)} which is not self-dual. Its orthogonal code is {(0, 0), (1, 2), (2, 1)}. Hence the image is formally self-dual with respect to the Hamming weight enumerator but not with respect to the complete weight enumerator. Lemma 5.2. If C is a self-dual code of odd length over Rq,∆ , then 2 divides ∆. Proof. Let C be a self-dual code of length n. Then |C||C ⊥ | = q n∆ , which n∆ gives that |C|2 = q n∆ and |C| = q 2 . By Theorem 2.5, n∆ 2 is an integer, therefore 2 divides ∆ since n is odd. Theorem 5.3. There exists a self-dual code of length one over Rq,∆ if and only if 2 divides ∆. Proof. If 2 divides ∆, by Proposition 5.1, the code C = hu2,1 i is a self-dual code of length one. By Lemma 5.2, the result follows.

10

We shall now describe a technique for producing more self-dual codes of length one when q = 2. and let J ⊆ {1, · · · , k1 }. Then the Proposition 5.4. Let ∆ = 2k1 pk22 · · · pkt t P length one code over R2,∆ generated by h i∈J u2,i i is a self-dual code. P Proof. Let C = h i∈J u2,i i. Since X X ( u2,i )2 = u22,i + 2 i∈J

i∈J

X

u2,i u2,j = 0,

i,j∈J ,i
we have that C ⊆ C ⊥ . If |J | = 1 then we have proven it in Proposition 5.1. If |J | > 1, then ⊥ the code P C is an ideal of R2,∆ P . Moreover |R2,∆ /C| = |R2,∆ |/|C| = |C |. If i∈J uP /C. Let a be least in J . 2,i ∈ C, then i∈J u2,i = 0 in R2,∆P Here, u2,a + i∈J −{a} u2,i = 0 and write u2,a = i∈J −{a} u2,i . Therefore, ∆

in R2,∆ /C we lose one indeterminant of order 2, u2,a , and |R2,∆ /C| = 2 2 . ∆ Hence |C ⊥ | = 2 2 and C is self-dual. Theorem 5.5. Let ∆ = 2k1 pk22 · · · pkt t and q = 2. Let a ∈ I = {1, · · · , k1 }. let S1 , · · · , Sl be a collection of subsets of I − {a} and b be a unitof R2,∆ . P let Q l Then the length one code over R2,∆ generated by h i=1 j∈Si u2,j +u2,a bi is a self-dual code. P Q  l Proof. Let C = h u i=1 j∈Si 2,j + u2,a bi. Since l Y l Y l Y X X X (( u2,j )+u2,a b)2 = ( u2,j )2 +(u2,a b)2 +2( u2,j )(u2,a b) = 0, i=1 j∈Si

i=1 j∈Si

i=1 j∈Si

we have that C ⊆ C ⊥ . Without loss of generality, we can suppose that a = k1 . Then we have P Q  l that i=1 j∈Si u2,j +u2,k1 b = m+u2,k1 b, where m is a non-unit satisfying m2 = 0 and b is a unit in R2, ∆ . By multiplying by the inverse of b we assume 2 that the code is generated by m0 + u2,k1 . 2 . We have that Ψ Consider the Gray map Ψ∆,∆0 : R2,∆ → R2, ∆,∆0 (m + ∆ 2

bu2,k1 ) = (m + b, b). Since the map is linear, the image is generated by ∆ (m0 + 1, 1) which is a free code and has cardinality 2 2 . ∆ Therefore |C| = |C ⊥ | = 2 2 and the code C is self-dual. Notice that the codes given in Propositions 5.1 and 5.4, are included in the ones obtained in Theorem 5.5. We shall now show that in fact, for q = 2, these are the only self-dual codes of length one. 11

Theorem 5.6. Let ∆ = 2k1 pk22 · · · pkt t and q = 2. The only length one self-dual codes over R2,∆ are the ones described in Theorem 5.5. Proof. Assume we have a non-unit generator g of a self-dual code not of this form, i.e. each summand is the product of at least 2 elements of the form u2,i . Consider the monomials that annihilates each summand in g. Note that these monomials are the product of u2,i , containing at least one element of each summand in g. Since each monomial satisfying this is finite, Q there must be one where the number of u2,i is a minimum. Define uA = i∈J u2,i for some J ⊂ {1, . . . , k1 } these minimal monomial. We shall show that uA is not in C even though it is in C ⊥ . 0 If uA = u2,i , for some i ∈ {1, . . . , k1 }, then necessarely, Q g = u2,i g , for 0 g ∈ m. In that case, uA 6∈ hgi = C. Then assume uA = i∈J u2,i for some J ⊂ {1, . . . , k1 } with |J | > 1. Order the elements of uA as u2,i1 , u2,i2 , · · · , u2,ij . If uA ∈ C, then rg = uA . For this to be true r must have u2,ij as a divisor. Then by writing r = r0 u2,ij we proceed by induction showing r = uA . But then uA g 6= uA . Theorem 5.7. Let ∆ = pe ∆0 , for e > 1. Let C be a self-dual code generated by v1 , . . . , vk over Rq,∆0 , then v1 , · · · vk generates a self-dual code over Rq,∆ . Proof. Let e1 the maximum exponent of p in ∆ and, e2 = e − e1 . Then we need to add to Rq,∆0 the indeterminants up,e2 +1 , up,e2 +2 , · · · , up,e1 to obtain e −e Rq,∆ . Then |Rq,∆ | = q p 1 2 |Rq,∆0 |. Let C1 be the code over Rq,∆0 generated by v1 , · · · , vs and let C2 be the code over Rq,∆ generated by v1 , · · · , vs . We have p−1 e1 X X

 C2 = C1 , ujp,i C1 . i=e2 +1 j=0 ∆0

∆

Then |C2 | = q 2 q e1 −e2 = q 2 . Self-orthogonality follows by induction noting that for v1 , v2 , w1 , w2 ∈ C1 we have that [v1 + ujp,i w1 , v2 + ujp,i w2 ] = [v1 , v2 ] + ujp,i [v1 , w2 ] + ujp,i [v2 , w1 ] + (ujp,i )2 [w1 , w2 ] = 0. Therefore, C2 is self-dual. Corollary 5.8. Let ∆0 divide ∆ and let C be a self-dual code generated by v1 , . . . , vk over Rq,∆0 , then v1 , · · · vk generates a self-dual code over Rq,∆ . As a consequence of Lemma 3.2, and the fact that in the Frobenius ring Rq,∆ we have |C||C ⊥ | = |Rq,∆ |n , we obtain the following result. Theorem 5.9. Let ∆ = 2k1 ∆0 , such that k1 ≥ 1 and 2 6 |∆0 , and let q = 2. Let C be a linear code of length n over Rq,∆ . Then Ψ∆,∆0 (C ⊥ ) = (Ψ∆,∆0 (C))⊥ . 12

Corollary 5.10. Let ∆ = 2k1 ∆0 , such that k1 ≥ 1 and 2 6 |∆0 , and let q = 2. Let C be a self-dual linear code of length n over Rq,∆ . Then Ψ∆,∆0 (C) is a self-dual code over Rq,∆0 of length 2k1 n. The following example shows that the previous results are not true for q= 6 2. Example 6. Consider the ring R2,2∆0 where ∆0 is an odd integer. Then the code C = hu2,1 i generates a self-dual code over R2,2∆0 with cardinality 0 q ∆ . By Theorem 5.9, the code Ψ∆,∆0 (C) is a self-dual code of length 2 over 0 Rq,∆0 . In fact, Ψ∆,∆0 (C) = h(1, 1)i, which obviously has cardinality q ∆ and is self-dual. This example, also shows why this fails when q 6= 2 since the vector (1, 1) is not self-orthogonal when q 6= 2. Consider Π∆,∆0 the projection defined in (2). If we apply this projection to a self-dual code, we obtain the following result. Theorem 5.11. Let C be a self-dual code over Rq,∆ with ∆0 |∆. Then Π∆,∆0 (C) is a self-orthogonal code. Proof. Since Π∆,∆0 is a ring homomorphism it follows that if [v, w] = 0 then [Π∆,∆0 (v), Π∆,∆0 (w)] = 0. Note that the projections of self-dual codes over Rq,∆ may not be selfdual over Rq,∆ . Example 7. The self-dual code C = hu2,3 i over R2,24 projects to the zero ideal in R2,6 . Therefore, Π24,6 (C) is self-orthogonal but not self-dual. Self-orthogonal are abundant since an element of the form uapi ,j , for a ≥ pi +1 2 , is self-orthogonal. We say that a code C is maximal self-orthogonal if C ( C ⊥ and there is no self-orthogonal code D such that C ( D. Theorem 5.12. Let C be a self-orthogonal code of length n in Rq,∆ with 2|∆. Then hC, u2,i C ⊥ i is a self-dual code of length n. Proof. Any two elements of hC, u2,i C ⊥ i are of the form v1 + v2,i w1 and v2 +u2,i w2 . Then [v1 +v2,i w1 , v2 +u2,i w2 ] = [v1 , v2 ]+u2,i [v1 , w2 ]+u2,i [v2 , w1 ]+ (u2,i )2 [w1 , w2 ] = 0, since [v1 , v2 ] = [vi , wj ] = 0 and u22,i = 0. Therefore the code is self-orthogonal. Assume |C| = q s then |C ⊥ | = (q ∆ )n /q s = q n∆−s . n∆−2s Then |C ⊥ /C| = q n∆−2s and |u2,i (C ⊥ /C)| = q 2 . Then |hC, u2,i C ⊥ i| = n∆−2s n∆ |C||u2,i (C ⊥ /C)| = q s q 2 = q 2 and therefore the code is self-dual. Corollary 5.13. Maximal self-orthogonal codes over Rq,∆ exist if and only if 2 does not divide ∆.

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pi +1 2 Proof. Assume 2 - ∆. Then hupi ,j i is a maximal self-orthogonal code of length 1. If 2|∆ then, by Theorem 5.12, any self-orthogonal code can be extended to a self-dual code.

Example 8. Let ∆ = 30 then C = h(u23,1 , u35,1 )i is a self-orthogonal code over Rq,30 . It follows that C ⊥ = h(u3,1 , 0), (0, u5,1 )i. Then hC, u2,1 C ⊥ i is a self-dual code. The standard proof gives the following lemma. Lemma 5.14. If C and D are self-dual codes of length n and m, respectively, over Rq,∆ . Then C × D is a self-dual code of length n + m over Rq,∆ . The product of free-codes is free, but if one of them is a non-free then the product is non-free. Corollary 5.15. If 2|∆, then there are self-dual codes over Rq,∆ of all lengths for all q. Proof. By Theorem 5.3, if 2|∆, then there exist self-dual codes of length 1 for all q. Therefore, by Lemma 5.14 we have self-dual codes of all lengths for any value of q. Theorem 5.16. If q = pe with p = 2 or p ≡ 1 mod 4, then there are self-dual codes over Rq,∆ of all even length. Proof. If p = 2, then for β ∈ F2e , the code of length 2 generated by (β, β) is self-dual over F2e and over R2e ,∆ , for any ∆. If p ≡ 1 mod 4 then there exist α ∈ Fq such that α2 = −1. So, there is a self-dual code over Fq of length 2 generated by (1, α), and hence over Rq,∆ . Then, by Lemma 5.14 we have the result. Theorem 5.17. If q = pe with p ≡ 3 mod 4, then there are self-dual codes over Rq,∆ for all length n such that n ≡ 0 mod 4. Proof. Let q = pe and p ≡ 3 mod 4. There exist α, β ∈ Fq such that  1 0 α β 2 2 α + β = −1. Then the code of length 4 generated by is 0 1 −β α self-dual over Rq,∆ . Then, by Lemma 5.14 we have the result. Notice that the code in the previous theorem is a free code. In the next example we shall construct a non-free code of length 4 over rings of characteristic 3 (mod 4).

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Example 9. Take q = 3e , then the code over Rq,3 with generator matrix   u3,1 u3,1 u3,1 0  0 u3,1 −u3,1 u3,1     0 0 u23,1 0  0 0 0 u23,1 is a self-orthogonal code. The cardinality of the ambient space is q 12 and the cardinality of the code is q 6 . Therefore it is a non-free self-dual code of length 4. From Corollary 5.15, we may construct non-free self-dual codes over Rq,∆ of all lengths and all values of q as long as 2 divides ∆. If p = 2 or p ≡ 1 mod 4, then we can construct free self-dual codes over Rq,∆ for all even lengths by Theorem 5.16. Finally, if p ≡ 3 mod 4, by Theorem 5.17, we can construct free self-dual codes over Rq,∆ for all length n ≡ 0 mod 4.

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