Constructing Formally Self-Dual Codes over Rk Suat Karadeniz Department of Mathematics Fatih University 34500 Istanbul, Turkey e-mail: [email protected] Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA e-mail: [email protected] Bahattin Yildiz ∗ Department of Mathematics Fatih University 34500 Istanbul, Turkey e-mail: [email protected] May 21, 2013

Abstract In this work, we study construction techniques of formally self-dual codes over the infinite family of rings Rk = F2 [u1 , u2 , . . . , uk ]/⟨u2i = 0, ui uj = uj ui ⟩. These codes give rise to binary formally self-dual codes. Using these constructions, we obtain a number of good formally self-dual binary codes including even formally self-dual binary codes of parameters [72, 36, 14], [56, 28, 12], [44, 22, 10] and odd formally self-dual binary codes ∗

Corresponding author Key words: formally self-dual codes, codes over Rk , extremal codes, codes over rings. MSC 2000 Classification: Primary 94B05, Secondary 13M05

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of parameters [72, 36, 13], all of which have better minimum distances than the best known self-dual codes of the same lengths.

1

Introduction

Formally self-dual codes are an interesting type of codes that have been studied quite extensively by many researchers. For some of these works we refer the reader to [4], [7], [8], [9], [10], [13] and [14]. Formally self-dual codes can have larger minimum distances than self-dual codes, which makes them of interest in searching for good codes. Since the weight enumerators of formally self-dual codes come from the same ring of invariants as the weight enumerators of self-dual codes, the Assmus-Mattson theorem can often be used to construct many new designs. There are a few different constructions for binary formally self-dual codes. Of particular interest is a series of constructions given as an exercise in [11]. These and variations of these were used in the literature by different researchers. Codes over rings have been a topic of great interest in the last two decades. Certain rings have been successfully used to obtain good binary codes with different properties. The ring F2 + uF2 was generalized first to F2 + uF2 + vF2 + uvF2 in [17] and then to Rk in [2]. The rich algebraic structure of these rings have been used quite effectively to obtain good binary codes with large automorphism groups as well as some new binary self-dual codes (see [12]). In this work, we apply the construction methods given in [11] to the ring Rk to construct formally self-dual codes over Rk . These codes result in binary formally self-dual codes with good parameters after taking the image under a weight-preserving Gray map. The rest of the paper is organized as follows: In section 2, we give the preliminaries about codes over the ring Rk as well as some of the definitions associated with formally self-dual codes. In section 3, we give constructions of formally self-dual codes from special types of matrices and prove the theoretical results. Section 4 includes the computational results about the codes constructed via the methods given in the previous section. The results are given in the form of tables.

2 2.1

Preliminaries Formally self-dual codes

We begin with the following definitions. On the binary space Fn2 , with v, w ∈ Fn2 , attach ∑ the usual inner-product [v, w] = vi wi and define C ⊥ = {w | [w, v] = 0 ∀v ∈ C}. We make the usual definition of the Hamming weight enumerator, namely WC (x, y) =

2

∑ v∈C

xn−wt(v) y wt(v) where wt(v) is the the number of non-zero coordinates of v.

Definition 1. A binary code C is called self-dual if C = C ⊥ . It is called isodual if C is equivalent to C ⊥ . The code C is called formally self-dual if WC (x, y) = WC ⊥ (x, y), that is C and C ⊥ have the same weight enumerators. A code is called even if all the weights are even, it is called odd (Type 0) otherwise. If an even code has all weights 0 (mod 4) then the code is said to be doubly-even (Type II), otherwise it is said to be singly-even (Type I). From the definitions it follows immediately that self-dual and isodual codes are formally self-dual. But, there are formally self-dual codes which are not self-dual. Note however that the weight enumerator of an even formally self-dual code is held invariant by the same matrices, and hence the same Gleason theorem applies. Namely we have the following. Let C be a formally self-dual code. Then, • WC (x, y) ∈ C[x2 + y 2 , y(x − y)], if C is Type 0, • WC (x, y) ∈ C[x2 + y 2 , x8 + 14x4 y 4 + y 8 ], if C is Type I, • WC (x, y) ∈ C[x8 + 14x4 y 4 + y 8 , x4 y 4 (x4 − y 4 )4 ], if C is Type II. For the remainder we let x = 1 when giving the Hamming weight enumerator. From [5], we know that if C is a binary formally self-dual code of length 2n, and d is the minimum Hamming weight of C, then ⌊n⌋ d≤2 + 2. 4 Formally self-dual codes meeting this bound are called extremal. Formally self-dual codes ⌊ ⌋ for which d = 2 n4 are called near-extremal. In [14], the authors conjecture that there are no near-extremal formally self-dual even binary codes of length n ≥ 48 with 8 | n.

2.2

The ring Rk and the properties

Define the following ring for k ≥ 1. Let Rk = F2 [u1 , u2 , . . . , uk ]/⟨u2i = 0, ui uj = uj ui ⟩. For any subset A ⊆ {1, 2, . . . , k} let uA :=

∏

ui

(1)

(2)

i∈A

with the convention that u∅ = 1. Then any element of Rk can be represented as ∑ cA uA , c A ∈ F2 . A⊆{1,...,k}

3

(3)

The ring Rk is a local ring with maximal ideal ⟨u1 , u2 , . . . , uk ⟩ and | Rk | = 2(2 ) . The ring is neither a principal ideal ring nor a chain ring when k ≥ 2. The ring is however a Frobenius ring. The rings R0 = F2 and R1 = F2 + uF2 have been studied quite extensively in the literature of coding theory. The ring R2 = F2 + uF2 + vF2 + uvF2 was first introduced by Yildiz and Karadeniz in [17]. In [2], it is shown that an element of Rk is a unit if and only if the coefficient of u∅ is 1 and that each unit is also its own inverse. We also have the following: { u1 u2 . . . uk if a is a unit ∀ a ∈ Rk , a · (u1 u2 . . . uk ) = (4) 0 otherwise. k

{ Also, ∀a ∈ Rk a2 =

1 if a is a unit 0 otherwise.

(5)

See [2] for proofs of these and other foundational results for the ring Rk . A linear code of length n over Rk is defined to be an Rk -submodule of Rkn . We denote a vector by a. We attach the usual inner product on this ambient space Rkn , ∑ that is ⟨a, b⟩k = ai bi . The dual code C ⊥ is defined by C ⊥ = {y ∈ Rkn | ⟨y, x⟩k = 0 for all x ∈ C}. We say that a code is self-orthogonal if C ⊆ C ⊥ and self-dual if C = C ⊥ . Since the ring Rk is Frobenius, for any linear code C we have that |C | |C ⊥ | = | Rk |n , see [16]. We say that a code has free rank s if it is isomorphic to s direct products of Rk , that is it is isomorphic to Rks . We define the Gray map inductively, extending it naturally from the Gray map on R1 as follows. n , then we can define For c ∈ Rkn , write c = c1 + uk c2 with c1 , c2 ∈ Rk−1 ϕk (c) = (ϕk −1 (c2 ), ϕk −1 (c1 ) + ϕk −1 (c2 )) , with ϕ0 being the identity map on F2 . For example on R1 we have ϕ1 (c1 + uc2 ) = (c2 , c1 + c2 ) and on R2 , ϕ2 (a + ub + vc + uvd) = (d, c + d, b + d, a + b + c + d). The Lee weight wL of a codeword is the Hamming weight of the image of the codeword k under ϕk . Then the Gray map is a linear weight preserving map from Rkn to F22 n and it is immediate that ϕk is one-to-one and that wL (uA ) = 2 |A| for each A ⊆ {1, 2, . . . , k}, see [2] for details. For a code C over Rk we define the Lee weight enumerator to be W LC (y) = ∑ wL (v) . v∈C y The subgroup Ub = ⟨1 + u1 , 1 + u2 , · · · , 1 + uk ⟩ of the unit group of Rk is called the subgroup of basic units. In [2], it is shown that multiplying by basic units corresponds to a combination of swap maps in the gray image, thus we have the following lemma:

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Lemma 2.1. (1) The Lee weight of each basic unit is 1. In fact any element in Rk of Lee weight 1 must be a basic unit. (2) If for α, β ∈ Rk , we have α = r · β for some r ∈ Ub , then wL (α) = wL (β). (3) If for α, β ∈ Rk , we have α = r · β for some unit r in Rk and wL (α) = wL (β), then r ∈ Ub . The following lemma, which was generalized from [17] in [3] serves as a fundamental tool for our work: Lemma 2.2. ([3]) If C is a linear code over Rk of length n, then ϕk (C ⊥ ) = ϕk (C)⊥ . Using the facts that ϕk is a distance-preserving map and that the ring Rk is a Frobenius ring which implies it has MacWilliams identities, we have the following corollary. Corollary 2.3. If C is a formally self-dual code over Rk of length n, then ϕk (C) is a binary formally self-dual code of length 2k n with W LC (y) = Wϕk (C) (y), that is the Hamming distribution of the Gray image is the same as the Lee distribution of C. We end this section by the following corollary of Lemma 2.2: Corollary 2.4. If C is an isodual code over Rk of length n, then ϕk (C) is a binary isodual code of length 2k n. Proof. We will just give the proof for the case when k = 1. The general case follows easily from the inductive construction of ϕk . Assume that a + ub ∈ R1n and let σ ∈ Sn be a permutation of {1, 2, . . . , n}. Then ϕ1 (σ(a + ub)) = ϕ1 (σ(a) + uσ(b)) = (σ(b), σ(a) + σ(b)) = (σ(b), σ(a + b)) which means ϕ1 (σ(c)) = τ (ϕ1 (c)) where τ ∈ S2n acts on F2n 2 as τ (x, y) = (σ(x), σ(y)). n This tells us that if c1 and c2 in R1 are equivalent, then their binary images under ϕ1 are equivalent. The result then follows from Lemma 2.2.

3

Constructing Formally Self-Dual codes over Rk

In [11], page 378, Huffman and Pless give an exercise asking to prove the following for binary codes.

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Theorem 3.1. Let A be a binary matrix and let Ik be the k by k identity matrix, then the following hold: (a) A code of even length n with generator matrix [In/2 | A] where A = AT is isodual. (b) A code with a double-circulant construction, i.e., a code with generating matrix [In/2 | A], A being a circulant matrix, is isodual. (c) A code with a bordered double-circulant construction, i.e., a code with generating matrix   α β β . . . β   γ     γ       G= In/2 . , A   .       .   γ where A is circulant is isodual provided β = γ = 0 or both β and γ are non-zero. In general, the proof of Theorem 3.1 for binary codes are much easier in that every code has a generator matrix that is in the standard form and from that form we can deduce the generator matrix of its orthogonal. However this is not the case for codes over Rk , but we can generalize the proofs in our case. We now generalize these constructions of formally self-dual codes to the ring Rk : Theorem 3.2. (Construction A) Let A be an n × n matrix over Rk such that AT = A. Then the code generated by the matrix [In | A] is an isodual code and hence a formally self-dual code of length 2n. Proof. Consider the matrix [AT | In ] = [A | In ] = G′ and let G = [In | A]. Both G and G′ generate codes with free rank n. Moreover the codes generated are permutation equivalent. So if C = ⟨G⟩ and C ′ = ⟨G′ ⟩, all we need to do to finish the proof is to show that C ′ = C ⊥ . Let v be the i-th row of G and w be the j-th row of G′ . Then ⟨v, w⟩k = Aji + Aij = 0 since Rk has characteristic 2 and AT = A. Therefore C ′ = C ⊥ and C is equivalent to C ′ = C ⊥. Combining this with Corollary 2.3, we immediately obtain the following result. Corollary 3.3. If C is a linear code over Rk of length 2n, generated by [In | A], where A = A⊥ , then ϕk (C) is a binary formally self-dual code of length 2k+1 n. Theorem 3.4. (Construction B) Let M be a circulant matrix over Rk of order n. Then the matrix [In | M ] generates an isodual code and hence a formally self-dual code over Rk .

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Proof. Let C be the code generated by [In | M ] = G and let C ′ be the code generated by [M T | In ] = G′ . Both codes have free rank n. We will first show that C ′ = C ⊥ . Let v be the i-th row of G and w be the j-th row of G′ . Then ⟨v, w⟩k = MjiT + Mij = Mij + Mij = 0 since the ring Rk has characteristic 2. Therefore C and C ′ are orthogonal, but since they both have free rank n, we have C ′ = C ⊥ . Next we show that C is equivalent to C ′ . There exists a permutation σ of rows such that after applying it to G′ , the first column of σ(M T ) is the same as the first column of M . Namely, T T T (M11 , M21 , · · · , Mn1 ) = (Mσ(1)1 , Mσ(2)1 , · · · , Mσ(n)1 ) = (M1σ(1) , M1σ(2) , · · · , M1σ(n) ).

Since the matrix M is circulant, every column of M is then equal to a column of σ(M T ). Then apply the necessary column permutation τ so that τ (σ(M T )) = M . We apply another column permutation ν so that the monomial matrix found by σ(In ) is returned to identity. Note that τ does not affect this part. Thus we obtain G from G′ by the consecutive applications of the permutations σ, τ and ν, which means that C is equivalent to C ′ = C ⊥ , hence the codes are isodual. Corollary 3.5. If C is a linear code over Rk of length 2n, generated by [In | M ], where M is a circulant matrix, then ϕk (C) is a binary formally self-dual code of length 2k+1 n. Remark 1. The codes obtained from Construction B are also quasi-cyclic codes of order 2. Theorem 3.6. (Construction C) Let M be a circulant matrix over Rk of order n − 1. Then the matrix   α β β ... β   γ       I γ M n , G=   .     .   γ where α ∈ Rk , ⟨γ⟩ = ⟨β⟩, and wL (γ) = wL (β) generates a formally self-dual code over Rk . Proof. Let G′ be given as

     G′ =     

α γ γ ... γ β β MT . . β



In

    .    

Set C = ⟨G⟩ and C ′ = ⟨G′ ⟩. Both C and C ′ are codes of free rank n. Let Gi denote the i-th row of G and G′j denote the j-th row of G′ . Then ⟨G1 , G′1 ⟩k = α + α = 0. For j > 1, we have 7

⟨G1 , G′j ⟩k = β + β = 0. Similarly, ⟨Gj , G′1 ⟩k = γ + γ = 0 for any j > 1. Finally, for i, j ≥ 0, we have ⟨Gi+1 , G′j+1 ⟩k = MjiT + Mij = Mij + Mij = 0. Hence we see that C ′ = C ⊥ . By the same method that was used in the previous construction, the parts of G and G′ except the γ and β can be made equivalent. Hence we can assume that C ⊥ is equivalent to a code C ∗ generated by a matrix of the form   α γ γ ... γ   β       I β M n ∗ . G =   .     .   β Now since ⟨γ⟩ = ⟨β⟩ we have γ = x · β and β = y · γ for some x, y ∈ Rk . Thus we get γ = xy · γ or γ(1 + xy) = 0 in Rk . Hence, if we assume γ ̸= 0, this implies 1 + xy must be a non-unit in Rk , implying that both x and y must be a unit. Hence we have γ = r · β for some unit r in Rk . But since wL (β) = wL (γ), this implies by Lemma 2.1, that r ∈ Ub . Now, multiply all but the first row of G∗ by r. The resulting matrix will generate the same code as C ∗ . Take any linear combination of the rows of this matrix and take the same linear combination of the rows of G, call the resulting codewords c∗ and c, respectively. Because wL (r · α) = wL (α) for all α ∈ Rk , we see that the first n + 1 coordinates of c and c∗ have the same Lee weights. Now, a typical element in one of the last n − 1 coordinates of c is of the form x · β + y · m, and the corresponding element of c∗ is x · γ + y · rm, where x, y ∈ Rk and m is a linear combination of the elements of the matrix M in the same coordinate. But x · γ + y · rm = r(x · β + y · m) and thus by Lemma 2.1, they have the same Lee weight. This shows that the weight enumerator of C and C ∗ are the same. But since C ∗ is equivalent to C ⊥ , this proves that C and C ⊥ have the same weight enumerator. Corollary 3.7. If C is a linear code over Rk of length 2n, generated by the matrix given in Theorem 3.6, then ϕk (C) is a binary formally self-dual code of length 2k+1 n that is not necessarily isodual. Remark 2. The binary images of codes obtained from constructions A and B are isodual because of Corollary 2.4. However as the codes obtained in construction C are not isodual over Rk , their binary images are not isodual in general. Remark 3. It is worth noting that the proofs of Constructions A-C are also valid over any ring of characteristic 2. This makes the constructions applicable to many other rings. Our next theorem gives a slightly different construction than the previous ones. Note that this is the only construction in this work that will give us odd length formally self-dual codes over Rk . 8

Theorem 3.8. (Construction D) Let G be the matrix  x  .    In . M   .   x  α . . . α . . . α

     ,    

where M is a circulant matrix of order n, x ∈ ⟨α⟩, and α ∈ Rk is chosen so that α generates a self-dual code of length 1. Moreover, we assume that the row sum of M is in the coset 1 + ⟨α⟩. Then G generates an isodual code of length 2n + 1 over Rk . Proof. Let C be the code generated by G and assume C ′ is the code generated by the matrix   x   .     T   M . I n ′ . G =   .     x   α . . . α . . . α We first observe that both C and C ′ are of length 2n + 1 and of size | Rk | n+1/2 = √ | Rk | 2n+1 . To prove that C ′ = C ⊥ , all we need to do is to show that they are orthogonal. By Gi and G′j we denote the i-th and j-th row of G and G′ respectively. Then we see that ⟨Gn+1 , G′n+1 ⟩k = 0 since ⟨α⟩ is self-dual. For i < n + 1, we have ⟨Gi , G′n+1 ⟩k

= α + αx +

n ∑

αMiℓ = α + α2 t + α(1 + αs) = α + α = 0

ℓ=1

because x ∈ ⟨α⟩ and the all α2 = 0. Similarly, because every column of M also consists of the same elements as the first row of M , we have ⟨Gn+1 , G′i ⟩k = 0,

i = 1, 2, . . . , n.

Finally for i, j < n + 1, we have ⟨Gi , G′j ⟩k = MjiT + x2 + Mij = 0 since x is a non-unit and the characteristic of the ring is 2. Thus we have shown that C ′ = C ⊥ . Now, by applying row and column permutations similar to the previous theorem, we can obtain G from G′ , which means that both C and C ′ = C ⊥ are equivalent, thus completing the proof. 9

Corollary 3.9. Let C be a code over Rk of length 2n + 1, generated by the matrix given in theorem 3.8. Then ϕk (C) is a binary formally self-dual code of length (2n + 1)2k . Moreover, the conditions given in the theorem imply that there has to be an even number of units in every row of G, thus the formally self-dual codes obtained via this construction are all even codes. Corollary 3.10. Let C be a code over R1 = F2 + uF2 of length 2n + 1, generated by   u   .       I . M n , G=   .     u   u . . . u . . . u where M is a circulant matrix of order n, whose first row consists of an odd number of units. Then C is an isodual code of length 2n + 1. Remark 4. Note that it is relatively easy to determine when the formally self-dual codes obtained by these constructions are even or odd. It is well known that units in Rk have odd weights and non-units have even weights ([2]). Since the parity of the Lee weight is preserved in linear operations, we know that a code generated by a matrix G over Rk is even if and only if all the rows of G have an even number of units. So for example, in the case of Construction B, we can easily say that a formally self-dual code obtained from Construction B is even if and only if the first row of M has an odd number of units. Similar characterizations are possible for the other constructions as well. By using the identity (4), what is explained in the previous remark can be reformulated in the following theorem: Theorem 3.11. Let C be a formally self-dual code over Rk of length n. Then C is even if and only if the all (u1 u2 · · · uk )-vector is in C.

4

Computational Results

In what follows we will give some of the computational results that we have obtained. We have used the magma computation algebra ([1]) to search for good formally self-dual codes coming from the constructions that we have described in the previous section. For computational purposes we only focus on Rk for k = 1 and k = 2. Note that we denote R1 = F2 + uF2 and R2 = F2 + uF2 + vF2 + uvF2 . We will divide the constructions into sections according to the construction method used. 10

4.1

Codes from Construction A

The symmetric matrices have many parameters, consequently expanding the search field. Our searches on symmetric matrices did result in a substantial number of formally self-dual codes of different parameters. Since the results obtained from the other constructions had better parameters, we did not put these in a table. For example, we did obtain odd binary formally self-dual codes with parameters [64, 32, 10], [60, 30, 10], [56, 28, 9] and [40, 20, 7].

4.2

Good formally self-dual codes from Construction B

We present in the following, some good binary formally self-dual codes obtained from formally self-dual codes constructed by Construction B over R1 and R2 , i.e., by the purely double circulant constructions. So the code C is generated over Rk (k = 1, 2) by a matrix of the form [In |M ] where M is a circulant matrix, hence in the tables we will only give the first row of M . We indicate with a ∗ next to the parameters to imply that the code obtained is optimal(i.e. theoretical upper bounds are achieved), and b next to the parameter denotes that the binary code is the best known linear code of that length and dimension, according to the latest database kept in [6].

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Table 1: Good f.s.d binary codes obtained from R1 by Construction B First Row of M ϕ1 (C) |Aut(ϕ1 (C))| Partial Weight Dist. ∗ (0, u, u, u, u, 1 + u, u, 1, [44, 22, 10] 22 1 + 342 z 10 + 10285 z 12 + · · · 1 + u, 1, 1) (0, 0, 1, u, 1 + u, 1, 1, 1, [44, 22, 9] 22 1 + 132 z 9 + 638 z 10 + · · · 1 + u, 1, 1 + u) (0, 0, u, u, 0, 0, 1 + u, 0, [52, 26, 10]b 26 1 + 338 z 10 + 6643 z 12 + · · · 1, 1, 1 + u, 0, 1) (u, 0, 0, 0, 0, u, 1, 0, 1, [52, 26, 10]b 52 1 + 416 z 10 + 6409 z 12 + · · · 1 + u, 1, 0, 1) (u, u, 0, u, 0, 0, 1, u, 1 + [52, 26, 10]b 26 1 + 182 z 10 + 806 z 11 + · · · u, 1, 1 + u, 1 + u, 1 + u) (0, u, u, u, u, 0, 1 + u, 0, [56, 28, 12]b 28 1 + 4704 z 12 + 44268 z 14 + · · · 1 + u, 1, 1, 1 + u, 1 + u, 1) (0, u, 0, 0, u, 1, 0, 0, 1, [56, 28, 11] 28 1 + 504 z 11 + 2352 z 12 + · · · 1 + u, 1, 1, u, 1 + u) (0, u, 0, 0, u, u, u, u, 1, [60, 30, 12]b 30 1 + 2735 z 12 + 32520 z 14 + · · · 1, 0, 0, 1 + u, 1, 1) (0, 0, u, u, u, u, 0, u, 1, [60, 30, 11] 30 1 + 360z 11 + 1110z 12 + · · · 0, 1, 1, 1 + u, 1 + u, 1) (u, u, u, u, 0, u, u, 1 + u, [64, 32, 12]b 32 1 + 1744 z 12 + 21312 z 14 + · · · 0, 1 + u, 1 + u, u, 1 + u, 1 + u, 1 + u, 1) (0, 0, 0, 0, 1 + u, 0, u, 1, [64, 32, 12]b 32 1 + 928 z 12 + 2464 z 13 + · · · 1 + u, 1 + u, 1, 1, 1, u, 0, 1) (u, u, u, 0, u, u, u, u, u, [72, 36, 14] 36 1 + 8748 z 14 + · · · 1 + u, 0, 1 + u, 1, 1, 1, 0, 1 + u, 1) (0, 0, 0, u, 0, 0, u, 0, 1+u, [72, 36, 13] 36 1 + 828 z 13 + 4482 z 14 + · · · 0, 1 + u, 1, 1 + u, 1 + u, 1 + u, u, 1 + u, 1 + u)

12

E-O E O E E O E

O E O E

O

E

O

Table 2: Good First Row of M (uv, u + v + uv, u + v, 1 + u + v, u, 1 + uv, 1 + u + v + uv) (v, u + v, u + v + uv, 1 + v + uv, 1 + u + uv, 1, 1 + u + v) (uv, uv, v + uv, 1 + u, u + v + uv, 1 + u + uv, u+v+uv, 1+u+v+uv) (v + uv, u + uv, u + uv, v, 1 + u, u + uv, 1, 1 + u + v) (v, u, 0, 1 + u, u + v + uv, 1 + uv, 1 + u + uv, 1) (v, v, 1 + u + v + uv, u + v, uv, 1 + v, 1 + v, 1 + v, 1 + u + v) (u + v, u, v, u + uv, 1 + v, u + v, 1 + uv, 1 + uv, 1 + v)

f.s.d binary codes obtained from R2 by Construction B ϕ2 (C) |Aut(ϕ2 (C))| Partial Weight Dist. b [56, 28, 12] 28 1 + 4718 z 12 + 44160 z 14 + · · ·

E-O E

[56, 28, 11]

28

1 + 588 z 11 + 2240 z 12 + · · ·

O

[64, 32, 12]b

64

1 + 1504 z 12 + 20736 z 14 + · · ·

E

[64, 32, 12]b

32

1 + 14888 z 12 + 22336 z 14 + · · ·

E

[64, 32, 12]b

64

1 + 616 z 12 + 3392 z 13 + · · ·

O

[72, 36, 14]

36

1 + 9180 z 14 + · · ·

E

[72, 36, 13]

36

1 + 1296 z 13 + 4662 z 14 + · · ·

O

Remark 5. In some lengths we obtained a lot more codes than that are displayed here. For example in lengths 52 and 64 we obtained thousands of inequivalent formally self-dual codes with the same parameters, with mostly different weight enumerators. We managed to obtain these numerous results in a relatively short time (in minutes) from both R1 and R2 . This shows the strength of our constructions. Remark 6. Note that in the case of the lengths 44, 56 and 72 we have managed to obtain formally self-dual codes with better minimum distances than best known self-dual codes of the same length.([15]).

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4.3

Good formally self-dual codes from Construction C

We let C be a code generated by a matrix of the form   α β β ... β   γ       I γ M n  ,   .     .   γ where M is an (n − 1) × (n − 1) circulant matrix. We take β and γ to be both units in Rk . In R1 we have no further constraints, however in R2 , we require that β and γ have the same Lee weight. The tables are similar to the ones we gave in the previous section with the exception that we have added (α, β, γ) as one of the columns:

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Table 3: Good f.s.d binary codes obtained from R1 by Construction C (α, β, γ) First Row of M ϕ1 (C) |Aut(ϕ1 (C))| Partial Weight E-O Dist. ∗ (1 + u, 1 + u, 1 + u) (u, u, 0, u, 1, 0, u, 1 + [44, 22, 10] 20 1 + 1310 z 10 + E 12 u, 1 + u, 1 + u) 10541 z + · · · (0, 1, 1) (u, u, 0, 0, 1, 1, 0, 1 + [44, 22, 9] 20 1 + 120 z 9 + O u, 1, 1 + u) 660 z 10 + · · · (0, 1 + u, 1) (u, u, 0, 1, 0, 1, 1 + u, [48, 24, 10] 22 1 + 880 z 10 + E 12 u, 1 + u, 1 + u,1) 7792 z + · · · (0, 1 + u, 1 + u) (0, u, 0, u, 0, 1 + u, 1, [48, 24, 10] 22 1 + 550 z 10 + O 0, 1, 1, 1 + u) 1122 z 11 + · · · (u, 1 + u, 1 + u) (u, u, 0, 0, u, u, 0, 1 + [52, 26, 10]b 24 1 + 228 z 10 + O u, u, 1 + u, 1, 1) 582 z 11 + · · · (1 + u, 1, 1) (u, u, u, 0, u, u, u, 1 + [52, 26, 10]b 24 1 + 474 z 10 + E 12 u, u, 1 + u, 1, 1) 6076 z + · · · b (u, 1, 1 + u) (u, 0, u, u, u, 0, u, u, [56, 28, 12] 26 1 + 4693 z 12 + E 1, 1, 0, 1 + u, 1 + u) 44358 z 14 + · · · (1 + u, 1, 1 + u) (0, 0, 0, 0, 1, 0, u, u, [60, 30, 12]b 28 1 + 2646 z 12 + E 14 1 + u, u, 1 + u, 1 + u, 33040 z + · · · 1, u) (1, 1, 1 + u) (u, u, 0, 0, u, 1 + u, u, [60, 30, 11] 28 1 + 308 z 11 + O u, 1 + u, 0, u, 1, 1, 1) 1568 z 12 + · · · (0, 1, 1) (u, u, 0, 0, u, u, 0, 1, [64, 32, 12]b 30 1 + 1206 z 12 + E 14 0, 1, 1 + u, 0, 1, 1, 1) 23500 z + · · · b (u, 1, 1) (u, u, 0, 1+u, 0, 1+u, [64, 32, 12] 30 1 + 1140z 12 + O 1, 1 + u, u, 1 + u, u, 0, 2160z 13 + · · · 1, 1 + u, u) (0, 1, 1 + u) (u, u, 0, 0, 0, u, u, u, [72, 36, 14] 34 1+9350 z 14 +· · · E 0, 1, 0, 1, 1 + u, 0, 1, 1 + u, 1) (0, 1, 1) (u, u, 0, u, 0, 0, 0, u, [72, 36, 13] 34 1 + 612 z 13 + O 1 + u, u, 1 + u, 1, u, 6086 z 14 + · · · 1 + u, 1 + u, 1, 1 + u)

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Table 4: Good f.s.d binary codes obtained from R2 by Construction C (α, β, γ) First Row of M ϕ2 (C) |Aut(ϕ2 (C))| Partial Weight E-O Dist. (u + v + uv, 1 + (v + uv, 1 + u, 1 + uv, [48, 24, 10] 20 1 + 624 z 10 + E 12 u + v, 1 + uv) 1, 1 + u + v + uv) 9520 z + · · · (1 + u, 1 + v + (v, 1 + v + uv, 1 + uv, [48, 24, 10] 20 1 + 404 z 10 + O uv, 1 + v + uv) 1 + u + v, 1) 1524 z 11 + · · · (1 + v, 1 + u + (u + uv, uv, 1 + u + uv, [56, 28, 11] 48 1 + 352 z 11 + O uv, 1 + u + uv) u + v + uv, 1 + u + v, 2268 z 12 + · · · 1 + v) (1+u+v+uv, 1+ (v, u + v + uv, v + uv, [64, 32, 12]b 28 1 + 788 z 12 + O u + uv, 1 + u + v) 1 + u + v, 0, 1 + u + v, 2888 z 13 + · · · 1 + v + uv) (u + v + uv, 1 + (u, 1 + uv, 1 + u, 1, [64, 32, 12]b 28 1 + 1940 z 12 + E 14 u + v, 1 + u + v) 1 + u + v + uv, 1 + uv, 20272 z + · · · 1 + uv) Remark 7. There is not much work about formally self-dual codes of the lengths that we have studied in the literature to make a comparison. It is computationally quite difficult to obtain binary formally self-dual codes of high lengths. The ease with which we obtained the codes given in these tables illustrates another advantage of our constructions and the rings that we use.

4.4

Formally self-dual codes from Construction D

We take the matrix

     G=    

In

u .

.

u . . M . u . u . . . u

         

over R1 and use this to generate formally self-dual codes of odd lengths over R1 , leading to binary formally self-dual codes of length 4n + 2. However we have to point out that, as we are taking an odd number of units in the first row of M , we will have an odd number of units in each column of M as well. Thus, when the sum of the first n rows of G is multiplied by u, we will get the codeword (u, u, . . . , u, 0, u, u, . . . , u). Adding this to the last row will result in a codeword of the form (0, 0, . . . , 0, u, 0, . . . , 0). Hence all formally self-dual codes obtained over R1 from Construction D will have minimum weight 2. Similarly when construction D 16

is applied to R2 , the minimum weights will always be 4. Although we do not get binary formally self-dual codes with high minimum distances, we do however see a curious situation. In all the constructions that we have, there is only one codeword of weight 2, and then there is a large jump to the next weight. We illustrate this on certain examples: Table 5: Even First Row of M (0, 0, u, 0, 0, 1, u, 1 + u, 1, 1, 1 + u) (0, u, 0, 1 + u, u, 1, 1 + u, 1, 1 + u, 0, 1 + u, 1) (0, 0, u, 0, u, 0, u, 1, 0, 1, 1, 1, 1) (u, 0, u, u, 0, 1 + u, u, u, 1, u, 0, 1, 1 + u, 1 + u)

f.s.d binary codes obtained from R1 by Construction D ϕ1 (C) |Aut(ϕ1 (C))| Partial Weight Dist. [46, 23, 2] 44 1 + z 2 + 1342z 10 + 11627z 12 + · · · [50, 25, 2]

48

1 + z 2 + 624z 10 + 10336z 12 + · · ·

[54, 27, 2]

52

1 + z 2 + 624z 10 + 6045z 12 + · · ·

[58, 29, 2]

56

1 + z 2 + 4956z 12 + 47212z 14 + · · ·

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[7] S. Han and J. B. Lee, Nonexistence of near-extremal formally self-dual even codes of length divisible by 8, Discrete Appl. Math., Vol 155, 1031–1037, 2007. [8] S. Han, H. Lee and Y. Lee, Binary formally self-dual odd codes, Des. Codes Cryptogr., Vol. 61, No. 2, 141–150, 2011. [9] S. Han and J-L.Kim, Formally self-dual additive codes over F4 , J. Symbolic Computation, Vol. 45, No. 7, 787–799, 2010. [10] S. Han and J-L.Kim, The non-existence of near extremal formally self-dual codes, Des. Codes Cryptogr., Vol. 51, No. 1, 69–77, 2009. [11] W.C. Huffman and V. Pless, Fundamentals of Error Correcting Codes. Cambridge: University Press, 2003. [12] S. Karadeniz, B. Yildiz, Double-circulant and bordered-double-circulant constructions for self-dual codes over R2 , Adv. Math. Commun., Vol. 6, No. 2, pp. 193-202, 2012. [13] G. T. Kennedy and V. Pless, On designs and formally self-dual codes, Des. Codes Cryptogr., Vol 4, 43–55, 1994. [14] J-L. Kim and V. Pless, A Note on formally self-dual even codes of length divisible by 8, Finite Fields Appl. Volume 13, No 2, 224–229, 2007.

[15] A. Munemasa, Database of self-dual codes, http://www.math.is.tohoku.ac.jp/∼munemasa/selfdualc [16] J. Wood, Duality for modules over finite rings and applications to coding theory. Amer. J. Math., Volume 121, 555-575, 1999. [17] B. Yildiz and S. Karadeniz, Linear codes over F2 + uF2 + vF2 + uvF2 , Des. Codes Cryptogr., Volume 54, Number 1, 61-81, 2010.

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