Constructions of Self-Dual Codes and Formally Self-Dual Codes Over Rings Steven T. Dougherty Department of Mathematics University of Scranton Scranton PA 18510, USA Abidin Kaya Department of Mathematics Fatih University Istanbul 34500, Turkey Eseng¨ ul Salt¨ urk Department of Mathematics University of Scranton Scranton PA 18510, USA Abstract We shall describe several families of X-rings and construct self-dual and formally self-dual codes over these rings. We then use a Gray map to construct binary formally self-dual codes from these codes. In several cases, we produce binary formally self-dual codes with larger minimum distances than known self-dual codes. We also produce non-linear codes which are better than the best known linear codes.

1

Introduction

We take the standard definitions for codes over rings. Namely a code over a ring R is a subset of Rn and it is said to be a linear code if it is a submodule of Rn . We attach to the P ambient space the inner-product [v, w] = vi wi and define the orthogonal as C ⊥ = {v |v ∈ Rn , [v, w] = 0, ∀w ∈ C}. We assume that all rings in this paper are finite, commutative and Frobenius. It follows that for a linear code in this setting we have |C||C ⊥ | = |R|n . A code is said to be self-dual if C = C ⊥ . 1

The Hamming weight enumerator of a code C is defined to be X WC (x, y) = xn−wtH (c) y wtH (c) ,

(1)

c∈C

where wtH (c) is the number of non-zero coordinates in a vector c and n is the length of the code. A code C is said to formally self-dual (FSD) if C and C ⊥ have the same weight enumerator. In [4], X-rings were defined. These rings have a canonical Gray map to the Hamming space. In this paper, we shall examine some families of these rings for whom the binary MacWilliams theorems apply to their Lee weight enumerators. This will allow us to construct formally self-dual codes over these rings whose images under the Gray map have weight enumerators which are held invariant by the action of the MacWilliams relations. This Gray may, in fact, be non-linear. The aim is to find examples that have better minimum weight than any corresponding binary linear code. The following definition of an X-ring can be found in [4]. Definition 1. A ring R is an X-ring if it is a finite commutative local Frobenius Y ring with |R| 2k ui 6= 0 maximal ideal m = hu1 , u2 , . . . , uk i such that |R| = 2 , and m = 2 , where uA = i∈A

for all A ⊆ {1, 2, . . . , k}. k

Let R be an X-ring of order 22 . Then any element of R can be written in the form X cA uA , cA ∈ {0, 1}, A⊆{1,2,...,k}

where uA :=

Y

ui with the convention that u∅ = 1. Moreover, it is easily seen that these cA

i∈A

are unique. It is also shown in [4] that an element of the ring is a unit if and only if c∅ = 1. k The following Gray map was defined over rings of order 22 recursively in the following way in [4]. Let φ1 be the map defined on a local ring of order 4. Namely, φ1 (a+ub) = (b, a+b) where u is either 2 in Z4 or u in F2 [u]/hu2 i. Then let c ∈ R. We can write c = c1 + uk c2 with k−1 c1 , c2 are elements of the X-ring of order 22 , then we define φk (c) = (φk−1 (c2 ), φk−1 (c1 ) + φk−1 (c2 )).

(2)

We then extend the map coordinatewise to Rn . It is shown that if R is an X-ring of order k k 22 then the map φk : R → F22 is a bijection. Furthermore, it is linear if and only if the characteristic of the ring is 2. See [4] for a complete description of X-rings and their Gray k maps. The Lee weight of a vector c in an X-ring of order 22 is the Hamming weight of φk (c).

2

2

Families of Rings

Recall that the Jacobson radical of a ring is defined as the intersection of all maximal ideals of the ring and the socle is defined as the sum of all the minimal one sided ideals of the ring. For Frobenius local rings the Jacobson radical is necessarily the unique maximal ideal and the socle is its orthogonal when the Jacobson radical is considered as a code of length 1. In [5] and [6], the following family of rings was studied: Rk = F2 [u1 , u2 , . . . , uk ]/hu21 , u22 , . . . , u2k , ui uj = uj ui i. It is shown that for a linear code C over this ring φk (C ⊥ ) = φk (C)⊥ . The next theorem follows immediately from this fact. Theorem 2.1. Let C be a self-dual code of length n over Rk . Then φk (C) is a binary selfdual code of length 2k n. If C is a formally self-dual code of length n over Rk then φk (C) is a binary formally self-dual code of length 2k n. We shall now define some new families of rings. Let Sk = Z4 [u1 , u2 , . . . , uk ]/hu21 − 2u1 , u22 − 2u2 , . . . , u2k − 2uk , ui uj = uj ui i. The ring S1 is a local Frobenius ring of order 16 and it was studied in [3]. For any k ≥ 1, the Jacobson radical for the ring Sk is J(R) = h2, u1 , u2 , . . . , uk i which is also the maximal ideal. The socle of the ring is Soc(Sk ) = h2u1 u2 · · · uk i. Since Sk /J(Sk ) ∼ = 2k Soc(Sk ), Sk is a Frobenius ring with |Sk | = 4 . Then it follows immediately that Sk is an k+1 X-ring of order 22 of characteristic 4. Define another family of rings. Let Tk = Z4 [u1 , u2 , . . . , uk ]/hu21 − 2, u22 − 2, . . . , u2k − 2, ui uj = uj ui i. The ring T1 is a local Frobenius ring of order 16 and it was studied in [3]. For any k ≥ 1, the Jacobson radical for the ring Tk is J(R) = h2, u1 , u2 , . . . , uk i which is also the unique maximal ideal. The socle of the ring is Soc(Tk ) = h2u1 u2 · · · uk i. Since k Tk /J(Tk ) ∼ = Soc(Tk ), Tk is a Frobenius ring with |Tk | = 42 . Then it follows immediately k+1 that Tk is an X-ring of order 22 of characteristic 4. We can also define both Sk and Tk recursively as follows: Sk = Sk−1 [uk ]/hu2k − 2uk , uk uj = uj uk i = Sk−1 + uk Sk−1 . Tk = Tk−1 [uk ]/hu2k − 2, uk uj = uj uk i = Tk−1 + uk Tk−1 . k+1

Note that both Sk and Tk are X-rings of order 22 and so it is the Gray map φk+1 that maps codes over these rings into the Hamming space. We shall define another Gray map which maps to the quaternary space to study codes over this space. 3

Define the following Gray map ψ1 : Rn → Z2n 4 such that ψ1 (a + ub) = (b, a + b), a, b ∈ Zn4 , where R = S1 or R = T1 . Let R be Sk or Tk and d = d1 + uk d2 ∈ R where d1 , d2 are the elements of the rings Sk−1 or Tk−1 , respectively. Then we define a Gray may recursively in the following way k ψk : Rn → Z24 where R is Sk or Tk such that ψk (d) = (ψk−1 (d2 ), ψk−1 (d1 ) + ψk−1 (d2 )). We then extend this map coordinatewise to Skn and Tkn . For a code C of length n over Rk , Sk or Tk we define the Lee weight enumerator as X xN −wtL (c) y wtL (c) , (3) LC (x, y) = c∈C

where N = 2k n for Rk and N = 2k+1 n for Sk and Tk . With the definition of the Gray map ψk , we have the following theorem. Theorem 2.2. Let C be a linear code over R of length n, where R is Sk or Tk . Then ψk (C) is a linear code over Z4 of length 2k n. Proof. The proof of both can be done by induction. We show the proof for Tk and it is similar for Sk . For ψ1 , we let a, b, c, d ∈ Z4 and we have ψ1 ((a+u1 b)+(c+u1 d)) = ψ1 ((a+c)+u1 (b+d)) = (b + d, a + b + c + d) = (b, a + b) + (c, c + d) = ψ1 (a + u1 b) + ψ1 (c + u1 d). Note that this is true precisely because the characteristic of both Z4 and S1 are 4. When this same map is applied from Z4 to F22 it is non-linear because the characteristic changes. The inductive step has the same computation, namely: we let a, b, c, d ∈ Si−1 and we have ψi ((a+ui b)+(c+ui d)) = ψi ((a+c)+ui (b+d)) = (b+d, a+b+c+d) = (b, a+b)+(c, c+d) = ψi (a + ui b) + ψi (c + ui d).

The following theorem describes the images of the orthogonals of codes over Sk . Theorem 2.3. Let C be a linear code over Sk of length n, then ψk (C ⊥ ) = ψk (C)⊥ . Proof. We prove the theorem by induction on k. First, for k = 1, all we need to show is that the Gray images of orthogonal vectors in S1 are orthogonal in Z4 . Let a1 + ub1 , a2 + ub2 be two orthogonal vectors of length n in S1 where a1 , a2 , b1 , b2 ∈ Zn4 . We have [a1 + ub1 , a2 + ub2 ] = a1 a2 + u(a1 b2 + b1 a2 + 2b1 b2 ) = 0. 4

This implies a1 a2 = a1 b2 + b1 a2 + 2b1 b2 = 0. Now consider the inner product of the Gray images: ψ1 (a1 + ub1 ) · ψ1 (a2 + ub2 ) = (b1 , a1 + b1 ) · (b2 , a2 + b2 ) = b1 b2 + a1 a2 + a1 b2 + b1 a2 + b1 b2 = 0. This proves that ψ1 (C ⊥ ) ⊆ ψ1 (C)⊥ . But since ψ1 is an isometry and C and ψ1 (C) have the same size, we have that |C| = |ψ1 (C)| = 16n/2 = 4n . It follows that ψ1 (C ⊥ ) = ψ1 (C)⊥ . Assume the theorem is true for Si−1 . We know that Si = Sk [ui ]/hu2i − 2ui i. Any element of Si is of the form a + ui b, where a, b ∈ Si−1 . Take x1 + ui y1 , x2 + ui y2 ∈ Si , x1 , x2 , y1 , y2 ∈ (i−1) n Z24 , we have [x1 + ui y1 , x2 + ui y2 ] = 0. From here, we get x1 x2 + ui (x1 y2 + x2 y1 + 2y1 y2 ) = 0. It follows that x1 x2 = x1 y2 + x2 y1 + 2y1 y2 = 0.

(4)

Since ψi is a linear map, from Equation (4), we get ψi (x1 + ui y1 ) · ψi (x2 + ui y2 ) = (ψi−1 (y1 ), ψi−1 (x1 ) + ψi−1 (y1 )) · (ψi−1 (y2 ), ψi−1 (x2 ) + ψi−1 (y2 )) = 2ψi−1 (y1 )ψi−1 (y2 ) + ψi−1 (x1 )ψi−1 (x2 ) + ψi−1 (x1 )ψi−1 (y2 ) + ψi−1 (y1 )ψi−1 (x2 ) = 0. Hence, we get ψi (C ⊥ ) ⊆ ψk (C)⊥ . Combining this with the fact that in Frobenius rings |C| · |C ⊥ | = |Sk |n , we get ψi (C ⊥ ) = ψi (C)⊥ . This completes the proof. The following theorem describes the images of the orthogonals of codes over Tk . Theorem 2.4. Let C be a linear code over Tk of length n, then ψk (C ⊥ ) = ψk (C)⊥ .

5

Proof. We prove the theorem by induction on k. First for k = 1, all we need to show is that the Gray images of orthogonal vectors in T1 are orthogonal in Z4 . Let a1 + ub1 , a2 + ub2 be two orthogonal vectors of length n in T1 where a1 , a2 , b1 , b2 ∈ Zn4 . We have [a1 + ub1 , a2 + ub2 ] = a1 a2 + 2b1 b2 + (a1 b2 + a2 b1 ) = 0. This implies a1 a2 + 2b1 b2 = a1 b2 + a2 b1 = 0. Now consider the inner product of Gray images: ψ1 (a1 + ub1 ) · ψ1 (a2 + ub2 ) = (b1 , a1 + b1 ) · (b2 , a2 + b2 ) = b1 b2 + a1 a2 + a1 b2 + b1 a2 + b1 b2 = 0. This proves that ψ1 (C ⊥ ) ⊆ ψ1 (C)⊥ . But since ψ1 is an isometry and C and ψ1 (C) have the same size, we have that |C| = |ψ1 (C)| = 16n/2 = 4n . It follows that ψ1 (C ⊥ ) = ψ1 (C)⊥ . Assume the theorem is true for Ti−1 . We know that Ti = Tk [ui ]/hu2i −2i. Any element of Ti (i−1) n is of the form a + ui b, where a, b ∈ Ti−1 . Take x1 + ui y1 , x2 + ui y2 ∈ Ti , x1 , x2 , y1 , y2 ∈ Z42 . We have [x1 + ui y1 , x2 + ui y2 ] = 0. From here, we get x1 x2 + 2y1 y2 + ui (x1 y2 + x2 y1 ) = 0. It follows that x1 x2 + 2y1 y2 = x1 y2 + x2 y1 = 0.

(5)

Since ψi is a linear map, from Equation (5), we get ψi (x1 + ui y1 ) · ψi (x2 + ui y2 ) = (ψi−1 (y1 ), ψi−1 (x1 ) + ψi−1 (y1 )) · (ψi−1 (y2 ), ψi−1 (x2 ) + ψi−1 (y2 )) = 2ψi−1 (y1 )ψi−1 (y2 ) + ψi−1 (x1 )ψi−1 (x2 ) + ψi−1 (x1 )ψi−1 (y2 ) + ψi−1 (y1 )ψi−1 (x2 ) = 0. Hence, we get ψi (C ⊥ ) ⊆ ψk (C)⊥ . Combining this with the fact that in Frobenius rings |C| · |C ⊥ | = |Tk |n , we get ψi (C ⊥ ) = ψi (C)⊥ . This completes the proof.

We have the following corollary: 6

Corollary 2.5. Let C be a self-dual code over R of length n, where R is Sk or Tk . Then ψk (C) is a self-dual code over Z4 of length 2k n. Proof. The proof follows immediately from Theorem 2.3 and Theorem 2.4.

We note that over Sk and Tk we have that φk+1 = φ ◦ ψk where φ is the standard Gray map from Z4 to F22 . This gives that the Lee weight enumerator of the quaternary code ψk (C) is the same as the Lee weight enumerator of φk+1 (C), where C is a code over Sk or Tk . We say that a code C over an X-ring is formally self-dual with respect to the Lee weight if LC (x, y) = LC ⊥ (x, y). Note that a self-dual code is necessarily formally self-dual. Theorem 2.6. Let C be a formally self-dual code of length n over Tk or Sk then ψk (C) is a formally self-dual code over Z4 of length 2k n and φk+1 (C) is a binary formally self-dual code. Proof. By the above discussion we have the following. If LC (x, y) = LC ⊥ (x, y) then Lψk (C) (x, y) = Lψk (C ⊥ ) (x, y) and Lφk+1 (C) (x, y) = Lφk+1 (C ⊥ ) (x, y).

This allows us to find the MacWilliams Identities for the Lee weight enumerators for codes over Tk and Sk . Theorem 2.7. Let C be a code over Sk or Tk . Then LC ⊥ (x, y) =

1 L (x |C| C

+ y, x − y).

Proof. We know that for quaternary codes (see [2]), the Lee weight enumerator satisfies the binary MacWilliams relations. Then using that fact that φk+1 = φ◦ψk and ψk (C ⊥ ) = ψk (C)⊥ we have the result. This gives the following corollary. Corollary 2.8. Let C be a formally self-dual code with respect to the Lee weight over Sk or Tk . Then the Hamming weight enumerator of φk+1 (C) is held invariant by the action of the MacWilliams relations. Notice that the image code φk+1 (C) is not necessarily formally self-dual as a binary code in the traditional sense since it may be non-linear. However, it has a weight enumerator which is held invariant by the action of the MacWilliams relations.

7

3

Self-dual codes

Now we have three families of rings which we can use to construct binary codes which have weight enumerators held invariant by the MacWilliams relations. Namely, Rk , Sk and Tk . In this section, we study the structure of self-dual codes over the family of rings Rk , Sk and Tk . We begin by some well-known definitions. A formally self-dual code is said to be Type II if the Lee weight of all elements of the code are 0 (mod 4). A formally self-dual code is said to be Type I if the Lee weight of all elements of the code are 0 (mod 2) but there exists at least one element whose Lee weight is not 0 (mod 4). A formally self-dual code is said to be Type 0 if it contains at least one codeword with odd Lee weight. Note that if the code is self-dual then it must be either Type I or Type II. However, the code C = {(x, 0)|x ∈ R} is an odd formally self-dual code of length 2 for any ring R where R is Rk , Sk or Tk since C ⊥ = {(0, x)|x ∈ R}. Hence Type 0 formally self-dual codes exist for all even lengths over Rk , Sk or Tk . A code C is free if it is isomorphic to Rm for some integer m where R is Rk , Sk or Tk . The following lemma is well-known. Lemma 3.1. Let R be one of the following: Rk , Sk or Tk . 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. We shall now determine when self-dual codes exist over these rings. Lemma 3.2. The code h2i is a self-dual code of length 1 over Sk or Tk . P P Proof. We prove the result for Tk , the proof for Sk is similar. Let cA uA , bB uB ∈ Tk . P P P P Then, [ cA uA , bB uB ] = dD uD where dD = cA0 bB 0 for some A0 , B 0 ∈ 2Z4 . But the product of any two elements in 2Z4 is 0. Therefore dD = 0, for all D. Then |h2i| = |2Tk | = |T2k | . Hence h2i is a self-dual code of length 1. It is Type I since 2 ∈ 2Tk and wtL (2) 6≡ 0 (mod 4). So h2i is a Type I self-dual code of length 1 over R. We also know from [6] that there is a self-dual code of length 1 over Rk . In fact, over Rk , hui i is a self-dual code of length 1 for i = 1, . . . , k. This is not the case for Sk or Tk since u2i 6= 0 in either case and hence not self-orthogonal. Theorem 3.3. There exist self-dual codes of all lengths over Rk , Sk or Tk . Proof. By Lemma 3.2, there exits a self-dual code of length 1. The result follows by taking cross products and applying Lemma 3.1. Note that these codes which are constructed in Theorem 3.3 are not free codes.

8

Consider the maximal ideal m in either Sk or Tk . Denote the ring by A. Then for α ∈ m, α ∈ 2A since u2i is either 2 or 2ui , and 22 = 0. Therefore m2 has characteristic 2. Then by Theorem 5.19 in [4], the code C = h{(a, a)|a ∈ m}, (0, 2u1 u2 · · · uk )i is a Type II self-dual code over Sk where u2i = 2ui . The code D = h{(a, a)|a ∈ m}, (0, 2u1 u2 · · · uk )i is a Type II self-dual code over Tk where u2i = 2. For the ring Rk , h1, 1 + u1 u2 + u3 · · · uk i generates a Type II self-dual code. This gives the following. 2

Theorem 3.4. Type II self-dual codes exist for all even length over Rk , Sk and Tk . Proof. The result follows from the previous explanation and taking cross products and applying Lemma 3.1. Note that this is not necessarily true for all X-rings but rather only for those X-rings with the property that m2 has characteristic 2. Consider the ring Sk . The following matrix generates a free self-dual code of length 8 over Sk :    A= 

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 3 0 1

1 1 3 0

1 0 1 3

0 1 1 1.

   . 

(6)

Note that the ring Sk does not have α, β with α2 + β 2 = −1. Theorem 3.5. There exist free self-dual codes of all lengths n ≡ 0 (mod 8) over Sk . Proof. There exits a free self-dual code of length 4 over Sk , given by (6). The result follows by taking cross products and applying Lemma 3.1. In the ring Tk we have u2i = 2 and 12 = 32 = 1. Therefore u2i + 12 = −1. Hence the matrix A=

1 0 ui 1 0 1 3 ui

! (7)

generates a free self-dual code of length 4 over Tk . Theorem 3.6. There exist free self-dual codes of all lengths n ≡ 0 (mod 4) over Tk . Proof. There exits a free self-dual code of length 4 over Tk , given by (7). The result follows by taking cross products and applying Lemma 3.1.

9

4

Constructions

The double circulant constructions for FSD codes were given in [8]. In [1], these constructions were extended to λ-circulant matrices. An n × n square matrix A is called λ-circulant if it is in the following form:   a1 a2 a3 · · · an   a1 a2 · · · an−1   λan   . λa λa a · · · a A= n−1 n 1 n−2   .. .. .. ..   .. . . . . .   λa2

λa3 λa4 · · ·

a1

In the sequel, let S be a commutative ring with identity, where the weight used satisfies wt (a) = wt (−a) for any element a ∈ S. The constructions are given in the following theorems. Theorem 4.1. [1] Let A be an n × n λ-circulant matrix then the code generated by G =  In A is a formally self-dual code over S. Theorem 4.2. [1] Let A be an n × n λ-circulant matrix then the code generated by   α β ··· β     β ∗  G =   In+1 .. . A   β

is a formally self-dual code over S. The construction given in Theorem 4.1 can be generalized further as follows: Theorem 4.3. Let λ be a unit in S. Let r be a vector of length n containing exactly one unit and let A be formedby λ-cyclic  shifts of r and B be an n × n λ-circulant matrix then the code generated by G = A B is a formally self-dual code over S. The code is self-dual when AAT + BB T = 0. 



then |C| = S n since A is equivalent  to In . The dual C ⊥ of C is generated by G0 = B T −AT since G (G0 )T = AB − BA = 0; 00 λ-circulant matrices   commute with each other. On the other hand the code C generated by G00 = B T AT is permutation equivalent to C since a λ-circulant matrix is permutation equivalent to its transpose when the permutation Proof. Let C be the code generated by G =

A B



πn = (1, n) (2, n − 1) · · · (k − 1, n − k + 2) (k, n − k + 1) 10

is applied to the rows and then to the columns of the matrix. The code C 00 and C ⊥ have the same weight enumerator since wt (a) = wt (−a) for any element a ∈ S. Hence C is a formally self-dual code. We shall be concerned with applying these constructions to the rings described earlier. We have the following lemma. Lemma 4.4. If a is an element of Rk , Sk or Tk then wtL (a) = wtL (−a). Proof. The result follows from a simple computation. Example 1. Let A and B be 1 + u-circulant matrices with first rows rA = (1 + u, u + v + uv, u + v, u, v) and rB = (v, 1 + u, uv, u, 1 + u) ,   respectively. Then by Theorem 4.3 the code over R2 that is generated by A B is a formally self-dual code by Theorem 4.3. The binary image of the code is a [40, 20, 9]b2 code with partial weight distribution 1 + 340z 9 + 972z 10 + 2120z 11 + · · · . The minimum distance of the code is better than the minimum distance of comparable self-dual codes. A non-linear binary code is said to be better than linear (BTL) if its minimum distance is better than the minimum distance of the best known linear code of the corresponding parameters. In the following example we give a construction for Nordstorm-Robinson code which is a (16, 28 , 6)2 BTL code. Example 2. Let A and B be 3-circulant matrices over T1 with first rows rA = (1 + u, 2 + u) and rB = (1 + u, 2) . Then, the code generated by 

A B



=

1+u 2+u 2 + 3u 1 + u

1+u 2 2 1+u

!

over T1 is a formally self-dual code. Moreover, AAT + BB T = 0 so by Theorem 4.3 the code is a self-dual code over T1 . The Z4 -image of the code is also self-dual with minimum Lee distance 6 and Lee weight enumerator 1 + 112z 6 + 30z 8 + 112z 10 + z 16 . The binary image of the code is Nordstrom-Robinson code, which is BTL. In the following tables we present computational results by applying these constructions to the rings described here. For the elements of R2 , u1 and u2 are replaced with u and v respectively. Similarly, u1 , u2 and u3 are replaced with u, v and z respectively in R3 . Throughout the tables ∗ is used to denote that the code is optimal as a linear code and b is used to denote its distance is the same as the best known linear code with respect to 11

Table 1: Formally self-dual codes over R2 by Theorem 4.1 n 4 5 6 7 8

λ 1+v 1+v 1+u 1+v 1+u

rA (v + uv, u + uv, u + v + uv, 1) (v, 0, 1 + u, 1 + u + v, u + uv) (1, 1 + u + v, u + v, uv, 1, u + uv) (1 + u + uv, 1, u + v + uv, 1 + u + uv, 0, v, uv) (1, u + v + uv, 1 + uv, u + v, 1, 1 + u, 0, 1 + u) (1, 1 + v, 1 + u, 1 + u, u + uv, 1 + uv, 9 1+v v + uv, v + uv, 1 + u + v) (u + v + uv, u + v, 1 + u, u, uv, 9 1+v 1 + u + v + uv, 1 + u, 1 + u + uv) (1 + u + v, 1 + v + uv, 0, 1 + u + v + uv, u, 10 1 + u 0, 1 + v + uv, 1 + uv, 1 + u + v + uv, 1 + v) (u, 1 + v + uv, u + v + uv, 1 + uv, 1 + uv, 11 1 + v 1 + v, u + v, 0, u + v, 1 + u, v + uv) (v, u + uv, 1, u + v + uv, 1 + u + v + uv, 11 1 1, 1 + uv, 1 + v, 1 + u + uv, 1, 1 + u) (u + v, 1 + u + v, 0, 1, 1, 0, 1 + uv, 0, 12 1 + v u + v + uv, u + uv, 1 + v, u + uv)

|Aut| 64 20 24 28 64

φ (C) [32, 16, 8]∗2 [40, 20, 9]b2 [48, 24, 10]2 [56, 28, 12]∗2 [64, 32, 12]b2

Paartial w.d. 01 8332 102304 . . . 01 9280 101052 . . . 01 10768 128624 . . . 01 124494 1445948 . . . 01 121488 1422528 . . .

E-O E O E E E

36

[72, 36, 13]2

01 131296 144626 . . .

O

36

[72, 36, 14]2

01 149180 16121194 . . .

E

40

[80, 40, 14]

01 142400 1649640 . . .

E

44

[88, 44, 14]

01 14704 1616698 . . .

E

44

[88, 44, 15]

01 151452 168646 . . .

O

48

[96, 48, 16]b2

01 164350 1696528 . . .

E

the online database [7]. As binary formally self-dual codes we were able to obtain optimal linear codes of lengths 32 and 56, best known linear codes of lengths 40, 64 and 96. More importantly, binary formally self-dual codes of lengths 40 and 72 with a better minimum distance than the self-dual codes at these lengths are constructed. Remark 1. Note that the formally self-dual codes constructed in Example 1 and Table 1 with parameters [40, 20, 9]2 , [72, 36, 13]2 and [72, 36, 14]4 have better minimum distances than the best known self-dual codes of these lengths.

Table 2: Formally self-dual codes over R3 by Theorem 4.1 n λ 4 1 + v + uv + uz 4 1+v 5

1 + v + z + vz + uvz

6

1+v

7

1 + u + z+ uz + vz + uvz)

rA |Aut| φ (C) (1 + v + z + uv + uvz, u, v, v + z + uv + uz) 16 [64, 32, 12]b2 (1 + v + z + uz + vz, 1 + u + v + vz, z + uv, z + uz + uvz) 32 [64, 32, 12]b2 (1 + v + z + uvz, v + vz, u + v + z + uv + vz + uvz, 16 [80, 40, 14]2 u + z + vz, u + v + uz + uvz) (1 + z + uz + uvz, u + v + uv + uz + uvz, uv + uz + vz, 96 [96, 48, 16]b2 u + uv + vz, u + v + z + uv + vz, z + uz + uvz) (u + v + z, uv + uz + uvz, 1 + z + uz + uvz, 1 + vz, 8 [112, 56, 16] u + uz + vz + uvz, 1 + uv + uz + uvz, u + v + z + uv + vz, v + uv + uz + uvz)

Partial w.d. 01 122016 1418688 . . . 01 12664 133232 . . .

E-O E O

01 142560 1648445 . . .

E

01 165562 1888256 . . .

E

01 16244 187760 . . .

E

In order to fit the upcoming tables, we abbreviate an element a + bu of T1 as ab and an element a + bu1 + cu2 + du1 u2 of S2 or T2 as abcd where a, b, c, d ∈ Z4 . Codes over S1 have been studied in detail in [9, 3] and the rings Sk and Tk are not suitable for computation when k > 2 due to their sizes. Therefore, we give computational results for T1 , T2 and S2 . 12

Table 3: Formally self-dual DC codes over Sk and Tk by Theorem 4.1 n 2 3 4 5 6 7 8 1 1 2 4 5 6 1 1 2 6

R T1 T1 T1 T1 T1 T1 T1 S2 S2 S2 S2 S2 S2 T2 T2 T2 T2

λ 32 11 32 30 30 10 10 − − 1223 3000 1000 1333 1020 3310

rA (32|03) (02|01|12) (10|01|03|02) (12|12|03|20|03) (00|23|31|21|30|11) (23|23|23|10|20|32|00) (31|01|23|01|10|03|10|20) (3311) (0112) (3223|2230) (2212|3321|3222|3120) (1333|0111|0133|3030|1110) (331|0321|0021|232|2321|031) (1230) (2232) (3232|0320) (2033|2111|2131|3111|0313|0330)

φZ4 (C) (8, 44 , 6L ) (12, 46 , 6L ) (16, 48 , 8L ) (20, 410 , 9L ) (24, 412 , 10L ) (28, 414 , 11L ) (32, 416 , 12L ) (8, 44 , 4L ) (8, 44 , 4L ) (16, 48 , 8L ) (32, 416 , 12L ) (40, 420 , 14L ) (48, 424 , 16L ) (8, 44 , 4L ) (8, 44 , 4L ) (16, 48 , 8L ) (48, 424 , 14L )

φ (C) BT L (16, 28 , 6)2 (24, 212 , 6)2 ∗ (32, 216 , 8)2 b (40, 220 , 9)2 (48, 224 , 10)2 (56, 228 , 11)2 b (64, 232 , 12)2 [16, 8, 4]2 (16, 28 , 4)2 ∗ (32, 216 , 8)2 b (64, 232 , 12)2 (80, 240 , 14)2 b (96, 248 , 16)2 [16, 8, 4]2 (16, 28 , 4)2 ∗ (32, 216 , 8)2 (96, 248 , 14)

Partial Lee w.d. 01 6112 830 . . . 01 668 8351 . . . 01 8348 102176 . . . 01 9280 101032 . . . 01 10768 128592 . . . 01 11560 122177 . . . 01 121456 1422272 . . . 01 428 8198 1228 . . . 01 414 58 624 . . . 01 8428 101536 . . . 01 121568 1422400 . . . 01 141960 1651190 . . . 01 164410 1894912 . . . 01 416 648 . . . 01 42 524 636 . . . 01 8396 101792 . . . 01 14164 164780 . . .

E-O E E E O E O E E O E E E E E O E E

In Table 4 formally self-dual non-linear binary codes are constructed as Gray images of FSD codes over Sk and Tk by bordered λ-circulant construction. Table 4: Formally self-dual codes over Sk and Tk by Theorem 4.2 n 1 1 2 3 4 5 6 7 1 3 1

R T1 T1 T1 T1 T1 T1 T1 T1 S2 S2 T2

λ − − 10 12 32 10 10 10 − 1002 −

rA (23) (21) (23|31) (00|21|01) (20|11|00|02) (01|00|31|12|02) (02|31|22|21|23|23) (13|11|12|13|22|20|03) (3231) (1110|0320|1000) (1020)

α, β 01, 23 23, 21 32, 23 20, 12 12, 21 02, 32 13, 12 22, 12 1121, 0033 2221, 1013 3002, 2322

φZ4 (C) (8, 44 , 4L ) (8, 44 , 6L ) (12, 46 , 6L ) (16, 48 , 8L ) (20, 410 , 8L ) (24, 412 , 10L ) (28, 414 , 10L ) (32, 416 , 12L ) (16, 48 , 8L ) (32, 416 , 12L ) (16, 48 , 8L )

φ (C) (16, 28 , 4)2 BT L (16, 28 , 6)2 (24, 212 , 6)2 ∗ (32, 216 , 8)2 (40, 220 , 8)2 (48, 224 , 10)2 (56, 228 , 10)2 b (64, 232 , 12)2 ∗ (32, 216 , 8)2 b (64, 232 , 12)2 ∗ (32, 216 , 8)2

Partial Lee w.d. 01 414 58 . . . 01 6112 8130 . . . 01 656 8423 . . . 01 8372 101984 . . . 01 8173 101632 . . . 01 10730 128836 . . . 01 10124 11564 . . . 01 121390 1422712 . . . 01 8380 101920 . . . 01 121576 1421792 . . . 01 8316 102432 . . .

E-O O E E E E E O E E E E

References [1] A. Batoul, K. Guenda, A. Kaya, B. Yildiz, Cyclic Isodual and Formally Self-dual Codes over Fq + vFq , European Journal of Pure and Applied Mathematics, 8(1), 64 - 80, 2015. 13

[2] A.R. Calderbank, A.R. Hammons, P. V. Kumar, N.J.A. Sloane, P. Sol´e, A linear construction for certain Kerdock and Preparata codes. Bull. Amer. Math. Soc., 29(2), 218222, 1993. [3] S.T. Dougherty, E. Salturk and S. Szabo, Codes Over Local Rings of Order 16 and Binary Codes, in submission. [4] S.T. Dougherty, E. Salturk Codes Over Local Rings, Gray Maps and Self-Dual Codes, in submission. [5] S.T. Dougherty, B. Yildiz, and S. Karadeniz, Codes over Rk , Gray maps and their Binary Images, Finite Fields and their Applications, 17(3), 205 - 219, 2011. [6] S.T. Dougherty, B. Yildiz, and S. Karadeniz, Self-dual Codes over Rk and Binary SelfDual Codes, European Journal of Pure and Applied Mathematics, 6(1), 2013. [7] M. Grassl, “Bounds on the minimum distance of linear codes and quantum codes”, Online available at http://codetables.de, accessed on 20.05. 2015. [8] S. Karadeniz, B. Yildiz, Double-Circulant and Bordered-Double-Circulant Constructions for self-dual codes over R2 , Advances in Mathematics of Communications, 6(2), 193 - 202, 2012. [9] E.M. Moro, S. Szabo, B. Yildiz, Linear Codes Over Z4 [x] / hx2 + 2xi, Int. J. Information and Coding Theory, 3(1), 78-96, 2015.

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