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New extremal binary self-dual codes of length 66 as extensions of self-dual codes over R_k
Abstract
We obtain extremal binary self-dual codes of parameters [64,32,12] as binary images of self-dual codes over R1, R2 and R3 by employing different methods. We then apply the extension theorem to these codes to obtain a number of extremal binary self-dual codes of length 66 with trivial automorphism groups. Fifteen of the codes we obtain have new β values in W66,3, of which only three were known to exist before. We also find nine codes with new β values in W66,1, thus updating the list of such known codes.
... They contain five codes which were not known to exist before. We also obtained four additional codes which were discovered only recently in [14] using a different method. ...
... We have been able to construct 5 such codes. Additionally, we found 4 codes that have been recently obtained in [14] from the ring R 3 . It is well-known that there are three possibilities for the weight enumerators of extremal self-dual codes of length 66 [4]. ...
... A substantial number of codes with weight enumerator W 66,1 are obtained in [3], [11], [12] and [20]. Codes with weight enumerator W 66,3 are found by Tsai et al. in [21] [14]. In this work, we obtain new extremal binary self-dual codes with β = 1, 30, 34, 84, 94, 25, 28, 39, 48 in W 66,1 . ...
... All such codes are classified up to length 50 [21]. In recent works [16], [14], [15] some self-dual codes with automorphisms of order 2 r p for primes p = 5, 7, and 11 were constructed using codes over rings ...
... and consider all permutations of the 11-cycles in F σ (C) that can generate different binary codes C. We construct [n, k] self-dual codes with minimum distance d = 12 with an automorphism of order 11 with 6 cycles. There exists a [76, 38, 14] binary self-dual code (see [1]) so 66 ≤ n ≤ 74. In the case of a [74, 37, 12] binary self-dual code we have automorphism of type 11-(6, 8). ...
... , 74, 76, 77, 78, 80, 83, 86, 87, 92; and with W 66,3 for β = 28, 33, 34, 54, 56, . . . , 59, 62 and 66 (see [18], [10] and [14]). For a binary [66, 33, 12] self-dual codes with an automorphism of type 11-(6, 0) by Theorem 3.1 the subcode C π is the unique [6, 3] binary self-dual code 3i 2 [19]. ...
Using a method for constructing self-dual codes having an automorphism of odd prime order, we classify up to equivalence all binary self-dual codes with an automorphism of order 11 with 6 cycles and minimum distance 12. This classification gives new [72, 36, 12] codes with weight enumerator that was previously not obtained as well as many [66, 33, 12], [68, 34, 12], and [70, 35, 12] codes with new values of the parameters in their respective weight enumerators.
... They contain five codes which were not known to exist before. We also obtained four additional codes which were discovered only recently in [14] using a different method. ...
... We have been able to construct 5 such codes. Additionally, we found 4 codes that have been recently obtained in [14] from the ring R 3 . It is well-known that there are three possibilities for the weight enumerators of extremal self-dual codes of length 66 [4]. ...
... A substantial number of codes with weight enumerator W 66,1 are obtained in [3], [11], [12] and [20]. Codes with weight enumerator W 66,3 are found by Tsai et al. in [21] [14]. In this work, we obtain new extremal binary self-dual codes with β = 1, 30, 34, 84, 94, 25, 28, 39, 48 in W 66,1 . ...
A classification of all four-circulant extremal codes of length 32 over F-2 + uF(2) is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with beta = 80 in W-64,W-2 is found. Then applying an extension method from the literature to extremal self-dual codes of length 64, we have found many extremal binary self-dual codes of length 66. Among those, five of them are new codes in the sense that codes with these weight enumerators are constructed for the first time. These codes have the values beta = 1, 30, 34, 84, 94 in W-66,W-1.
... Recently certain binary rings (rings of characteristic 2) have been successfully used to obtain many new extremal binary self-dual codes using the construction methods mentioned above and extension theorems. Some of the examples of these constructions can be found in [9,10,11,12], etc. ...
... By considering lifts of [8,4,4] 2 Hamming code Karadeniz and Kaya constructed 11 new codes in W 58,2 in [9]. Together with the ones added from [19] and [9], the existence of such codes is known for β = 55 in W 58,1 and for β = 0 with γ ∈ {2m|m =0, 1,8,9,10,15,16,34,71,79 In this section, we obtain the codes in W 58,2 for β = 0 with γ = 28, β = 1 with γ = 28, 32, 34, 38, 40 and for β = 2 with γ = 46. ...
... First codes with a weight enumerator in W 66,3 were discovered in [18] and recently 14 codes were discovered in [10] together with these the existence of codes in W 66,3 is known for β =28, 29, 30, 31, 32, 33, 34, 49, 50, 54, 55 Remark 4.1. In order to apply the extension in Theorem 2.6 the generator matrices for the binary image of L i are converted into standard form [I 32 |A i ] and the matrices are available online at [13]. ...
In this work, quadratic double and quadratic bordered double circulant
constructions are applied to F_4 + uF_4 as well as F_4, as a result of which
extremal binary self-dual codes of length 56 and 64 are obtained. The binary
extension theorems as well as the ring extension version are used to obtain 7
extremal self-dual binary codes of length 58, 24 extremal self-dual binary
codes of length 66 and 29 extremal self-dual binary codes of length 68, all
with new weight enumerators, updating the list of all the known extremal
self-dual codes in the literature.
... , 92, 94, 100, 101, 115} \ {4, 7, 58, 70, 91} (see [5], [8], [10], [17] and [18]). Extremal singly even self-dual codes with weight enumerator W 66,2 are known (see [8] and [15] (see [9], [10], [11] and [12]). ...
For lengths 64 and 66, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new 40 inequivalent extremal doubly even self-dual [64,32,12] codes with covering radius 12 meeting the Delsarte bound.
... Together with the codes recently obtained in [1] and the ones from [12], [13] and [7], extremal singly even self-dual codes with weight enumerator W 66,1 are known for 182,187,189,191,192,193,195,198,200,201,202 Applying Theorem 2.2 over F 2 and F 2 + uF 2 (to the code constructed in Table 1), we construct self-dual codes of lengths 64, 66 and 68 (Tables 2, 3 and 4). We replace 3 with 1 + u to save space. ...
In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2 and F2 + uF2. Using extensions, neighbours and sequences of neighbours, we construct many new self-dual codes. In particular, we construct one new self-dual code of length 66 and 51 new self-dual codes of length 68.
... Extremal singly even self-dual codes with weight enumerators W 64,1 are known ( [1,16,27]): 14,16,18,19,20,22,24,25,26,28,29,30,32,34,35,36,38,39,44,46,49,53,54,58,59,60,64,74 and extremal singly even self-dual codes with weight enumerator W 64,2 are known for: 40,41,42,44,45,46,47,48,49,50,51,52,54,55,56,57,58,60,62,64,69,72,80,88,96,104,108,112,114,118,120,184 \ {31, 39}. ...
In this paper, we construct self-dual codes from a construction that involves 2x2 block circulant matrices, group rings and a reverse circulant matrix. We provide conditions whereby this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2, F2 + uF2 and F4 + uF4. Using extensions, neighbours and neighbours of neighbours, we construct 32 new self-dual codes of length 68.
... Together with the codes recently obtained in [1] and the ones from [11], [13], [14] and [15], extremal singly even self-dual codes with weight enumerator In this section, we construct the codes with weigth enumerators for β=65, 68, 69 and 72 in W 66,3 as extensions of codes in Section 4.1. ...
In this work we consider modified versions of quadratic double circulant and quadratic bordered double circulant constructions over the binary field and the rings F_2+uF_2 and F_4+uF_4 for different prime values of p. Using these constructions with extensions and neighbors we are able to construct a number of extremal binary self-dual codes of different lengths with new parameters in their weight enumerators. In particular we construct 2 new codes of length 64, 4 new codes of length 66 and 14 new codes of length 68. The binary generator matrices of the new codes are available online at [8].
... Dougherty, Yildiz and Karadeniz extended their work over the ring R(2, l) for an arbitrary integer l by defining a homogeneous weight on the ring and deriving an isometry from R(2, l) to a product of binary field elements under the homogeneous and Hamming weight, respectively. Other studies on R(2, l) are done in [7] and [15]. This paper is organized as follows: a brief discussion on Frobenius rings, trace functions on Galois fields and weight functions on a commutative ring is given in Section 2, ring structure and modular properties of R(q, l) in Section 3.1, and the derivation of weight functions on R(q, l) some of which are egalitarian, homogeneous or neither in Section 3.2. ...
Let q = p m be a power of a prime p and m, l 2 N. Denote by F q the Galois field of characteristic p and cardinality q. In this paper, the ring R(q, l) = F q [u1, u2,. .. , u l ]/ u 2 i which is a non-principal ideal ring Frobenius ring was examined. The ring has been shown to be isomorphic to a ring of poly-nomials over Fq and a subring of the ring of 2 l ⇥ 2 l upper triangular matrices over Fq. The latter isomorphism was then used to define a weight function on R(q, l) called the MB-weight some of which are egalitarian. Following the definition of the weight defined by Bachoc on R(p, 1), a Bachoc weight on R(2, l) was defined. Conditions on the parameters m and l of the ring were determined in order for the Bachoc weight to be homogeneous. Lastly, a generating character on R(q, l) was obtained in order to derive a homogeneous weight on the ring for any q and l.
... , 92, 94, 100, 101, 115} \ {4, 7, 58, 70, 91} (see [5], [8], [10], [17] and [18]). Extremal singly even self-dual codes with weight enumerator W 66,2 are known (see [8] and [15] (see [9], [10], [11] and [12]). ...
For lengths 64 and 66, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new 40 inequivalent extremal doubly even self-dual [64,32,12] codes with covering radius 12 meeting the Delsarte bound.
... , 74, 76, 77, 78, 80, 83, 84, 86, 87, 92, 94 in W 66,1 . For a list of known codes in W 66,3 we refer to [10]. ...
In this work, four circulant and quadratic double circulant (QDC)
constructions are applied to the family of the rings R_k,m. Self-dual binary
codes are obtained as the Gray images of self-dual QDC codes over R_k,m.
Extremal binary self-dual codes of length 64 are obtained as Gray images of
?-four circulant codes over R_2,1 and R_2,2. Extremal binary self-dual codes of
lengths 66 and 68 are constructed by applying extension theorems to the F_2 and
R_2,1 images of these codes. More precisely, 11 new codes of length 66 and 39
new codes of length 68 are discovered. The codes with these weight enumerators
are constructed for the ?first time in literature. The results are tabulated.
... This family of rings have provided an alternate method, to many existing ones, of constructing binary self-dual codes of different automorphism groups, and in many cases codes with new weight enumerators. (see [16], [17], [15], [26] for example). The common theme in these works is the presence of a duality and distance preserving Gray map and the intricate structure of the ring with a high number of units that lead to large automorphism groups. ...
In this work, we study codes over the ring
R_{k,m}=F_2[u,v]/<u^{k},v^{m},uv-vu>, which is a family of Frobenius,
characteristic 2 extensions of the binary field. We introduce a distance and
duality preserving Gray map from R_{k,m} to F_2^{km} together with a Lee
weight. After proving the MacWilliams identities for codes over R_{k,m} for all
the relevant weight enumerators, we construct many binary self-dual codes as
the Gray images of self-dual codes over R_{k,m}. In addition to many extremal
binary self-dual codes obtained in this way, including a new construction for
the extended binary Golay code, we find 175 new Type I binary self-dual codes
of parameters [72,36,12] and 105 new Type II binary self-dual codes of
parameter [72,36,12].
In this paper, we construct new self-dual codes from a construction that involves a unique combination; \begin{document}\end{document} block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings \begin{document}\end{document}, \begin{document}\end{document} and \begin{document}\end{document}. Using extensions and neighbours of codes, we construct \begin{document} 32 \end{document} new self-dual codes of length \begin{document} 68 \end{document}. We construct 48 new best known singly-even self-dual codes of length 96.
In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2 and F2+uF2. Using extensions, neighbours and sequences of neighbours, we construct many new self-dual codes. In particular, we construct one new self-dual code of length 66 and 51 new self-dual codes of length 68.
A great deal of attention has been given to codes over finite rings from the 1990s because of their new role in algebraic coding theory and their great application. One of the rings is called Frobenius ring. In this work, we characterize different properties of p-ary local Frobenius rings and their generating characters.
We find a general form for the homogeneous weights of such rings, using the generating character. In particular, we prove that the homogeneous weight has two non-zero values. We also find distance preserving, linear Gray maps for some classes of p-ary local Frobenius rings and using the Gray image, we construct many linear p-ary codes that attain the Griesmer bound
In this work, we study codes over the ring 𝓡k, m = 𝔽2[u, v]/ 〈uk, vm, uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡k, m to F2km together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡k, m. In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].
We define a class of finite Frobenius rings of order , describe their generating characters, and study codes over these rings. We define two conjugate weight preserving Gray maps to the binary space and study the images of linear codes under these maps. This structure couches existing Gray maps, which are a foundational idea in codes over rings, in a unified structure and produces new infinite classes of rings with a Gray map. The existence of self-dual and formally self-dual codes is determined and the binary images of these codes are studied.
The alphabet F2+uF2 is viewed here as a
quotient of the Gaussian integers by the ideal (2). Self-dual F<sub>2
</sub>+uF2 codes with Lee weights a multiple of 4 are called
Type II. They give even unimodular Gaussian lattices by Construction A,
while Type I codes yield unimodular Gaussian lattices. Construction B
makes it possible to realize the Leech lattice as a Gaussian lattice.
There is a Gray map which maps Type II codes into Type II binary codes
with a fixed point free involution in their automorphism group.
Combinatorial constructions use weighing matrices and strongly regular
graphs. Gleason-type theorems for the symmetrized weight enumerators of
Type II codes are derived. All self-dual codes are classified for length
up to 8. The shadow of the Type I codes yields bounds on the highest
minimum Hamming and Lee weights
It is shown that the minimal distance d of a binary
self-dual code of length n ⩾74 is at most
2[( n +6)/10]. This bound is a consequence of some new conditions
on the weight enumerator of a self-dual code obtained by considering a
particular translate of the code, called its shadow. These conditions
also enable one to find the highest possible minimal distance of a
self-dual code for all n ⩾60; to show that self-dual codes
with d ⩽6 exist precisely for n ⩾22, with d
⩾8 exist precisely for n =24, 32 and n ⩾26,
and with d ⩾10 exist precisely for n ⩾46; and to
show that there are exactly eight self-dual codes of length 32 with
d =8. Several of the self-dual codes of length 34 have trivial
group (this appears to be the smallest length where this can happen)
A number of new extremal self-dual codes of length 66, having weight enumerators for which extremal self-dual codes were not previously known to exist, are constructed.
We introduce codes over an infinite family of rings and describe two Gray maps to binary codes which are shown to be equivalent. The Lee weights for the elements of these rings are described and related to the Hamming weights of their binary image. We describe automorphisms in the binary image corresponding to multiplication by units in the ring and describe the ideals in the ring, using them to define a type for linear codes. Finally, Reed Muller codes are shown as the image of linear codes over these rings.
The focus in this work is on self-dual codes over the ring F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2. Type I and Type II codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 are defined and some of the techniques used in the literature are applied to get some theoretical results about self-dual codes on F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 and their binary images. In particular some extremal and optimal binary codes of Type I and Type II including the extended Golay code are obtained as the Gray images of codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2.
In this work, we investigate linear codes over the ring
. We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the
-linearity of binary codes under the Gray map and give a main class of binary codes as an example of
-linear codes. The duals and the complete weight enumerators for
-linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over
are obtained.
In this work, the double-circulant, bordered-double-circulant and stripped bordered-double-circulant constructions for self-dual codes over the non-chain ring R 2 = F 2 + uF 2 + vF 2 + uvF 2 are introduced. Using these methods, we have constructed three extremal binary Type I codes of length 64 of new weight enumerators for which extremal codes were not known to exist. We also give a double-circulant construction for extremal binary self-dual codes of length 40 with covering radius 7.
Recently extremal double circulant self-dual codes have been classified for lengths n ≤ 62. In this paper, a complete classification
of extremal double circulant self-dual codes of lengths 64 to 72 is presented. Almost all of the extremal double circulant
singly-even codes given have weight enumerators for which extremal codes were not previously known to exist.
We construct extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.
This article is a survey of the current status of the classification and enumeration of self-dual linear codes of small to moderate lengths over the fields F2, F3, and F4 and the rings Z4, F2+uF2, and F2+vF2. Self-duality is considered using a variety of inner products. We also examine formally self-dual binary codes and additive self-dual codes over F4.
A method for constructing binary self-dual codes having an automorphism of order p
2 for an odd prime p is presented in (S. Bouyuklieva et al. IEEE. Trans. Inform. Theory, 51, 3678–3686, 2005). Using this method, we investigate
the optimal self-dual codes of lengths 60 ≤ n ≤ 66 having an automorphism of order 9 with six 9-cycles, t cycles of length 3 and f fixed points. We classify all self-dual [60,30,12] and [62,31,12] codes possessing such an automorphism, and we construct
many doubly-even [64,32,12] and singly-even [66,33,12] codes. Some of the constructed codes of lengths 62 and 66 are with
weight enumerators for which the existence of codes was not known until now.
The binary optimal self-orthogonal codes of dimension 4⩽k⩽10 and length 15⩽n⩽k+16 are classified. They are used for constructing self-dual codes of lengths 40, 42, and 44. All optimal binary self-dual codes of lengths 42 and 44 which have an automorphism of order 5 with four independent cycles are obtained up to equivalence. All optimal self-dual codes of lengths 40, 42 and 44 which have an automorphism of order 3 with six independent cycles are constructed. Some of them are the first known codes with their weight enumerators.
In the rst of two papers on Magma, a new system for computational algebra, we present the Magma language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets. c 1997 Academic Press Limited Magma is a new software system for computational algebra, the design of which is based on the twin concepts of algebraic structure and morphism. The design is intended to provide a mathematically rigorous environment for computing with algebraic struc- tures (groups, rings, elds, modules and algebras), geometric structures (varieties, special curves) and combinatorial structures (graphs, designs and codes). The philosophy underlying the design of Magma is based on concepts from Universal Algebra and Category Theory. Key ideas from these two areas provide the basis for a gen- eral scheme for the specication and representation of mathematical structures. The user language includes three important groups of constructors that realize the philosophy in syntactic terms: structure constructors, map constructors and set constructors. The util- ity of Magma as a mathematical tool derives from the combination of its language with an extensive kernel of highly ecient C implementations of the fundamental algorithms for most branches of computational algebra. In this paper we outline the philosophy of the Magma design and show how it may be used to develop an algebraic programming paradigm for language design. In a second paper we will show how our design philoso- phy allows us to realize natural computational \environments" for dierent branches of algebra. An early discussion of the design of Magma may be found in Butler and Cannon (1989, 1990). A terse overview of the language together with a discussion of some of the implementation issues may be found in Bosma et al. (1994).
Construction methods for self-dual codes are given. By using these methods some new extremal self-dual [66, 33, 12] and [68, 34, 12] codes are obtained.
In this correspondence, new extremal self-dual codes of length 62 are constructed with weight enumerators of three different types. Two of these types were not represented by any known code up till now. All these codes possess an automorphism of order 15. Some of them are used to construct extremal self-dual codes of length 60 by the method of subtracting. By additional subtracting, an extremal self-dual [58, 29, 10] code was obtained having a weight enumerator which does not correspond to any code known so far.
We construct new extremal self-dual binary codes of lengths 36,
38, and 58. We show that there are at least 14 inequivalent extremal
self-dual [38,18,8] codes and that there are at least 368 inequivalent
extremal self-dual [38,19,8] codes. For length 58, we construct 11
extremal self-dual [58,29,10] codes whose weight enumerators were
previously unknown
New extremal self-dual codes of dimensions 33 and 34 are obtained
In this correspondence, we investigate binary extremal self-dual
codes. Numerous extremal self-dual codes and interesting self-dual codes
with minimum weight d=14 and 16 are constructed. In particular, the
first extremal Type I [86,43,16] code and new extremal self-dual codes
with weight enumerators which were not previously known to exist for
lengths 40,50,52 and 54 are constructed. We also determine the possible
weight enumerators for extremal Type I codes of lengths 66-100