ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.729402 J. Algebra Comb. Discrete Appl. 7(2) • 103–119 Received: 19 February 2019 Accepted: 22 September 2019 0 Journal of Algebra Combinatorics Discrete Structures and Applications A modified bordered construction for self-dual codes from group rings Research Article Joe Gildea, Abidin Kaya, Alexander Tylyshchak, Bahattin Yildiz Abstract: We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78. 2010 MSC: 94B05 Keywords: Group rings, Self-dual codes, Codes over rings, Bordered constructions 1. Introduction The standard form of the generator matrix of any binary self-dual code of length 2n is of the form (In|A) where A is an n × n matrix satisfying AAT = −In. When searching for self-dual codes, some special structure is imposed on the matrix A to make the search field more feasible. Taking A to be a circulant or a block circulant matrix is one of the methods that has been utilized in the literature. Sometimes, from a special such generator matrix, an extension can be achieved by modifying the matrix to get a new self-dual code of higher lengths. While these are generally known as extension methods in the literature, we can also view them as new construction methods for self-dual codes. One such example of a matrix, that we will extend in our constructions is defined in [13] as  1 0 x1 · · · xn 1 · · · 1 y1 y1 ... ... In A yn yn   , (1) Joe Gildea; University of Chester, UK (email: J.Gildea@chester.ac.uk). Abidin Kaya; Sampoerna University, Indonesia (email: nabidin@gmail.com). Alexander Tylyshchak; Uzhgorod State University, Ukraine (email: alxtlk@gmail.com). Bahattin Yildiz (Corresponding Author); Northern Arizona University, USA (email: bahattin.yildiz@nau.edu). 103 https://orcid.org/0000-0003-0175-1909 https://orcid.org/0000-0001-8106-3123 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 where yi = xi + 1 and (In|A) is the generator matrix of a self-dual code of length 2n, possibly coming from a special construction method described above. In this work, we shall extend the above construction, using matrices A that arise from group rings. Group rings have been used to construct self-dual codes on many occasions. In [2], certain ideals of F2S4 were used to construct the extended binary Golay code. In [15], an isomorphism from a group ring to a certain subring of the n × n matrices was described. This was used to construct self-dual codes in [16, 18, 19]. In [5], it is shown that zero divisors can’t be used to construct the putative [72, 36, 16] code. In [11], it is shown that unitary units can be used to construct self-dual codes under a certain construction, and using such units, many new extremal binary self-dual codes were obtained. In the same work, groups of different orders were used to describe many new construction methods for self-dual codes. In our work, we will extend the structure of the matrix given in (1) with the matrices that we get from group ring elements to find new methods for constructing self-dual codes. We will apply the constructions coming from groups of order 2p with p an odd number (using the cyclic group C2p and the Dihedral group D2p) over the binary field F2, F4 and the rings Rk and F4 + uF4 to obtain many extremal and best known binary self-dual codes of various lengths: 14, 28, 56, 44, 30, 38, 46, 54, 62, 70 and 78. In particular, we obtain a new extremal binary self-dual code of length 78. Many of these lengths are not well-known like the oft-studied lengths of 64, 66 and 68. The rest of the paper is organized as follows: In section 2, we give some definitionas and notations that will be used in subsequent sections. In section 3 we give the construction together with a special case when it produces self-dual codes. In section 4 we give the computational results. We finish the paper with some concluding remarks and directions for possible future research. 2. Definitions and notations 2.1. Codes In this paper, all rings are assumed to be commutative, finite, Frobenius rings with a multiplicative identity. A code over a finite commutative ring R is said to be any subset C of Rn. When the code is a submodule of the ambient space then the code is said to be linear. To the ambient space, we attach the usual inner-product, specifically [v,w] = ∑ viwi. The orthogonal with respect to this inner-product is defined as C⊥ = {w | w ∈ Rn, [v,w] = 0,∀v ∈ C}. Since the ring is Frobenius we have that for all linear codes over R, |C||C⊥| = |R|n. If a code satisfies C = C⊥ then the code C is said to be self-dual. If C ⊆ C⊥ then the code is said to be self-orthogonal. For binary codes, a self-dual code where all weights are congruent to 0 (mod 4) is said to be Type II and the code is said to be Type I otherwise. An upper bound on the minimum Hamming distance of a binary self-dual code finalized in [20]. Theorem 2.1. ([20]) Let dI(n) and dII(n) be the minimum distance of a Type I and Type II binary code of length n, respectively. Then dII(n) ≤ 4b n 24 c + 4 and dI(n) ≤ { 4b n 24 c + 4 if n 6≡ 22 (mod 24) 4b n 24 c + 6 if n ≡ 22 (mod 24). Self-dual codes meeting these bounds are called extremal. A best known self-dual code of a given 104 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 length is a self-dual code that has the highest possible known minimum distance, for a length for which the existence of an extremal code is not currently known. Throughout the paper, we will be constructing extremal or best known binary self-dual codes of different lengths. 2.2. Group rings and special matrices We shall use group rings in our construction so we give the standard definition of a group ring. Let G be a finite group or order n, then the group ring RG consists of ∑n i=1 αgigi, αgi ∈ R, gi ∈ G. Addition in the group ring is done by coordinate addition, namely n∑ i=1 αgigi + n∑ i=1 βgigi = n∑ i=1 (αgi + βgi )gi. (2) The product of two elements in a group ring is given by( n∑ i=1 αgigi ) n∑ j=1 βgjgj   = ∑ i,j αgiβgjgigj. (3) It follows that the coefficient of gi in the product is ∑ gigj=gk αgiβgj . We restrict ourselves to finite groups since we are mainly concerned with using these to construct codes whose lengths will be determined by the order of the group. The following construction of a matrix was first given for codes over fields by Hurley in [15] and extended to rings in [5]. Let R be a finite commutative Frobenius ring and let G = {g1,g2, . . . ,gn} be a group of order n. Let v = αg1g1 + αg2g2 + · · · + αgngn ∈ RG. Define the matrix σ(v) ∈ Mn(R) to be σ(v) =   αg−11 g1 αg−11 g2 αg−11 g3 . . . αg−11 gn αg−12 g1 αg−12 g2 αg−12 g3 . . . αg−12 gn ... ... ... ... ... αg−1n g1 αg−1n g2 αg−1n g3 . . . αg−1n gn   . (4) We note that the elements g−11 ,g −1 2 , . . . ,g −1 n are the elements of the group G in some given order. Lemma 2.2. If v = ∑n i=1 αgigi is unitary unit of RG and µ = ∑n i=1 αgi then µ 2 = 1. Proof. The map ∗ : RG → RG defined by  ∑ g∈G agg  ∗ = ∑ g∈G agg −1 is an antiautomorphism of RG of order 2. An element v of V (KG) satisfying vv∗ = 1 is called unitary. The homomorphism ε : RG → R given by ε ( n∑ i=1 αgigi ) = n∑ i=1 αgi is called the augmentation mapping of RG. Let v = ∑n i=1 αgigi, then v∗ = ∑n i=1 αgig −1 i and ε(v) = ε(v ∗) = ∑n i=1 αgi = µ. Therefore ε(vv ∗) = ε(v)ε(v∗) = µ2 = 1. 2.2.1. σ(v) for D2p and C2p In what follows, circ(a1,a2, . . . ,am) denotes the m×m criculant matrix whose first row is given by (a1,a2, . . . ,am). 105 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 Let G = C2p = 〈x |x2p = 1〉. If α = p−1∑ i=0 1∑ j=0 a1+i+pjx 2i+j ∈ RC2p, then σ(α) = ( A B B′ A ) . where A = circ(a1, . . . ,ap), B = circ(ap+1, . . . ,a2p) and B′ = circ(a2p,ap+1, . . . ,a2p−1). Let G = D2p = 〈x,y |xp = y2 = 1, xy = x−1〉. If α = p−1∑ i=0 1∑ j=0 a1+i+pjx iyj ∈ RD2p, then σ(α) = ( A B BT AT ) . where A = circ(a1, . . . ,ap) and B = circ(ap+1, . . . ,a2p), and AT represents the transpose of A. 2.3. Rings We shall use several alphabets in our constructions, including the binary field F2, the quaternary field F4 and rings Rk and F4 + uF4. 2.3.1. The ring family Rk The ring family Rk were defined in [8] and [9]. We briefly give the descriptions of these rings. For k ≥ 1, define Rk = F2[u1,u2, . . . ,uk]/〈u2i = 0,uiuj = ujui〉. (5) For k = 1, we denote the rings by F2 +uF2, and when k = 2, we denote them by F2 +uF2 +vF2 +uvF2, both of which have been considered in coding theory quite extensively. The rings can also be defined recursively as: Rk = Rk−1[uk]/〈u2k = 0,ukuj = ujuk〉 = Rk−1 + ukRk−1. (6) For any subset A ⊆{1, 2, . . . ,k} we will fix uA := ∏ i∈A ui (7) with the convention that u∅ = 1. Then any element of Rk can be represented as∑ A⊆{1,...,k} cAuA, cA ∈ F2. (8) It is shown in [8] that the ring Rk is a commutative ring with |Rk| = 2(2 k). It is also shown that ∀a ∈ Rk a2 = { 1 if a is a unit 0 otherwise. (9) 106 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 We shall now recall the Gray map from Rk to F2 k 2 . For R1 we have the following map: φ1(a + bu) = (b,a + b). Then let c ∈ Rk, c can be written as c = a + buk−1,a,b ∈ Rk−1. Then φk(c) = (φk−1(b),φk−1(a + b)). (10) The map φk is a distance preserving map and the following is shown in [9]. Theorem 2.3. Let C be a self-dual code over Rk, then φk(Rk) is a binary self-dual code of length 2kn. 2.3.2. The ring F4 + uF4 Let F4 = F2 (ω) be the quadratic field extension of F2, where ω2 + ω + 1 = 0. The ring F4 + uF4 is defined via u2 = 0. Note that F4 + uF4 can be viewed as an extension of R1 = F2 + uF2 and so we can describe any element of F4 + uF4 in the form ωa + ω̄b uniquely, where a,b ∈ F2 + uF2. A linear code C of length n over F4 + uF4 is an (F4 + uF4)-submodule of (F4 + uF4) n. In [10] and [6] the following Gray maps were introduced; ψF4 : (F4) n → (F2) 2n ϕF2+uF2 : (F2 + uF2) n → F2n2 aω + bω 7→ (a,b) , a,b ∈ Fn2 a + bu 7→ (b,a + b) , a,b ∈ Fn2 . Those were generalized to the following maps in [17]; ψF4+uF4 : (F4 + uF4) n → (F2 + uF2) 2n ϕF4+uF4 : (F4 + uF4) n → F2n4 aω + bω 7→ (a,b) , a,b ∈ (F2 + uF2) n a + bu 7→ (b,a + b) , a,b ∈ Fn4 These maps preserve orthogonality in the corresponding alphabets. The binary images ϕF2+uF2 ◦ ψF4+uF4 (C) and ψF4 ◦ ϕF4+uF4 (C) are equivalent. The Lee weight of an element is defined to be the Hamming weight of its binary image. Proposition 2.4. ([17]) Let C be a code over F4 + uF4. If C is self-orthogonal, so are ψF4+uF4 (C) and ϕF4+uF4 (C). C is a Type I (resp. Type II) code over F4 +uF4 if and only if ϕF4+uF4 (C) is a Type I (resp. Type II) F4-code, if and only if ψF4+uF4 (C) is a Type I (resp. Type II) F2 + uF2-code. Furthermore, the minimum Lee weight of C is the same as the minimum Lee weight of ψF4+uF4 (C) and ϕF4+uF4 (C). Corollary 2.5. Suppose that C is a self-dual code over F4 +uF4 of length n and minimum Lee distance d. Then ϕF2+uF2 ◦ψF4+uF4 (C) is a binary [4n, 2n,d] self-dual code. Moreover, C and ϕF2+uF2 ◦ψF4+uF4 (C) have the same weight enumerator. If C is Type I (Type II), then so is ϕF2+uF2 ◦ψF4+uF4 (C). 3. The construction Let v ∈ RG where R is a finite Frobenius ring of characteristic 2 and G is a finite group of order 2p and p is odd. Define the following matrix: α1 0 α2 + 1 α2 + 1 ... ... α2 + 1 α2 + 1 α3 + 1 α3 + 1 ... ... α3 + 1 α3 + 1 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 σ(v) Ip 0p 0p Ip M(σ) = . 107 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 Let Cσ be a code that is generated by the matrix M(σ) and µ = ∑n i=1 αgi. Let A1 = (α1, 0) ∈ R 2, A2 = (α2, . . . ,α2,α3, . . . ,α3) ∈ R2p, A3 = (α4, . . . ,α4) ∈ R2p and B1 =   α2+1 α2+1 ... ... α2+1 α2+1 α3+1 α3+1 ... ... α3+1 α3+1  . Then, M(σ)(M(σ))T = ( A1A T 1 + A2A T 2 + A3A T 3 A1B T 1 + A2 + A3σ(v) T B1A T 1 + A T 2 + σ(v)A T 3 B1B T 1 + I + σ(v)σ(v) T ) where • A1AT1 + A2AT2 + A3AT3 = α21 + pα22 + pα23 + 2pα24 = α21 + p(α2 + α3)2, • A1BT1 + A2 + A3σ(v)T = (α1(α2 + 1), . . . ,α1(α2 + 1),α1(α3 + 1), . . . ,α1(α3 + 1)) + (α2, . . . ,α2,α3, . . . ,α3)+(α4, . . . ,α4)σ(v) = (α1(α2+1)+α2+µα4, . . . ,α1(α2+1)+α2+µα4,α1(α3+ 1) + α3 + µα4, . . . ,α1(α3 + 1) + α3 + µα4) and • B1BT1 + I + σ(v)σ(v)T = I + σ(v)(σ(v)T = I + σ(vv∗). Theorem 3.1. Let R be a finite commutative Frobenius ring of characteristic 2, G be a finite group of order 2p where p is odd and µ = ∑2p i=1 αgi. If α 2 1 + p(α2 + α3) 2 = 0, α1(α2 + 1) + α2 + µα4 = 0, α1(α3 + 1) + α3 + µα4 = 0 vv ∗ = 1 and (α1,p(α2(α2 + 1) + α3(α3 + 1)),α2 + 1,α3 + 1) has free rank 1 then Cσ is a self-dual code of length 4p + 2. Proof. If vv∗ = 1, then I +σ(vv∗) = 0. Additionally, let α21 +p(α2 +α3) 2 = 0, α1(α2 +1)+α2 +µα4 = 0 and α1(α3 + 1) + α3 + µα4 = 0, Therefore Cσ is self-orthogonal. It remains to show that the M(σ) has free rank 2p + 1. rank(M(σ)) = rank   α1 0 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 α2 + 1 α2 + 1 ... ... Ip 0p α2 + 1 α2 + 1 σ(v) α3 + 1 α3 + 1 ... ... 0p Ip α3 + 1 α3 + 1   = rank   α1 0 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 α2 + 1 ... ... Ip 0p 0 α2 + 1 σ(v) 0 α3 + 1 ... ... 0p Ip 0 α3 + 1   108 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 = rank   α1 γ1 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 0 ... ... Ip 0p 0 0 σ(v) 0 α3 + 1 ... ... 0p Ip 0 α3 + 1   = rank   α1 γ2 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 0 ... ... Ip 0p 0 0 σ(v) 0 0 ... ... 0p Ip 0 0   = rank     α1 γ2 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 0 ... ... Ip 0p 0 0 σ(v) 0 0 ... ... 0p Ip 0 0    I2 0 00 Ip 0 0 σ(v)T Ip     = rank   α1 γ2 γ3 · · · γ3 γ4 · · · γ4 α4 · · · α4 α4 · · · α4 0 0 ... ... 0 0 I + σ(v)σ(v∗) σ(v) 0 0 ... ... 0 0   = rank   α1 γ2 γ3 · · · γ3 γ4 · · · γ4 α4 · · · α4 α4 · · · α4 0 0 ... ... 0p 0p 0 0 σ(v) 0 0 ... ... 0p 0p 0 0   109 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 where γ1 = pα2(α2 + 1), γ2 = pα2(α2 + 1) +pα3(α3 + 1), γ3 = α2 +µα4 = α2 + 1 and γ4 = α3 +µα4 = α3 + 1. Finally, rank(M(σ)) =   α1 γ2 α2 + 1 · · · α2 + 1 α3 + 1 · · · α3 + 1 γ5 · · · γ5 γ5 · · · γ5 0 0 ... ... 0p 0p 0 0 σ(v) 0 0 ... ... 0p 0p 0 0   where γ5 = α4 +µ2α4 = α4 + (1)α4 = 0 by Lemma 2.2. Therefore Cσ is self if (α1,p(α2(α2 + 1) +α3(α3 + 1)),α2 + 1,α3 + 1) has free rank 1. The family of rings Rk is particularly well suited for this construction. Corollary 3.2. Let R = Rk and let G be a finite group of order 2p where p is odd. Let v ∈ RG be a unitary unit. Then if α2 + α3 is any unit then Cσ is a self-dual code of length 4p + 2. Proof. We will show that this case satisfies the hypotheses of Theorem 3.1. If v is a unitary unit then vv∗ = 1. If α2 + α3 is a unit then (α2 + α3)2 = 1 by (9). Then 1 + p(α2 + α3)2 = 1 + p(1) = 0 and vv∗ = 1. 4. Computational results In this section, we apply the constructions discussed in the previous section over particular groups and rings that have been described before. The sizes of the groups and the alphabets used lead to particular lengths for self-dual codes. In all the subsequent subsections, we tabulate the extremal binary self-dual codes or the best-known (if the existence of extremal self-dual codes is not known for that length) self-dual codes of the certain lengths. 4.1. Constructions from groups of order 6 We first apply construction D6 over the binary field and F4. Table 1. Extremal binary self-dual code of length 14 from D6 (α1,α2,α3,α4) (a1, . . . ,a6) |Aut(C)| (1,0,1,1) (0,1,1,1,1,1) 27 ·32 ·72 Table 2. Extremal self-dual code of length 28 coming from applying D6 over F4 (α1,α2,α3,α4) (a1, . . . ,a6) |Aut(C)| (1,ω,ω + 1,1) (0,0,1,1,ω,ω + 1) 25 ·3 ·7 110 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 In [14], the possible weight enumerators for a self-dual Type I [56, 28, 10] code were obtained in two forms as: W56,1 = 1 + (308 + 4α) y 10 + (4246 − 8α) y12 + (40852 − 28α) y14 + · · · W56,2 = 1 + (308 + 4α) y 10 + (3990 − 8α) y12 + (42900 − 28α) y14 + · · · where α is an integer. Applying the constructions over the ring F4 + uF4, we will be able to get binary self-dual codes of length 56. For brevity of notation, we need a brief notation for the elements of F4 + uF4. 0 ↔ 0000, 1 ↔ 0001, 2 ↔ 0010, 3 ↔ 0011, 4 ↔ 0100, 5 ↔ 0101, 6 ↔ 0110, 7 ↔ 0111, 8 ↔ 1000, 9 ↔ 1001, A ↔ 1010, B ↔ 1011, C ↔ 1100, D ↔ 1101, E ↔ 1110, F ↔ 1111. We use the ordered basis {uω,ω,u, 1} to express the elements of F4 + uF4. For instance, 1 + uω corresponds to 1001, which is represented by the hexadecimal 9. In the following tables α = mi denotes that the weight enumerator of the code has parameter α = m in Wi, i = 1, 2. Table 3. [56,28,10] codes over F4 + uF4 from D6 where (α1,α2,α3,α4) = (1,6,F,1) (a1, . . . ,a6) |Aut(C)| Type (a1, . . . ,a6) |Aut(C)| Type (A,A,1,1,4,5) 23 ·3 α = −322 (A,A,1,3,C,F) 22 ·3 α = −82 (A,A,1,B,6,D) 24 ·3 ·7 α = −562 (A,8,3,1,6,7) 2 ·3 α = −262 (A,8,3,3,E,D) 2 ·3 α = −202 (8,8,1,1,4,5) 24 ·3 α = −82 (8,8,1,3,C,F) 22 ·3 α = −322 Table 4. [56,28,10] codes over F4 + uF4 from D6 where |Aut(C)| = 2 ·3 (α1, . . . ,α4) (a1, . . . ,a6) Type (a1, . . . ,a6) (α1, . . . ,α4) Type (1,4,7,1) (0,0,9,9,C,D) α = −82 (1,4,7,1) (0,8,3,1,4,F) α = −242 (1,4,7,1) (0,0,B,1,C,7) α = −182 (1,4,7,1) (0,0,B,3,4,D) α = −122 (1,4,7,1) (2,0,B,3,C,7) α = −202 (1,4,7,1) (2,0,B,B,6,5) α = −142 (1,4,7,1) (2,A,1,3,6,D) α = −262 (1,4,7,9) (A,8,3,3,C,7) α = −62 (1,4,7,9) (8,2,9,3,E,7) α = −382 (1,4,7,9) (8,2,B,3,4,F) α = −302 (1,4,D,1) (0,8,1,1,E,7) α = −322 (1,4,D,1) (A,0,9,9,E,5) α = −321 (1,4,D,1) (A,2,B,B,4,D) α = −261 (1,4,D,1) (A,A,3,3,6,7) α = −141 (1,4,D,9) (0,0,9,1,4,5) α = −261 (1,4,D,9) (A,0,3,9,4,D) α = −381 (1,4,F,1) (2,8,9,B,6,F) α = −281 (1,4,F,1) (A,8,1,1,C,F) α = −221 (1,4,F,9) (0,0,9,3,C,F) α = −161 (1,4,F,9) (0,8,1,B,E,5) α = −281 (1,4,F,9) (8,2,3,9,4,D) α = −401 (1,4,F,9) (8,2,3,B,C,7) α = −461 We can also construct self-dual codes of length 56 from applying the construction D6 over the ring F2 + uF2 + vF2 + uvF2 as well. This is a ring of size 16, so in the same way as was done for F4 + uF4, we can use hexadecimals to shorten the notations. The correspondence between the binary 4 tuples and 111 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 the hexadecimals is as follows: 0 ↔ 0000, 1 ↔ 0001, 2 ↔ 0010, 3 ↔ 0011, 4 ↔ 0100, 5 ↔ 0101, 6 ↔ 0110, 7 ↔ 0111, 8 ↔ 1000, 9 ↔ 1001, A ↔ 1010, B ↔ 1011, C ↔ 1100, D ↔ 1101, E ↔ 1110, F ↔ 1111. The ordered basis {uv,v,u, 1} is used to express elements of F2 +uF2 +vF2 +uvF2 For instance, 1 +u+v is represented as 0111 which corresponds to hexadecimal 7. Table 5. [56,28,10] codes over F2 + uF2 + vF2 + uvF2 from D6 where (α1,α2,α3,α4) = (1,6,B,1) (a1, . . . ,a6) |Aut(C)| Type (a1, . . . ,a6) |Aut(C)| Type (4,1,9,1,7,B) 23 ·3 α = −282 (4,5,D,1,F,3) 23 ·3 α = −522 (4,1,9,1,B,7) 23 ·3 α = −42 (4,5,D,9,7,3) 22 ·3 α = −42 (4,5,D,9,3,7) 22 ·3 α = −162 (4,7,F,1,3,F) 22 ·3 α = −282 (4,7,F,9,F,B) 22 ·3 α = −402 Table 6. [56,28,10] codes over F2 + uF2 + vF2 + uvF2 from D6 where (α1,α2,α3,α4) = (1,2,5,1) (a1, . . . ,a6) |Aut(C)| Type (a1, . . . ,a6) |Aut(C)| Type (6,1,9,1,B,5) 22 ·3 α = −141 (6,1,9,1,D,3) 22 ·3 α = −261 (0,1,9,5,3,F) 22 ·3 α = −181 (0,1,9,5,7,B) 22 ·3 α = −421 (0,1,9,7,D,3) 22 ·3 α = −301 Table 7. [56,28,10] codes over F2 + uF2 + vF2 + uvF2 from D6 where (α1,α2,α3,α4) = (1,2,7,1) (a1, . . . ,a6) |Aut(C)| Type (a1, . . . ,a6) |Aut(C)| Type (C,7,F,9,7,B) 22 ·3 α = −381 (C,7,F,9,B,7) 22 ·3 ·13 α = −381 4.2. Constructions from groups of order 10 Table 8. Extremal binary self-dual code of length 22 from D10 (α1,α2,α3,α4) (a1, . . . ,a10) |Aut(C)| (1,0,1,1) (0,0,0,1,1,0,1,0,1,1) 28 ·32 ·5 ·7 ·11 In [7], the possible weight enumerators for a self-dual Type I [44, 22, 8] code were obtained in two forms as: W44,1 = 1 + (44 + 4β)y 8 + (976 − 8β)y10 + · · · where 10 ≤ β ≤ 122 and 112 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 W44,2 = 1 + (44 + 4β)y 8 + (1232 − 8β)y10 + · · · where 10 ≤ β ≤ 154. Table 9. Extremal self-dual code of length 44 over F4 from D10 (α1,α2,α3,α4) (a1, . . . ,a6) |Aut(C)| Type (1,ω,ω + 1,1) (0,0,0,1,1,1,ω,ω,ω + 1,ω + 1) 24 ·5 W44,2 (β = 14) (1,ω,ω + 1,1) (0,0,0,ω,ω + 1,0,ω,ω + 1,ω,ω + 1) 2 ·5 W44,2 (β = 4) (1,ω,ω + 1,1) (0,0,0,ω,ω + 1,1,1,ω,1,ω + 1) 2 ·5 W44,2 (β = 4) (1,ω,ω + 1,1) (0,0,1,1,1,0,ω,ω,ω + 1,ω + 1) 23 ·5 W44,2 (β = 34) (1,ω,ω + 1,1) (0,0,1,1,1,1,1,1,ω,ω + 1) 23 ·5 W44,2 (β = 34) (1,ω,ω + 1,1) (0,0,1,ω,ω + 1,0,1,1,ω + 1,ω) 5 W44,2 (β = 14) (1,ω,ω + 1,1) (0,0,1,ω,ω + 1,0,1,ω,1,ω + 1) 5 W44,2 (β = 9) (1,ω,ω + 1,1) (0,0,1,ω,ω + 1,1,ω,ω,ω,ω) 2 ·5 W44,2 (β = 4) (1,ω,ω + 1,1) (0,0,ω,1,ω,0,1,1,ω + 1,ω + 1) 22 ·5 W44,2 (β = 4) (1,ω,ω + 1,1) (0,0,ω,1,ω,1,ω,ω,ω,ω + 1) 2 ·5 W44,2 (β = 4) (1,ω,ω + 1,1) (0,0,ω,1,ω + 1,1,1,1,1,1) 216 ·32 ·52 W44,2 (β = 74) (1,ω,ω + 1,1) (0,0,1,ω,ω,1,0,ω,ω,ω + 1,ω) 2 ·5 W44,2 (β = 4) (1,ω,ω + 1,1) (0,ω,ω,ω + 1,ω + 1,1,ω,ω + 1,ω,ω + 1) 2 ·5 W44,2 (β = 4) (1,ω,ω + 1,1) (1,1,1,ω,ω + 1,1,ω,ω + 1,ω,ω + 1) 22 ·5 W44,2 (β = 14) Table 10. Extremal self-dual code of length 44 over F2 + uF2 from C10 (α1,α2,α3,α4) (a1, . . . ,a6) |Aut(C)| Type (1,u,1,1) (u,u,u,1,1,1,1,0,u + 1,0) 22 ·5 W44,1 (β = 32) (1,u,1,1) (u,u,u,1,1,u + 1,u + 1,0,u + 1,0) 215 ·34 ·52 ·72 W44,1 (β = 122) (1,u,1,1) (u,0,u,1,1,1,1,0,1,0) 24 ·5 W44,2 (β = 10) (1,u,1,1) (u,0,u,1,1,u + 1,u + 1,0,1,0) 23 ·5 W44,2 (β = 30) (1,u,1,1) (0,u,0,1,1,u + 1,u + 1,u,u + 1,u) 23 ·5 W44,1 (β = 12) (1,u,1,1) (0,0,0,1,1,1,1,u,1,u) 216 ·32 ·52 W44,2 (β = 90) (1,u,u + 1,1) (u,0,u,1,1,1,1,0,1,0) 24 ·5 W44,2 (β = 14) (1,u,u + 1,1) (u,0,u,1,1,u + 1,u + 1,0,1,0) 23 ·5 W44,2 (β = 34) (1,u,u + 1,1) (0,0,0,1,1,1,1,u,1,u) 216 ·32 ·52 W44,2 (β = 74) (1,0,u + 1,1) (u,0,u,1,1,1,1,0,1,0) 24 ·5 W44,2 (β = 14) (1,0,u + 1,1) (u,0,u,1,1,u + 1,u + 1,0,1,0) 23 ·5 W44,2 (β = 34) (1,0,u + 1,1) (0,0,0,1,1,1,1,u,1,u) 216 ·34 ·52 ·72 ·112 W44,2 (β = 154) 113 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 4.3. Constructions from groups of order 14 Table 11. Extremal binary self-dual code of length 30 from D14 (α1,α2,α3,α4) (a1, . . . ,a14) |Aut(C)| (1,0,1,1) (0,0,0,0,0,0,1,0,0,1,0,1,1,1) 211 ·3 ·7 (1,0,1,1) (0,0,0,0,0,1,1,0,0,1,0,0,1,1) 28 ·7 (1,0,1,1) (0,0,0,1,0,1,1,0,1,1,1,1,1,1) 27 ·32 ·5 ·7 Table 12. Extremal binary self-dual code of length 30 from C14 (α1,α2,α3,α4) (a1, . . . ,a14) |Aut(C)| (1,0,1,1) (0,0,0,0,0,1,1,0,1,1,0,0,1,0) 28 ·7 Table 13. Extremal binary self-dual code of length 60 over F4 from D14 (α1,α2,α3,α4) (a1, . . . ,a14) |Aut(C)| Type (1,ω,ω + 1,1) (0,0,1,ω,1,ω,1,0,1,ω + 1,0,ω,ω + 1,ω + 1) 22 ·7 W60,1 (β = 0) 4.4. Constructions from groups of order 18 Table 14. Extremal binary self-dual code of length 38 from D18 (α1,α2,α3,α4) (a1, . . . ,a18) |Aut(C)| (1,0,1,1) (0,0,0,0,0,0,1,1,1,0,0,1,1,0,1,1,1,1) 2 ·32 (1,0,1,1) (0,0,0,0,1,0,1,1,1,0,0,1,1,0,1,0,1,1) 2 ·32 ·19 4.5. Constructions from groups of order 22 Table 15. [46,22,8] codes from D22 (α1,α2,α3,α4) (a1, . . . ,a22) |Aut(C)| (1,0,1,1) (0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,1,0,1,1,1,1,1) 2 ·11 (1,0,1,1) (0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,0,1,0,1,1) 2 ·11 (1,0,1,1) (0,0,0,0,0,1,0,0,1,1,1,0,0,0,1,0,0,1,1,0,1,1) 11 (1,0,1,1) (0,0,0,1,0,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1) 215 ·34 ·52 ·72 ·112 ·232 114 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 Table 16. [46,22,8] codes from C22 (α1,α2,α3,α4) (a1, . . . ,a22) |Aut(C)| (1,0,1,1) (0,0,0,0,0,0,0,1,0,1,1,0,1,1,1,0,1,1,0,0,0,1) 2 ·11 (1,0,1,1) (0,0,0,0,0,0,1,0,1,1,1,0,0,1,1,0,1,1,0,1,0,0) 2 ·11 (1,0,1,1) (0,0,0,0,1,0,1,1,1,1,1,0,1,1,0,1,1,1,1,0,1,0) 2 ·11 (1,0,1,1) (0,0,0,1,0,0,1,1,1,1,1,1,1,0,1,1,0,1,1,0,0,1) 2 ·11 (1,0,1,1) (0,0,0,0,1,0,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1) 215 ·34 ·52 ·72 ·112 ·232 4.6. Constructions from groups of order 26 From [3], it is known that the weight enumerator of a [54, 27, 10] self-dual code can be of the follwing form: W54,1 = 1 + (351 − 8β)y10 + (5031 + 24β)y12 + . . . W54,2 = 1 + (351 − 8β)y10 + (5543 + 24β)y12 + . . . In the following tables, we consturct inequivalent self-dual codes of parameters [54, 27, 10] from D26 and C26. Table 17. Inequivalent [54,27,10] codes from D26 (α1,α2,α3,α4) (a1, . . . ,a26) |Aut(C)| Type (1,0,1,1) (0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1,0,1,1,1,0,1) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,1,0,1,0,1,0,1,1,1) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0,1,1) 13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,0,0,1,1,1,1,0,0,1) 2 ·3 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,1,0,0,0,1,1,1,1) 13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,1,0,0,1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,1,1,1) 2 ·34 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0,1,1,0,1,1,1,1) 2 ·3 ·13 W54,1 (β = 0) Table 18. Inequivalent [54,27,10] codes from C26 (α1,α2,α3,α4) (a1, . . . ,a26) |Aut(C)| Type (1,0,1,1) (0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,1,0) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,0,0,1,1,0,1,1,1,0) 2 ·3 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,0,1,0,0,1,1,1,1,1,1) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,1,0,1,0,1,1,1,1,0,1,1,0,1,1,1,1,0,1,0) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,1,0,1,0,1,1,1,1,1,0,0,1,0,1,1,1,1,0,1,0) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,1) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,1,0,0,1,1,1,0,1,1,0,1,1,1,0,0,1,0,1,1,0,1) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,1,0,1,1,1,1,1,1,1,0,0,0,1,0,1,1,1,1,0,0,0) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,1,1,0,1,1,1,1,1,1,0,1,1,0,1,1,1,1,1,1,1,0) 2 ·13 W54,1 (β = 0) (1,0,1,1) (0,0,0,1,0,0,1,1,0,1,1,1,1,0,1,1,0,1,1,1,1,1,1,1,1,0) 2 ·3 ·13 W54,1 (β = 0) 115 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 4.7. Constructions from groups of order 30 There are two possibilities for the weight enumerators of extremal singly-even [62, 31, 12]2 codes ([3]): W62,1 = 1 + 2308y 12 + 23767y14 + · · · W62,2 = 1 + (1860 + 32β) y 12 + (28055 − 160β) y14 + · · · , : 0 ≤ β ≤ 93. only codes with weight enumerator for β = 0, 2, 9, 10, 15, 16 in W62,2 known to exist. Table 19. [62,31,12] codes from D30 where (α1,α2,α3,α4) = (1,0,1,1) (a1, . . . ,a30) |Aut(C)| Type (0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,1) 2 ·3 ·5 W62,2 (β = 10) (0,0,0,0,0,0,0,0,1,0,1,1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,1,0,1) 2 ·3 ·5 W62,2 (β = 10) (0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,0,1,1,1,0,1,1,1,0,1,1,1) 22 ·3 ·5 W62,2 (β = 0) (0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,0,0,1,1,1,1,0,0,1,1,1,1,1,1,1) 2 ·3 ·5 W62,2 (β = 0) (0,0,0,0,0,1,0,0,1,0,0,1,1,1,1,0,1,0,1,1,0,1,1,1,1,1,1,0,1,1) 2 ·3 ·5 W62,2 (β = 10) (0,0,0,0,0,1,1,0,0,1,0,1,1,1,1,0,0,1,1,0,1,1,1,0,1,1,0,1,1,1) 3 ·5 W62,2 (β = 0) (0,0,0,0,0,1,1,0,1,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,1,1,1,1,1,1) 2 ·3 ·5 W62,2(β = 0) (0,0,0,0,0,1,1,1,0,1,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1) 2 ·3 ·5 W62,2(β = 0) (0,0,0,0,1,1,0,0,1,1,1,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,1) 2 ·3 ·5 W62,2(β = 0) 4.8. Constructions from groups of order 34 The weight enumerator of a self-dual [70, 35, 12]2 code is in one of the following forms ([13]): W70,1 = 1 + 2β + (11730 − 2β − 128γ) y14 + (150535 − 22β + 896γ) y16 + · · · W70,2 = 1 + 2β + (9682 − 2β) y14 + (173063 − 22β) y16 + · · · The code with weight enumerator for γ = 1,β = 416 is constructed in [13]. Together with the results from [4] and [12], the existence of codes with weight enumerators γ = 0 in W70,1 is known for many β values. In the following tables we tabulate the [70, 35, 12] self-dual codes from D34 and C34 together with their β values and automorphism groups. Note that the automorphism groups all have an element of order 17 in them. Naturally, these have the same parameters as the ones obtained in [12]. However, here we have given an alternative construction to those codes. 116 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 Table 20. [70,35,12] codes from D34 where (α1,α2,α3,α4) = (1,0,1,1) (a1, . . . ,a34) |Aut(C)| W70,1 (γ = 0) (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,0,0,0,1,1,0,1,0,1,1,0,1,0,1,1) 2 ·17 β = 102 (0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,0,0,0,0,0,1,0,1,1,0,0,1,1,0,1,1,1) 2 ·17 β = 136 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0,0,1,0,0,1,0,1,1,0,1,1) 17 β = 170 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,1,0,0,1,0,1,0,0,1,1,0,0,1,1,1) 17 β = 204 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,1,0,0,1,0,1,1,0,1,1,0,1,0,1,1) 17 β = 238 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,1,0,1,1,0,1,0,1,0,1,1,0,0,1,1) 17 β = 272 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,0,1,0,0,0,1,1,0,1,1,0,1,1,1,1) 17 β = 306 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,0,0,0,0,1,0,1,0,1,0,0,1,1,0,0,1,1,1) 17 β = 340 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0,1,1,1,0,0,0,1,1,1) 17 β = 374 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,1,1) 17 β = 408 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,0,1,0,1,0,1,1,0,0,0,1,0,1,1,1) 17 β = 442 (0,0,0,0,0,0,0,0,1,1,0,1,1,1,1,1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,1) 2 ·17 β = 476 (0,0,0,0,0,0,0,0,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,1,1,1,1) 17 β = 510 Table 21. Extremal [70,35,12] codes from C34 where (α1,α2,α3,α4) = (1,0,1,1) (a1, . . . ,a34) |Aut(C)| W70,1 (γ = 0) (0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0,1,1,0) 2 ·17 β = 102 (0,0,0,0,0,0,0,1,0,1,1,0,1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,1,1,1,1) 2 ·17 β = 136 (0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0,1,1,0,0,0,1,1) 2 ·17 β = 238 (0,0,0,0,0,0,1,0,0,1,1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,1,1,1) 2 ·17 β = 272 (0,0,0,0,0,0,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,1,0,1,0,1) 2 ·17 β = 306 (0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1) 2 ·17 β = 374 (0,0,0,0,0,0,1,0,0,0,1,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,1,1,1) 2 ·17 β = 442 4.9. Constructions from groups of order 38 The possible weight enumerators for self-dual codes of parameters [78, 39, 14] are given as follows ([7]): W78,1 = 1 + (3705 + 8β)y 14 + (62244 + 512α− 24β)y16 + . . . , 0 ≤ α ≤ −1 16 β ≤ 28, W78,2 = 1 + (3750 + 8α)y 14 + (71460 − 24α)y16 + . . . ,−486 ≤ α ≤−135. For many of these parameters, the existence of a code with that weight enumerator is not known. Together with the ones that were recently found in [1, 21, 22], the existence of codes which have W78,1 with α = 0 and β = 0,−13,−19 − 26,−38,−39,−52,−65,−78,−104,−117 and which have α = −135 in W78,2. In the following table we construct [78, 39, 14] self-dual codes from D38. The one with β = −57 is a new code. 117 J. Gildea et al. / J. Algebra Comb. Discrete Appl. 7(2) (2020) 103–119 Table 22. Extremal [78,39,14] codes from D38 where (α1,α2,α3,α4) = (1,0,1,1) and a1 = a2 = a3 = a4 = 0 (a5, . . . ,a34) |Aut(C)| W78,1 (α = 0) (1,1,1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0,1,1,1,1,1,1,0,1,1,1,1,1) 2 ·19 β = 0 (1,1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0,0,1,0,1,1,0,0,1,1,1,1,0,1,1,1,1,1) 19 β = 0 (1,1,1,1,0,1,0,0,0,0,1,1,0,0,0,1,0,0,1,0,1,1,0,0,0,1,1,0,1,0,1,0,1,1) 19 β = −19 (1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,1,1) 19 β = −38 (1,1,1,1,0,0,1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,1,1,0,1,1) 19 β = −57 5. Conclusion We have integrated a modified bordered construction with the matrices corresponding to a group ring element in RC2p and RD2p where R is a commutative Frobenius ring of characteristic 2 and as a result we have been able to obtain many extremal binary self-dual codes. The structure of the groups has allowed us to look at such lengths as 62, 70, 78, etc. that are different than the oft-studied lengths of 64, 66 and 68. 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Ge, On the existence of certain optimal self–dual codes with lengths between 74 and 116, The Electronic Journal of Combinatorics 22(4) (2015) 1–25. 119 https://doi.org/10.1016/j.ffa.2018.01.002 https://doi.org/10.1016/j.ffa.2018.01.002 https://hrcak.srce.hr/157710 https://hrcak.srce.hr/157710 https://doi.org/10.1006/ffta.1996.0174 https://doi.org/10.1006/ffta.1996.0174 https://arxiv.org/abs/0711.3983 https://arxiv.org/abs/0711.3983 https://doi.org/10.1006/eujc.2001.0509 https://doi.org/10.1007/s10623-011-9530-0 https://doi.org/10.1007/s10623-011-9530-0 https://doi.org/10.1109/TIT.2008.928260 https://doi.org/10.1109/TIT.2008.928260 https://doi.org/10.1109/18.651000 http://dx.doi.org/10.3934/amc.2017047 http://dx.doi.org/10.3934/amc.2017047 https://doi.org/10.37236/5213 https://doi.org/10.37236/5213 Introduction Definitions and notations The construction Computational results Conclusion References