Codes Over a Family of Local Frobenius Rings, Gray Maps and Self-Dual Codes Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA Eseng¨ ul Salt¨ urk ∗ Department of Mathematics University of Scranton Scranton, PA 18510 USA September 10, 2016

Abstract k

We define a class of finite Frobenius rings of order 22 , describe their generating characters, and study codes over these rings. We define two conjugate weight preserving Gray maps to the binary space and study the images of linear codes under these maps. This structure couches existing Gray maps, which are a foundational idea in codes over rings, in a unified structure and produces new infinite classes of rings with a Gray map. The existence of self-dual and formally self-dual codes is determined and the binary images of these codes are studied.

Keywords Codes over rings; Gray Maps; Self-Dual Codes MSC: 94B05, 11T71, 13M05 ∗

Eseng¨ ul Salt¨ urk would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their support while writing this paper.

1

1

Introduction

The foundational idea for codes over rings was the existence of a Gray map from the ring to the Hamming space. This Gray map was used to describe non-linear binary codes as linear codes over the ring Z4 , see [15] for a complete description of this work. Following this paper, the notion of a Gray map was extended to all rings of order 4 in [3] and [5]. Some of these maps were extended to infinite families of rings in [14], which built on the definitions in [4] and [23]. We shall take as a definition of Gray map, a map from Rn to the Hamming space where weight is preserved. Following early work on codes over rings, it was shown that the largest class of rings for which it is natural to define codes is the class of Frobenius rings. In particular, it was shown that both MacWilliams theorems hold for codes over these rings. Specifically, that every Hamming isometry can be extended to a monomial transformation and that the weight distribution of the orthogonal is determined by the weight distribution of the code. For commutative Frobenius rings, one can use the Chinese Remainder Theorem to decompose the rings into the product of local Frobenius rings, namely Frobenius rings with a unique maximal ideal. As a generalization of the rings of order 4, the rings of order 16 were studied. In [8], the authors constructed a standard form of the generator matrix for codes over the local rings of order 16 and, in [9], a Gray map, which need not be linear, was described from the local rings of order 16 to the binary Hamming space. See [6] and [7] for applications of these maps. This Gray map can be used in a manner similar to the existing Gray maps of order 16. In this paper, we can extend the work that was begun in [8] and [9] by describing a class of rings which admit a weight preserving map from the ring to the Hamming space. This definition allows us to consider several infinite families of rings at once and understand codes in the binary space as images of linear codes over these families of rings. This allows us to place existing Gray maps in a unified setting. Using this setting we can study numerous rings which fall into this class and study the images of linear codes over these rings via the Gray map. Our structure not only covers existing Gray map structures but extends the ideas to numerous other rings as well. We then study formally self-dual codes with respect to the weight enumerator for the Lee weight, where the Lee weight is the Hamming weight of its binary image.

2 2.1

Definitions and Notations Rings and Codes

We begin by recalling some standard definitions from coding theory and ring theory. We assume that all rings have a multiplicative identity and that all rings are commutative. 2

The Jacobson radical of a ring consists of all annihilators of simple left R-modules. It can be characterized as the intersection of all maximal ideals of the ring. The socle of the ring is the sum of all the minimal one sided ideals of the ring. For a ring R, we denote the Jacobson radical as J(R) and the socle as Soc(R). For a module M over a ring R define c = Hom (M, C∗ ). We note C∗ can be replaced with Q/Z if we want to use an additive M Z model rather than a multiplicative one. If R is a finite ring then the following statements b ∼ are equivalent: (1) R is a Frobenius ring; (2) as a left module, R = R R; (3) as a right ∼ b = RR . In this paper, we shall assume that all rings are Frobenius. A ring is a module R local ring if it has a unique maximal ideal. For a local Frobenius ring, the maximal ideal is necessarily the Jacobson radical and Soc(R) is the dual of the Jacobson radical and is the unique minimal ideal in the ring. Given a ring R we say that a code of length n over R is a subset of Rn . If the code is a submodule we say that the code is linear. If C is a code over a ring R then the weight enumerator with respect to a weight wt is X WC (x, y) = xN −wt(c) y wt(c) , c∈C

where N is the maximum weight for a vector of length n. P We attach to the ambient space the Euclidean inner-product, [v, w] = vi wi . We define an orthogonal with respect to this inner-product, namely C ⊥ = {v | [v, w] = 0, ∀w ∈ C}. The code C ⊥ is linear whether or not C is linear. It follows from the foundational results in [22] that |C||C ⊥ | = |R|n , when C is a linear code over a Frobenius ring. The Hamming weight of a codeword is the number of non-zero entries in the codeword. If C ⊆ C ⊥ we say that the code C is self-orthogonal. If C = C ⊥ we say that the code C is self-dual. If a code C satisfies WC (x, y) = WC ⊥ (x, y) then we say that the code is formally self-dual with respect to that weight enumerator. For binary codes, there is only the Hamming weight, so a binary formally self-dual code is formally self-dual with respect to the Hamming weight enumerator. A binary formally self-dual code is said to be Type II if the Hamming weights of all codewords of the code are congruent to 0 (mod 4). A binary formally self-dual code is said to be Type I if the Hamming weights of all codewords of the code are congruent to 0 (mod 2) but there exists at least one codeword whose Hamming weight is not congruent to 0 (mod 4). A binary formally self-dual code is said to be Type 0 if it contains at least one codeword with odd Hamming weight. Note that if the code is self-dual then it must be either Type I or Type II. However, one can have the code h(1, 0)i which is a Type 0 formally self-dual code of length 2. Often in the literature Type I and Type II refer only to self-dual codes but we will use them for formally self-dual codes which may not be self-dual as well. We say that two codes are equivalent if one is formed from the other by permuting the coordinates. Equivalent codes necessarily have the same weight enumerators for all weights. 3

This is why we use this definition rather than a more general definition of equivalence, because we use it to construct formally self-dual codes by showing that a code is equivalent to its dual. If a code is equivalent to its dual we say that it is isodual.

2.2

A Family of Rings

We shall now describe a family of rings for which we can define two conjugate Gray maps. First we need a notational convention. Let u1 , u2 , . . . , uk be a set of k elements of a ring, Q for k a positive integer. For any subset A ⊆ {1, 2, . . . , k} we denote by uA := i∈A ui with the convention that u∅ = 1. Notice that this does not allow for a ui to be represented more than once in the product. Definition 1. A ring R is an X-ring if it is a finite commutative Frobenius local ring with k , where uA 6= 0 for all maximal ideal m = hu1 , u2 , . . . , uk i such that |R| = 22 , and |m| = |R| 2 A ⊆ {1, 2, . . . , k}. It follows immediately that if R is an X-ring then m = J(R) and |Soc(R)| = 2. We denote the elements of the socle as {0, w}. In this case, we have that Soc(R) = m⊥ . Notice that we are not saying in the definition that the set {u1 , u2 , . . . , uk } is a minimal set of generators for the maximal ideal, but rather simply that they generate the maximal ideal, without their product being 0. The next two lemmas follow easily from the definition. k

Lemma 2.1. Let R be an X-ring of order 22 . Any element of R can be written in the form X cA uA A⊆{1,2,...,k}

with cA ∈ {0, 1}. Moreover, the cA are unique. Proof. We have that uA 6= 0 for all subsets A of {1, 2, . . . , k}. Hence taking k 22 elements.

P

cA uA gives

Note that we are not saying that this representation is a canonical representation for the group structure of the ring. For example, the ring Z4 is an X-ring in that each element can be written as a + b2 where a, b ∈ {0, 1} but the group structure is not Z2 × Z2 . Here u1 = 2 1 and the ring has 22 elements. The next lemma characterizes units in an X-ring. P k Lemma 2.2. Let R be an X-ring of order 22 . The element A⊆{1,2,...,k} cA uA is a unit in R if and only if c∅ = 1. Proof. Let f be the canonical map f : R → R/m ∼ = F2 . Then we have that r ∈ R is a unit if and only if r = 1 + a, a ∈ m. Let A ⊆ {1, 2, . . . , k}, then any element of R can be written as P Q P A⊆{1,2,...,k} cA uA where uA := i∈A ui and cA ∈ {0, 1}. This gives that A⊆{1,2,...,k} cA uA is a unit if and only if c∅ = 1. 4

Example 1. We shall now give examples of rings for various characteristics that are X-rings. • The commutative ring Rk = F2 [u1 , u2 , . . . , uk ]/hui 2 i is a non-chain ring which has characteristic 2 with maximal ideal m = hu1 , u2 , . . . , uk i and Soc(Rk ) = hu1 u2 · · · uk i. This ring has been studied extensively in [14]. It is also possible to describe this ring as a group algebra where the group is the k fold product of the cyclic group of order 2. k

• The ring Z22k is a chain ring which has characteristic 22 with maximal ideal m = k k−1 h2, 4, 16, . . . , 22 i and Soc(Z22k ) = h22 −1 i. Notice that m = h2i, that is, it is a principal ideal. However, the elements that we have generating the maximal ideal are such that no product of them is 0 and that every element in the ring has a unique representation of a sum of these generators with coefficients in the set {0, 1}. • The ring Z2s [x]/hxt i is a ring which has characteristic 2s where s = 2l , l ≥ 0, t = 2k−l l−1 k−l−1 with maximal ideal m = h2, 4, 16, . . . , 22 , x, x2 , x4 , . . . , x2 i and Soc(Z2s [x]/hxt i) = l k−l h22 −1 x2 −1 i. • The ring Z2s [x]/hxt − 2r xm i is a ring which has characteristic 2s where s = 2l , l ≥ 0, l−1 k−l−1 t = 2k−l , m < t, r ≥ 0 with maximal ideal m = h2, 4, 16, . . . , 22 , x, x2 , x4 , . . . , x2 i l −1 2k−l −1 t r m 2 and Soc(Z2s /hx − 2 x i) = h2 x i. We can use this example to get the following existence theorem. k

Theorem 2.3. There exist X-rings of order 22 with characteristic 2s where 1 ≤ s ≤ 2k , s = 2l , l ≥ 0. k

Proof. The ring Z2s [x]/hxt i in Example 1 gives rings of order 22 with characteristic 2s for each such s. The next theorem counts the number of rings in the previous example. k

Proposition 2.4. Let R be an X-ring with size 22 defined by Z2s [x]/hxt i, with m = l−1 k−l−1 l k−l h2, 4, 16, . . . , 22 , x, x2 , x4 , . . . , x2 i and Soc(R) = h22 −1 x2 −1 i where s = 2l , l ≥ 0 and t = 2k−l . Then for each positive integer k there are k + 1 rings of this form. Proof. We have that 2k = 2l 2k−l . Then there are k + 1 choices for l, 0 ≤ l ≤ k which gives the result. The rings in Proposition 2.4 can either be chain rings or non-chain rings. When s = 1 or t = 1 the ring is a chain ring, otherwise it is a non-chain ring. While we do have chain rings that are X-rings, the next theorem shows that non-trivial Galois rings, namely those not isomorphic to Zpe , cannot be X-rings. Of course, the ring Zpe is both an X-ring and a Galois ring. 5

Theorem 2.5. If R is a non-trivial Galois ring then it is not an X-ring. Proof. First we note that if the Galois ring is a field then it cannot be an X-ring since that would require it to have a non-trivial ideal. If R is a Galois ring with order a power of 2, then it is isomorphic to Z2e [x]/hg(x)i where g(x) is a monic basic irreducible polynomial of degree r in Z2e [x]. It is well known that a Galois ring R is a chain ring. Then any ideal of a Galois ring is of the form 2i with 1 ≤ i ≤ r and so h2i is the maximal ideal of R. But R is not an X-ring since its maximal ideal has (2e )r−1 elements and this number is not half of (2e )r = |R|. We illustrate the result with an example. Example 2. Let R = Z4 [x]/hx2 + x + 1i. Then R is a Galois ring of order 16 and the maximal ideal is generated by h2i, namely h2i = {0, 2, 2x, 2 + 2x}. Hence it is not an X-ring since its maximal ideal is equal to its socle and both are of size 4.

3

The Gray Map φk

In this section, we shall generalize the Gray map from the rings of order 4 to any X-ring. For k = 1, we have two local Frobenius rings that are X-rings, namely Z4 and F2 + uF2 where u2 = 0. Any element in Z4 can be written as a + b2, a, b ∈ {0, 1}. Similarly, any element in F2 + uF2 can be written as a + bu, a, b ∈ {0, 1}. Define the standard Gray map from the X-ring of order 4 to F22 as follows: φ1 (a + bx) = (b, a + b) where x is either 2 or u. For Z4 , this is the well known Gray map defined in [15] and for F2 + uF2 , it is the map defined in [5]. For Z4 , the map is non-linear and for F2 + uF2 , the map is linear. For k = 2, we have twelve local Frobenius rings of order 16. The rings of order 16 all satisfy the property that all elements can be written as a + bu + cv + dw where m = hu, vi and Soc(R) = hwi = {0, w}. These rings were classified in [21]. However, for these rings it is possible that uv = 0. For X-rings we are assuming that this is not possible. In [9], a Gray map which need not be linear was defined from Rn to F4n 2 . For these twelve rings, we have the following Gray map φ2 (a + bu + cv + dw) = (d, c + d, b + d, a + b + c + d) where a, b, c, d ∈ {0, 1}. This map was defined recursively using the Gray map on the rings of order 4. k We define a Gray map for local Frobenius X-rings of order 22 recursively in the following way. Let φ1 be the map defined above on a ring of order 4. Then let c ∈ R. We can write 6

c = c1 + uk c2 with c1 , c2 are elements of the X-ring of order 22 formed by modding out by uk . Then we define

k−1

, specifically, the X-ring

φk (c) = (φk−1 (c2 ), φk−1 (c1 ) + φk−1 (c2 )).

(1)

We then extend the map coordinatewise to Rn . k

k

Theorem 3.1. Let R be an X-ring of order 22 . The map φk : R → F22 is a bijection. Furthermore, it is linear if and only if the characteristic of the ring is 2. Proof. We prove the first part of the lemma by induction on k. For k = 1, it is obvious that k−1 φ1 is a bijective map. Assume the map φk−1 is bijective on a ring of order 22 formed from R by setting uk = 0. If (φk−1 (c2 ), φk−1 (c1 ) + φk−1 (c2 )) = (φk−1 (d2 ), φk−1 (d1 ) + φk−1 (d2 )) k−1 where c1 , c2 , d1 , d2 are elements of the X-ring of order 22 . Then by comparing the first coordinates we get φk−1 (c2 ) = φk−1 (d2 ) and by comparing the second coordinates we get φk−1 (c1 ) = φk−1 (d1 ). This gives that c1 = d1 and c2 = d2 since φk−1 is a bijection. Hence, we get c = d, which gives the result. When char(R) > 2, then the Gray map is not linear since φk (1 + 1) 6= φk (1) + φk (1) = 0. When char(R) = 2, it is easy to show by induction on k that φk is linear. Define the Lee weight of a codeword as the Hamming weight of the image of the codeword under the Gray map. We note that when applied to Z22k this is different than the usual definition of Lee weight on Zn . Given the Lee weight we can now define the following Lee weight enumerator: X k LC (x, y) = x2 n−wtL (c) y wtL (c) c∈C

where wtL (c) is the weight of a codeword c ∈ C and 2k n is the maximal Lee weight of any vector of length n. This weight enumerator coincides with the Hamming weight enumerator of the image, k φk (C), in the space F22 n . That is LC (x, y) = Wφk (C) (x, y). We can now determine the number of elements for each distinct Lee weight. ! k 2 k Lemma 3.2. Let R be an X-ring of order 22 . There are precisely elements of Lee i weight i in R, for i = 0, 1, 2, . . . , 2k . k

Proof. Since φk is a bijection from R to F22 , the Lee weight enumerator of R1 is precisely 2k 2k the Hamming weight ! enumerator of F2 which is (x + y) . This implies that, in R, there 2k are precisely elements of Lee weight i, for i = 0, 1, 2, . . . , 2k . i One of the primary reasons behind the definition of the Gray map is that the elements with even Lee weight are all contained in the maximal ideal as we prove in the following lemma. 7

k

Lemma 3.3. Let R be an X-ring of order 22 with maximal ideal m. If x ∈ m then wL (x) is even. Proof. If x ∈ m, then c∅ = 0 by Lemma 2.2. For k = 1, is it obvious that the elements in the maximal ideals of Z4 and F2 + uF2 have even Lee weights since φ1 (2) = φ1 (u) = (1, 1). Assume the lemma is true for k − 1, namely the elements in the maximal ideal of the X-ring k−1 all have even Lee weights. Take x ∈ m where m is the maximal ideal of the of order 22 k X-ring of order 22 . Then write x = x1 + uk x2 where x1 , x2 are elements of the X-ring of k−1 order 22 . Then φk (x) = (φk−1 (x2 ), φk−1 (x1 ) + φk−1 (x2 )). Since x ∈ m then x1 is a non-unit and so it is in the maximal ideal of the ring of order k−1 22 . Hence φk−1 (x1 ) has an even number of coordinates containing a 1. This gives that the Hamming weight of φk−1 (xi ) is even for i = 1. Let v = (vi ) and w = (wi ) be binary vectors and define |v ∧ w| to be the number of coordinates where vi = wi = 1. Then we have that wt(φk (x)) = wt(φk−1 (x2 )) + wt(φk−1 (x1 )) + wt(φk−1 (x2 )) − 2|φk−1 (x1 ) ∧ φk−1 (x2 )| = 2wt(φk−1 (x2 )) + wt(φk−1 (x1 )) − 2|φk−1 (x1 ) ∧ φk−1 (x2 )| ≡ 0

(mod 2).

Therefore φk (x) has even Hamming weight. The rank of a code C is the minimum number of generators of C. Let dL (C) denote the minimum Lee weight of all non-zero vectors in C. Using the same proof as was given in [10], we have the following. Theorem 3.4. Let C be a linear code over R, then b

4

dL (C) − 1 c ≤ n − rank(C). 22k

(2)

The Gray Map ψk

In this section, we describe another generalization of the Gray map which will be conjugate to the map φk . k Let R be an X-ring with |R| = 22 , m = hu1 , u2 , . . . , uk i. We shall establish a bijection k between R and F22 by viewing R as a vector space over F2 by indexing the coordinates by the subsets of {1, 2, . . . , k}. Namely, view R as a vector space over F2 with basis {uA : A ⊆ Q k {1, 2, . . . , k}} where uA = i∈A ui . Define ψk : R → F22 by ψk (uA ) = (cB ) where ( 1 B⊆A (cB ) = 0 otherwise. 8

Then extend ψk linearly to be defined over R, namely ψk (uA + uB ) = ψk (uA ) + ψk (uB ). For example, if k = 3, the subsets are ordered as  ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.  Then ψk (u1 u2 ) = 1 1 1 0 1 0 0 0 and ψk (u2 ) = 1 0 1 0 0 0 0 0 .   Then by linearity we have ψk (u2 + u1 u2 ) = 0 1 0 0 1 0 0 0 . It is immediate that the weight of ψk (x), x ∈ R, is odd if and only if x is a unit and that ψk (ui ) has weight 2 for all i, 1 ≤ i ≤ k. Lemma 4.1. The map ψk is conjugate to φk by the permutation k

k

k

k−1

(1, 2 )(2, 2 − 1)(3, 2 − 2) . . . (2

k−1

,2

+ 1) =

k−1 2Y

(i, 2k + 1 − i).

i=1

Proof. Notice that for k = 2, it is a simple computation to see that they are congruent via the permutation (1, 4)(2, 3). Then since the maps are defined recursively the result follows by induction. It follows immediately from the previous lemma that wL (ψk (v)) = wL (φk (v)). Hence we can use the Lee weight of a vector without specifying the particular Gray map. k

Lemma 4.2. Let R be an X-ring of order 22 and A ⊆ {1, 2, . . . , k}. If |A| > 1 then wL (uA ) ≡ 0 (mod 4). Proof. We have that wL (uA ) = 2|A| . Then since |A| > 1 we have that wL (uA ) ≡ 0 (mod 4).

The next lemma describes an orthogonality relation between the images of certain vectors. We shall use this lemma and the previous lemma to describe elements whose weight is 0 (mod 4). k

Lemma 4.3. Let R be an X-ring of order 22 and A, B ⊆ {1, 2, . . . , k}. If |A ∩ B| > 0 then [ψk (uA ), ψk (uB )] = 0. Proof. The number of coordinates where ψk (uA ) and ψk (uB ) are both 1 is 2|A∩B| . Since |A ∩ B| > 0, this number is even. Hence [ψk (uA ), ψk (uB )] = 0. k

Theorem 4.4. Let R be an X-ring of size 22 . If A ⊆ {1, 2, . . . , k} with |A| ≥ 2 and uA is such that uA uB = uC with |C| ≥ 2 for all subsets B ⊆ {1, 2, . . . , k}, then huA i is a self-orthogonal code whose elements all have doubly-even Lee weight.

9

P Proof. Consider any ideal of the form huA i where |A| ≥ 2. If x ∈ huA i then x = uA B∈I uB = P B∈I uA uB . Then notice that uA uB = uC for each B, with |C| ≥ 2. Then by Lemma 4.3, P P [uC , uC 0 ] = 0. Then any two elements in this ideal are necessarily orthogonal since [ uC , uC 0 ] = 0. Next we show that all of the weights are even. Recall that for binary vectors v, w wt(v + w) = wt(v) + wt(w) − 2|v ∧ w|. We note that the elements of the ideal are sums of uC . Also we have that the elements uC and uC 0 are orthogonal. Applying this to the definition of ψk we have that each ψk (uC ) is doubly-even and then to compute ψk of the sum we apply it linearly. For two vectors we P have wt(uC + uC 0 ) ≡ 0 + 0 − 2(2a) ≡ 0 (mod 4). Then by induction uC has doubly-even weight in this case. Note that over Rk the conditions of this theorem are true for all A ⊆ {1, 2, . . . , k}, with |A| ≥ 2. As an example of the theorem consider the ring Z256 . In this ring u1 = 2, u2 = 4 and u3 = 16. Note that if A = {1, 2}, the conditions of the previous theorem fails as u1 uA = u3 and |{3}| < 2. However, the conditions of the theorem are true for A = {1, 3}. Consider a chain ring which is an X-ring, with m = hγi. Then u1 = γ, u2 = γ 2 , u3 = k−1 γ 4 , . . . , uk = γ 2 . We note that huk i is a self dual code. Note that the conditions of Theorem 4.4 are met if k ∈ A and |A| > 1. Then applying the theorem we have the following corollary. k

Corollary 4.5. Let R be an X-ring of order 22 that is a chain ring. If s > 2k−1 then hγ s i is a self-orthogonal code of length 1 whose elements all have Lee weight a multiple of 4. Proof. The condition s > 2k−1 was necessary to ensure that uA uB = uC with |C| ≥ 2. What this corollary implies is that the farther one descends the chain of ideals, the more restrictive the weights become. That is at the top of the chain, namely the ring itself, the Lee weights can be odd or even. Once you are in the maximal ideal hγi then the weights must all be even. Then after you descend past the ideal which is the self-dual code of length 1 then all of the weights must be doubly-even. Extend the map Ψk to Rn by applying it coordinatewise. k

Lemma 4.6. Let R be an X-ring of order 22 . Then ψk (u1 u2 · · · uk , u1 u2 · · · uk , . . . , u1 u2 · · · uk ) is the all-one binary vector. Proof. Since every A is a subset of {1, 2, . . . , k} we have that the image of u1 u2 · · · uk is the all-one vector of length 2k . Hence the image of the vector is the all-one vector. 10

We have described a family of rings that admit two conjugate Gray maps. However, this does not imply that these are the only rings which admit a similar Gray map. Consider, for example, the Frobenius ring of order 16, F2 [u, v]/hu2 + v 2 , uvi. Here any element in this ring can be written as a + bu + cv + du2 . The socle of this ring is Soc(R) = {0, u2 }. We can then define the Gray map φ2 in the standard way using u2 instead of uv. The map ψ2 can be defined similarly replacing set theoretic containment with the partial order defined by 1 being the minimal element, u2 being the maximal element, and u and v being incomparable. However, this ring is not an X-ring as uv = 0. To include this type of ring for all k would overcomplicate the definition. But it is worthwhile to notice that many of the results in this paper can be extended to other families of rings by altering the definitions slightly.

4.1

MacWilliams Relations

Label the elements of R by 0 = r0 , r1 , . . . , rs−1 and let xi the indeterminate associated with ri . We can define the complete weight enumerator of a code C by cweC (x0 , x1 , . . . , xs−1 ) =

n XY

xci ,

c∈C i=1 k

where c = (c1 , c2 , . . . , cn ) and s = 22 = |R|. b = Hom (M, C∗ ). It is well known that a Frobenius ring has a generating character for R Z Denote the generating character for the Frobenius ring R by χ. It is shown in [22], that the following MacWilliams relations hold for linear codes over Frobenius rings: cweC ⊥ (x0 , x1 , . . . , xs−1 ) =

1 cweC (T · (x0 , x1 , . . . , xs−1 )), |C|

where T is an s by s matrix defined by Ta,b = χ(ab), with χ the generated character for b and a, b run over the elements of R. Here (T · (x0 , x1 , . . . , xs−1 )) indicates replacing xi R, with the i-th entry of T (x0 , . . . , xs−1 )t . Hence it is imperative to determine the generating character for a Frobenius ring. We shall describe what the generating characters are for the X-rings described in this paper. It is shown in [22] that χ is a generating character if and only if ker(χ) contains no non-trivial ideals. For an X-ring, if χ is non-trivial on the minimal ideal {0, w} then it is non-trivial on w, i.e. if χ(w) 6= 1. Hence, if we have a character of the ring such that χ(w) 6= 1 then necessarily χ is a generating character for the X-ring. Notice also that w + w = 0 hence χ(w) must be −1. Theorem 4.7. The X-rings R have the following generating characters: P • If R = Rk = F2 [u1 , u2 , . . . , uk ]/hu2i = 0, ui uj = uj ui i, then χR ( A⊆{1,2,...,k} cA uA ) = (−1)wt(c) , where c = (cA ). 11

k

• If R = Z22k , then χR (a) = ξ2a2k , where ξ22k is a primitive 22 -th root of unity. • If R = Z2s [x]/hxt i, then the ring has characteristic 2s where s = 2l , l ≥ 0, t = 2k−l , P Q and χR ( ai xi ) = ξ2asi where ξ2s is a primitive 2s -th root of unity. • If R = Z2s [x]/hxt − 2r xm i, then the ring has characteristic 2s where s = 2l , l ≥ 0, P Q t = 2k−l , m < t, r ≥ 0 and χR ( ai xi ) = ξ2asi where ξ2s is a primitive 2s -th root of unity. Proof. If R = Rk = F2 [u1 , u2 , . . . , uk ]/hu2i = 0, ui uj = uj ui i, then for the given χR we have χR (u1 u2 . . . uk ) = (−1)1 = −1 hence it is a generating character. If R = Z22k , then k χR (w) = χR (22 −1 ) = −1, and hence it is a generating character. If R = Z2s [x]/hxt i is a ring l k−l which has characteristic 2s where s = 2l , l ≥ 0, t = 2k−l , then χR (22 −1 x2 −1 ) = −1, and hence it is a generating character. If R = Z2s [x]/hxt −2r xm i is a ring which has characteristic l k−l 2s where s = 2l , l ≥ 0, t = 2k−l , m < t, r ≥ 0, then χR (22 −1 x2 −1 ) = −1, and hence it is a generating character.

4.2

Automorphism Group

In this section we examine the automorphism group of the binary image as determined by the multiplication of units in the ring. This has been done for the ring Rk in [14]. Of course, the results of this section can be generalized to other families of rings, we choose these as examples of the technique. We shall consider the group of automorphisms for the images of codes over the family of rings R = Z2s [x]/hxt i for s = 1 and s = 2. For s = 1, we get the family of chain rings, R1 = Z2 [x]/hxt i where t = 2k . For s = 2, we get the family of non-chain rings, R2 = Z4 [x]/hxt i where t = 2k−1 . We study the automorphisms which correspond to multiplication of units both in R1 and R2 . Define the map αk to be αk (x1 , . . . , x2k ) = (x2k , x1 , x2 , . . . , x2k −1 ) where xi are binary vectors of length n. Theorem 4.8. Let R = Z2 [x]/hxt i, k ≥ 1. The group of automorphisms for the binary images under the Gray map φk contains {1, αk } where αk is defined previously. Proof. We prove the theorem by induction on k. If k = 1 then t = 2, so we get the chain ring R1 = Z2 [x]/hx2 i with U (R1 ) = {1, 1 + x}. Since (1 + x)(a + xb) = a + x(a + b), we get φ1 ((1 + x)(a + xb)) = (a + b, b) = α1 (φ1 (a + xb)) where a, b are binary vectors of length n. Hence the automorphism group for the binary images under the Gray map φ1 contains {1, α1 }. Now assume the assertion to be true up to 12

k − 1. Take an element c = c1 + uk c2 in Rn = (Z2 [x]/hxt i)n where t = 2k and c1 and c2 are k−1 vectors in (Z2 [x]/hx2 i)n . By induction the binary images of c1 and c2 are invariant under the permutation group hαk−1 i. Since φk (c1 + uk c2 ) = (φk−1 (c2 ), φk−1 (c1 ) + φk−1 (c2 )), this is also invariant under the group hαk−1 i. We also have φk ((1 + x)(c1 + uk c2 )) = φk ((1 + x)(a1 + xa2 + · · · + x2

k −1

a2k )) k −1

= φk (a1 + x(a1 + a2 ) + x2 (a2 + a3 ) + · · · + x2

(a2k −1 + a2k ))

= (a2k −1 + a2k , a2k −2 + a2k , . . . , a2k ), where ai ∈ F2 . Then αk (φk ((1 + x)(c1 + uk c2 ))) = φk ((c1 + uk c2 )). Hence the binary images of linear codes over R are invariant under the group {1, αk }. Define the maps βk (x1 , x2 , . . . , x2k−1 , x2k−1 +1 , . . . , x2k ) = (x2k−1 +1 , x2k−1 +2 , . . . , x2k , x1 , x2 , . . . , x2k−1 ), and τk (x1 , x2 , . . . , x2k−1 , x2k−1 +1 , . . . , x2k ) = (x2k−1 , x1 , x2 , . . . , x2k−1 −1 , x2k , x2k−1 +1 , x2k−1 +2 , . . . , x2k −1 ) where xi are binary vectors of length n. The action of the map τk on X is the action of αk−1 on each part C and D. Theorem 4.9. Let R = Z4 [x]/hxt i, k > 1. The group of automorphisms for the binary images under the Gray map φk contains {1, βk , τk , βk τk } where βk and τk are defined previously. Proof. We prove the theorem by induction on k in a manner similar to the preceding proof. If k = 2 then t = 2, then we have the non-chain ring R1 = Z4 [x]/hx2 i with U (R1 ) = {1, 3, 1 + x, 1 + 2x, 1 + 3x, 3 + x, 3 + 2x, 3 + 3x}. Each unit corresponds to a permutation. However, not all the permutations act on vectors independently. Multiplying the elements in R1 by the units 3 and 1 + x corresponds to independent permutations. Let β2 and τ2 be permutations which act on vectors of length 4n as follows: β2 (x1 , x2 , x3 , x4 ) = (x3 , x4 , x1 , x2 ), τ2 (x1 , x2 , x3 , x4 ) = (x2 , x1 , x4 , x3 ) where x1 , x2 , x3 , x4 are binary vectors of length n. Since (3)(a + xb + 2c + 2xd) = a + xb + 2(a + c) + 2x(b + d) and (1 + x)(a + xb + 2c + 2xd) = a + x(a + b) + 2c + 2x(c + d) we get β2 (φ2 ((3)(a + xb + 2c + 2xd))) = φ2 (a + xb + 2c + 2xd) and τ2 (φ2 ((1 + x)(a + xb + 2c + 2xd))) = φ2 (a + xb + 2c + 2xd) 13

where a, b, c, d are binary vectors of length n and β2 and τ2 are as defined above. So the group of automorphisms for binary images under the Gray map φ2 contains hβ2 , τ2 i. Now assume the assertion to be true up to k −1. Take an element v = v1 +uk v2 in R = Z4 [x]/hxt i where t = 2k−1 and v1 and v2 are vectors in Z4 [x]/hxt i where t = 2k−2 . By the definition of the Gray map and since uk = 2 for each k for R, we have φk ((3)(v1 +uk v2 )) = φk (3v1 +2v2 ) = (φk (v1 +2(v1 +v2 )) = (φk−1 (v1 +v2 ), φk−1 (v1 )+φk−1 (v1 +v2 )). Then βk (φk ((3)(v1 + uk v2 ))) = φk ((v1 + uk v2 )). We also have φk ((1 + x)(v1 + uk v2 )) = φk (v1 + xv1 + 2(v2 + xv2 )). Here the last statement means that the unit 1+x acts on the vector of length 2k n by making one shift to each half of the vector. That is, if the vector of length 2k n is of the form (C|D), then the part C and D will cycle n shifts, separately. Then we have τk (x1 , x2 , . . . , x2k−1 , x2k−1 +1 , . . . , x2k ) = (x2k−1 , x1 , x2 , . . . , x2k−1 −1 , x2k , x2k−1 +1 , x2k−1 +2 , . . . , x2k −1 ). Hence we have the following: τk (φk ((1 + x)(v1 + uk v2 ))) = φk ((v1 + uk v2 )).

5

Self-Dual and Formally Self-Dual Codes

In this section, we shall examine the existence of formally self-dual and self-dual codes over X-rings. Recall that a code C is self-dual if C = C ⊥ and a code is said to be Lee formally self-dual if LC (x, y) = LC ⊥ (x, y). Definition 2. Let C be a Lee formally self-dual code over R. If all of the vectors in the code are doubly-even then we say that the code is Type I, if all of the vectors in the code are even then we say that it is Type I and we say that it is Type 0 otherwise. The following lemma is straightforward. Lemma 5.1. Let R be a Frobenius ring. If C and D are self-dual codes over R of length n and m respectively then C × D is a self-dual code of length nm. If C and D are formally self-dual codes of length n and m respectively then C × D is a formally self-dual code of length nm.

14

5.1

Type I Self-Dual Codes

We begin by studying Type I self-dual codes. Lemma 5.2. Let w = (w, w, . . . , w), where Soc(R) = {0, w} and let R be an X-ring of size k 22 . If C is a self-dual code then w ∈ C. Proof. If y ∈ R − m, then y = 1 + x for some x ∈ m. Then (1 + x)2 = 1 + 2x + x2 . Consider P 2 the vector z = (z1 , z2 , . . . , zn ) ∈ C. Then zi = 0 since C is self orthogonal. Here the number of units is a multiple of the characteristic. Then if zi ∈ m, zi w = 0. If zi ∈ / m then Pt zi w = (1 + xi )w, xi ∈ m which is equal to w. Then [z, w] = i=1 w where t is the number of units, but the characteristic divides t, so [z, w] = 0. Then w ∈ C ⊥ = C. Therefore the all w vector is in all self-dual codes. It follows from this lemma that the image of a self-dual code always contains the all-one vector, given that we have shown that w = u1 u2 · · · uk and the image of (u1 u2 · · · uk , u1 u2 · · · uk , . . . , u1 u2 · · · uk ) is the all-one vector. Lemma 5.3. Let R be a chain X-ring. There exists a Type I self-dual code of length 1 over R. k

k

Proof. Let R be a chain X-ring. We have that |R| = 22 with |m| = 22 −1 . Then there k k k−1 exists γ with γ 2 = 0 and |hγi| = 22 −1 . Take hγ 2 i = C. This ideal is self-orthogonal since k−1 k−1 γ 2 γ 2 = 0. All ideals are of the form hγ a i for some a since it is a chain ring. Let C ⊥ be k−1 k−1 k−1 hγ a i. If a < 2k−1 , then γ a γ 2 6= 0. If a > 2k−1 , then hγ 2 i ⊆ hγ a i. Hence C ⊥ = hγ 2 i and C is self-dual. Notice that all of the elements in the code are in the maximal ideal and hence have even weight, but the generator does not have doubly-even weight so the code is Type I. Using this theorem we have the following corollary. Corollary 5.4. Let R be a chain X-ring. Then there exist self-dual codes of all lengths over R. Proof. There exits a self-dual code of length 1 by Lemma 5.3. Then by Lemma 5.1 and noticing that the direct sum of Type I codes is Type I, we have self-dual codes of all lengths. We can now determine the existence of Type I self-dual codes for certain examples of X-rings. It has been shown in [13] that self-dual Type I codes exist for all lengths over the ring Rk . The next theorem considers other examples of X-rings. 15

l−1

Theorem 5.5. Let R be Z2s [x]/hxt i where s = 2l , l ≥ 0, t = 2k−l . Then the ideals h22 k−l−1 and hx2 i are Type I self-dual codes. l−1

l−1

l−1

l

i

k−l−1

= 22(2 ) = 22 = 0 and x2 · Proof. Consider the ring Z2s [x]/hxt i. Then 22 · 22 k−l−1 k−l−1 k−l 2 2(2 ) 2 2l−1 2k−l−1 x = x = x = 0. Then we have that both the ideals h2 i and hx i 2l−1 t−1 2l−1 are self-orthogonal. If 2 (β0 + β1 x + · · · + βt−1 x ) = 0 then βi ∈ h2 i for all i, hence k−l−1 l−1 l−1 i is a self-dual code in the h22 i⊥ = h22 i and it is a self dual code. The ideal hx2 same way. Since not all elements are doubly-even, the codes are Type I. Similarly we have the following. Theorem 5.6. Let R be the ring Z2s [x]/hxt − 2r xm i where s = 2l , l ≥ 0, t = 2k−l , m < t, l−1 r ≥ 0. Then the ideal h22 i is a Type I self-dual code of length 1. Proof. The proof is similar to the previous theorem. In the usual manner we can show existence for all lengths of these classes of rings using the previous theorem. Corollary 5.7. Let R be Z2s [x]/hxt i where s = 2l , l ≥ 0, t = 2k−l or Z2s [x]/hxt − 2r xm i where s = 2l , l ≥ 0, t = 2k−l , m < t, r ≥ 0. Then there exist self-dual codes of all lengths over R. Notice that the algebraic structure of the rings makes a difference about the existence of self-dual codes. For example, for the ring Rk , hu1 i is a self-dual code of length 1. However, for the chain ring Z22k , u1 = 2 but h2i is not a self-dual code. In fact, it is the maximal ideal. We shall now give a construction of self-dual codes from self-orthogonal codes. k

Theorem 5.8. Let C be a self-orthogonal code over an X-ring R of size 22 . If there exists a self-dual code of length 1 over R that is principally generated by α then C ⊕ (C ⊥ ∩ αRn ) is a self-dual code. Proof. Consider vectors v, v0 ∈ C and w, w0 ∈ C ⊥ . Then [v + αw, v0 + αw0 ] = [v, v0 ] + [v, αw0 ] + [αw, v0 ] + α[w, αw0 ] = 0 + 0 + 0 + α2 [w, w0 ] = 0, since α2 = 0. Hence the code is self-orthogonal. Then we have (C ⊕ (C ⊥ ∩ αRn ))⊥ = C ⊥ ∩ (C ⊥ ∩ αRn )⊥ = C ⊥ ∩ (C ⊕ αRn ) since (αRn )⊥ = αRn . Then C ⊥ ∩ (C ⊕ αRn ) = C ⊕ (C ⊥ ∩ αRn ). Therefore the code is self-dual. 16

Example 3. Let R be the ring Rk . Let C = h(1, 1, 1, . . . , 1)i. Then |C| = |R|. The orthogonal to this code is generated by the matrix   1 1 0 0 ... 0    0 1 1 0 ... 0     0 0 1 1 ... 0 .    ..  .   0 0 0 ... 1 1 Then C ⊕ (C ⊥ ∩ ui Rkn ) is a self-dual code of length n. We shall now investigate when free self-dual codes exist. A code C is free if it is isomorphic to Rm for some integer m. Lemma 5.9. Let R be an X-ring. Let C be a free self-dual code of length n over R. Then n is even. Proof. If C is a free self-dual code, then |C|2 = |R|n and |C| = |R|m for some integer m. This gives that |R|2m = |R|n , where m is an integer, which gives that n is even. As for Frobenius rings in general, the existence of free self-dual codes is linked to the existence of a square root of −1 as in the next theorem. Theorem 5.10. Let R be a local Frobenius ring. Free self-dual code exist over R for all even lengths if and only if there exists an element γ ∈ R with γ 2 = −1. Proof. If there exists such an element γ, then (1, γ) generates a free self-dual code of length 2. Then by Lemma 5.1 there exists self-dual codes of all even lengths. Next assume there exists a free self-dual code of all even lengths, hence there exists a free self-dual code of length 2. This implies there exists a vector (a, b) with |ha, bi| = |R|. This means that α(a, b) = (0, 0) if and only if α = 0. Assume that such an α exists, then α ∈ Ann(ha, bi). If the ideal generated by a and b is not R then ha, bi ⊆ m since it is the unique maximal ideal. This implies that α(a, b) = (0, 0) for the non-trivial element in Soc(R). Hence ha, bi must be R. If neither a nor b were a unit, then ha, bi would not be R, so at least one must be a unit. Without loss of generality assume b is a unit. Then a2 + b2 = 0 implies ( ab )2 = −1 and we have the element α described in the theorem. Note that if the X-ring R has characteristic 2 then it satisfies the conditions of the previous theorem. Theorem 5.11. If h2i is a prime ideal in the X-ring R, then R has an element a with a2 = −1 if and only if char(R) = 2.

17

Proof. If the characteristic of the ring is 2 then 12 = −1 and we have the result. Next assume the characteristic of the ring is not 2. If there exists a2 = −1 then a is a unit and hence of the form 1 + m where m ∈ m, where m is the maximal ideal of the ring. Then −1 = (1 + m)2 = 1 + 2m + m2 . This gives that 2(m + 1) = −m2 and hence m2 ∈ h2i. Since h2i is prime then m ∈ h2i. Then we have that 2(m + 1) ∈ h2i2 which gives that m + 1 ∈ h2i. Then we have h2i = R which is a contradiction since 2 is never a unit in an X-ring. Hence there does not exist an element a with a2 = −1 in this case. Theorem 5.12. Let R be an X-ring. ! If there exists elements α and β in a ring R with 1 0 α β α2 + β 2 = −1 then generates a free self-dual code of length 4. 0 1 −β α Proof. Each row in the matrix is self-orthogonal since 12 + α2 + β 2 = 0 and their innerproduct with each other is −αβ + αβ = 0. Hence the code is self-orthogonal and the form gives that the size of the generated code is |R|2 which gives that it is self-dual. The obvious corollary follows by taking direct products of these codes. Corollary 5.13. Let R be an X-ring. If there exist elements α and β in a ring R with α2 + β 2 = −1 then there exists self-dual codes of all lengths congruent to 0 (mod 4). Consider the X-ring R = Z4 [x]/hx2 − 2i, which is a ring of the form Z2s [x]/hxt − 2r xm i with s = 2, t = 2 and |R| = 16. In this ring 12 + x2 = −1. Hence it satisfies the conditions of Theorem 5.12 and so there are free self-dual codes of all lengths 0 (mod 4) over this ring. We can now show how self-dual codes over a subring can be used to generate free self-dual codes over an X-ring R. Theorem 5.14. Let R be an X-ring. Let S be a subring of R. If C is a self-dual code in S n with generator matrix of the form (I n2 | A) then hCi ⊆ Rn is a free self-dual code. Proof. Since the generator matrix is of the form (I n2 | A) then the generated code has n size |R| 2 . Then if vi , wj ∈ S n and αi , βj ∈ R we have that X X X [ αi vi , βj w j ] = αi βj [vi , wj ] = 0, giving that the code is self-orthogonal. Therefore, the code is self-dual. In Proposition 3.4 in [1], it is shown that there is a 4 × 4 matrix M such that (I4 |M ) is a free self-dual code of length 8 over Z2k . This then applies to the more specialized case of Z2k . Specifically,   a b c d  b −a −d c    M = ,  c d −a −b  d −c b −a 18

where 1 + a2 + b2 + c2 + d2 = 0. The existence of such elements is guaranteed by a theorem of Lagrange. Every X-ring has a characteristic of the form 2s for some s and hence contains Z2s as a subring. Then by applying Theorem 5.14 we have the following corollary. Corollary 5.15. Let R be an X-ring. There exist free self-dual codes of all lengths a multiple of 8 over R.

5.2

Formally Self-Dual Codes

In this section, we investigate the existence of formally self-dual codes which are not necessarily self-dual. We shall now determine the existence of formally self-dual codes of various types over X-rings. Theorem 5.16. Let R be an X- ring.There exist Type 0 Lee formally self-dual codes of all even lengths. Proof. The code h(1, 0)i is a Type 0 Lee formally self-dual code of length 2. Then apply the direct sum and we have Type 0 Lee formally self-dual codes of all even lengths. Of course, no Type 0 code is self-dual so all of the codes in the previous theorem are formally self-dual but not self-dual. The next theorem gives the existence of Type I formally self-dual codes. The codes described in this theorem are only self-dual when a is a self-dual code of length 1. Theorem 5.17. Let a be any ideal in an X-ring, then C = {(x, y) | x ∈ a, y ∈ a⊥ } is a Type I Lee formally self-dual code. Proof. We have that C ⊥ = {(x, y) | x ∈ a⊥ , y ∈ a}. Then we have that |C| = |R| = |a||a⊥ | = |C ⊥ |. We have that C and C ⊥ are equivalent codes and have the same Lee weight enumerator. Since all elements in an ideal are also in the maximal ideal they all have even Lee weight. Hence every vector has even Lee weight. Given any ideal either it or its orthogonal has a Lee weight that is not doubly-even, hence if this element is x then either (x, 0) or (0, x) is a singly-even Lee weight vector. The next theorem gives another construction of Type I formally self-dual codes. Theorem 5.18. The code h(1, 1)i is a Type I Lee formally self-dual code over any X-ring. Proof. The code C ⊥ = h(1, −1)i which has the same Lee weight enumerator as C. It is Type I since wtL ((1, 1)) = 2 6≡ 0 (mod 4) and every other vector is of the form (x, x) which has Lee weight 2wtL (x) ≡ 0 (mod 2). If the ring does not have characteristic 2 then the code produced in Theorem 5.18 is not self-dual. 19

5.3

Type II Self-Dual and Formally Self-Dual Codes

We now investigate the existence of Type II self-dual codes over X-rings. k

Theorem 5.19. Let R be an X-ring of size 22 , k ≥ 2, maximal ideal m and Soc(R) = {0, w}. If m2 has characteristic 2 then C = h{(a, a) |a ∈ m}, (0, w)i is a Type II self-dual code of length 2. Proof. If a ∈ m then wtL (a) ≡ 0 (mod 2). Therefore (a, a) has doubly-even Lee weight. The vector (0, w) has doubly-even Lee weight since k ≥ 2. Note that every other element is of the form (a, a + w) which also has doubly-even Lee weight since both elements are from the maximal ideal. Then a2 ∈ m2 which has characteristic 2 so a2 +a2 = 0. Therefore (a, a) is self-orthogonal when a ∈ m. Moreover, [(a, a), (0, w)] = 0 and (0, w) is self-orthogonal. Therefore C is selforthogonal with |C| = 2|m| = |R|. Therefore, C is a Type II self-dual code. Corollary 5.20. There exists Type II codes of all even lengths over Rk and Z2 [x]/hxt i and for all X-rings of characteristic 2. Proof. These rings satisfy the conditions of Theorem 5.19. Note also that every non-chain local ring of order 16 satisfies the condition of the theorem since m2 = Soc(R). Consider for example the ring of order 16, Z4 [x]/hx2 i. Here m = hx, 2i and Soc(R) = {0, 2x}. Notice that m2 = Soc(R) which has characteristic 2 and so there is a Type II code of length 2 over this ring. The code is h(2, 2), (x, x), (0, 2x)i. We now show an example of a Type II self-dual code over an X-ring not constructed with the method in Theorem 5.19. Example 4. The code over Z16 with generator matrix   2 2 2 2  0 4 4 0       0 0 4 4  0 0 0 8 has 8 · 42 · 2 = 162 elements and is self-orthogonal by verifying that any two rows are orthogonal. It is easy to verify that all elements have doubly-even Lee weight. Hence this matrix generates a Type II self-dual code over Z16 . We can use also the technique in Theorem 5.19 to construct Type II formally self-dual codes when this technique fails to produce a self-dual code. k

Theorem 5.21. Let R be an X-ring with size 22 and maximal ideal m. If char(m2 ) 6= 2 then C = h(a, a) | a ∈ m}, (0, w)i is a Type II formally self-dual code that is not self-dual. 20

Proof. Consider s = [φk ((a, a)), φk ((b, b))], a, b ∈ m. Then s = 2[φk (a), φk (b)] = 0. Hence φk [h{(a, a) | a ∈ m}i] is a binary self-orthogonal code. Then φk ((0, w)) = 02k−1 12k−1 which is orthogonal to φk ((a, a)) since wtH (φk (a)) is even. Moreover, it is immediate that φk ((0, w)) is self-orthogonal. Then φk (C) is self-dual, hence C is Lee formally self-dual. Since char(m2 ) 6= 2 then [(a, a), (a, a)] = 2a2 which is not necessarily 0 for all a so C is not self-dual. Then wtL ((a, a)) = 2wtL (a) ≡ 0 (mod 4) and wtL (a, a + w) ≡ 0 (mod 4). Hence C is a Lee formally self-dual Type II code which is not self-dual. ! 2 2 Example 5. Consider the code over Z16 generated by . This code is a Lee formally 0 8 ! 2 6 self-dual Type II code. Its orthogonal is generated by . 0 8 Codes over various examples of X-rings have been found which show why this class of rings is so important. For example, in [19], codes over R2 were used to produce new singly-even extremal binary self-dual codes of length 66 and 68. In [18], codes over R3 were used to get twenty-six non-equivalent binary Type I codes of length 64, ten of which have new weight enumerators. In [16], codes over Rk were used to get even formally self-dual codes with parameters [72, 36, 14], [56, 28, 12], [44, 22, 10] and odd formally self-dual binary codes of parameters [72, 36, 13], all of which have better minimum distances than the best known self-dual codes. In [17], self-dual [64, 32, 12] codes were produced as binary images of codes over R1 , R2 and R3 . In [6], the X-rings Rk , Sk and Tk were used to produce numerous formally self-dual codes (both even and odd), many of which were optimal and many of which were better than the best known self-dual codes. In [7], cyclic codes over X-rings of order 16 like F2 [u, v]/hu2 + v 2 , uvi, Z4 [x]/hx2 − 2i and Z16 , were used to produce numerous optimal binary codes. In fact, [92, 66, 8] and [84, 58, 8] quasi-cyclic codes were produced which were the first quasi-cyclic codes with these parameters.

6

Conclusion

The study of codes over rings began in earnest with the application of the Gray map from Zn4 to the binary Hamming space. This Gray map was generalized in numerous ways to various rings. In this work we have described a new family of rings called X-rings. These X-rings not only contain most of the families of rings for which Gray maps have been studied but they also expand the number of rings to which we can apply the techniques applied to the Gray map. To these Frobenius rings we attach two conjugate Gray maps to the binary space. These maps have different strengths in that various theorems are often more easily proven using one of the maps rather than the other. These maps allow us to define a Lee weight for elements in an X-ring. Finally, we study self-dual and formally self-dual codes over X-rings. 21

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