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New extremal binary self-dual codes of length 68 from R2-lifts of binary self-dual codes
Abstract
A lift of binary self-dual codes to the ring R2 is described. By
using this lift, a family of self-dual codes over R2 of length 17 are constructed.
Taking the binary images of these codes, extremal binary self-dual codes of
length 68 are obtained. For the �rst time in the literature, extremal binary
codes of length 68 with
= 4 and
= 6 in W68;2 have been obtained. In
addition to these, six new codes with
= 0 and fourteen new codes with
= 2
in W68;2 have also been found.
... Several construction methods have been employed for this purpose. Among the most common ones, we can mention double and bordered double-circulant constructions, constructions with a specific automorphism group, and recently ring constructions using different rings of characteristic 2. We refer the reader to [3,4,8,10,11,13,14,16,22] and [20] for more on these constructions. ...
... Tsai et al. constructed a substantial number of codes in both possible weight enumerators in [20]. Recently, 32 new codes are obtained in [15] and 28 new codes including the first examples with γ = 4 and γ = 6 in W 68,2 are obtained in [13]. Together with the ones in [13,15] codes exist for W 68,2 when γ = 0 and β = 38, 40, 44, 45, 47, Table 2. Throughout Tables 3-7, 9 the codes are generated over F 2 + uF 2 by the matrices of the following form; ...
... Recently, 32 new codes are obtained in [15] and 28 new codes including the first examples with γ = 4 and γ = 6 in W 68,2 are obtained in [13]. Together with the ones in [13,15] codes exist for W 68,2 when γ = 0 and β = 38, 40, 44, 45, 47, Table 2. Throughout Tables 3-7, 9 the codes are generated over F 2 + uF 2 by the matrices of the following form; ...
In this work, extension theorems are used for self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F2m+uF2m for m=1,2. The duality and distance preserving Gray maps from F4+uF4 to F2+uF2)2 and F24 are used to obtain self-dual codes whose binary Gray images are [64,32,12]-extremal self-dual. An F2+uF2-extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the first examples of codes with γ=3 and many codes with the rare γ=4,6 parameters are obtained. In addition to these, two hundred fifty doubly even self dual [96,48,16]-codes with new weight enumerators are obtained from four-circulant codes over F4+uF4. New extremal doubly even binary codes of lengths 80 and 88 are also found by the F2+uF2-lifts of binary four circulant codes and thus a lower bound on the number of non-isomorphic 3-(80, 16, 665) designs is modified.
... 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. ...
... Theorem 2.3. ( [11]) (F 2 replaced by F 4 ) Let C be a self-dual code of length n over F 4 + uF 4 then µ (C) is a self-orthogonal code of length n over F 4 . Definition 2.4. ...
... Tsai et al. constructed a substantial number of codes in both possible weight enumerators in [18]. Recently, 178 new codes are obtained in [15] and 28 new codes including the first examples with γ = 4 and γ = 6 in W 68,2 are obtained in [11]. Together with the ones in [11,15] [18]. ...
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.
... Tsai et al. constructed a substantial number of codes in both possible weight enumerators in [17]. Most recently, 28 new codes including the first examples with γ = 4 and γ = 6 in W 68,2 are obtained in [11]. For the list of codes with γ = 4 and γ = 6 in W 68,2 we refer to [11]. ...
... Most recently, 28 new codes including the first examples with γ = 4 and γ = 6 in W 68,2 are obtained in [11]. For the list of codes with γ = 4 and γ = 6 in W 68,2 we refer to [11]. Together with the ones in [11] [17]. ...
... For the list of codes with γ = 4 and γ = 6 in W 68,2 we refer to [11]. Together with the ones in [11] [17]. ...
In this work, quadratic residue codes over the ring F2+uF2+u2F2F2+uF2+u2F2 with u3=uu3=u are considered. A duality and distance preserving Gray map from F2+uF2+u2F2F2+uF2+u2F2 to F23 is defined. By using quadratic double circulant, quadratic bordered double circulant constructions and their extensions self-dual codes of different lengths are obtained. As Gray images of these codes and their extensions, a substantial number of new extremal self-dual binary codes are found. More precisely, thirty two new extremal binary self-dual codes of length 68, 363 Type I codes of parameters [72,36,12][72,36,12], a Type II [72,36,12][72,36,12] code and a Type II [96,48,16][96,48,16] code with new weight enumerators are obtained through these constructions. The results are tabulated.
... Several construction methods have been employed for this purpose. Among the most common ones, we can mention double and bordered double-circulant constructions, constructions with a specific automorphism group, and recently ring constructions using different rings of characteristic 2. We refer the reader to [3,4,7,9,10,12,13,15,20] and [21] for more on these constructions. ...
... Tsai et al. constructed a substantial number of codes in both possible weight enumerators in [21]. Recently, 32 new codes are obtained in [14] and 28 new codes including the first examples with γ = 4 and γ = 6 in W 68,2 are obtained in [12]. Together with the ones in [12,14] ...
... Recently, 32 new codes are obtained in [14] and 28 new codes including the first examples with γ = 4 and γ = 6 in W 68,2 are obtained in [12]. Together with the ones in [12,14] ...
In this work, extension theorems are generalized to self-dual codes over
rings and as applications many new binary self-dual extremal codes are found
from self-dual codes over F_2^m+uF_2^m for m = 1, 2. The duality and distance
preserving Gray maps from F4 +uF4 to (F_2 +uF_2)^2 and (F_4)^2 are used to
obtain self-dual codes whose binary Gray images are [64,32,12]-extremal
self-dual. An F_2+uF_2-extension is used and as binary images, 178 extremal
binary self-dual codes of length 68 with new weight enumerators are obtained.
Especially the ?rst examples of codes with gamma=3 and many codes with the rare
gamma= 4, 6 parameters are obtained. In addition to these, two hundred ?fty
doubly even self dual [96,48,16]-codes with new weight enumerators are obtained
from four-circulant codes over F_4 + uF_4. New extremal doubly even binary
codes of lengths 80 and 88 are also found by the F_2+uF_2-lifts of binary four
circulant codes and a corresponding result about 3-designs is stated.
... Recently, some rings of characteristic 2 have been used effectively to construct new extremal binary self-dual codes. Lifts were used in [11] and [13]. Extension theorems for self-dual codes were applied to codes over F 4 + uF 4 in [16]. ...
... Together with the codes obtained in [20,14] We obtain a code with a weight enumerator β = 169 in W 68,1 . First codes with γ = 4 and γ = 6 in W 68,2 are constructed in [11]. Recently, new codes in W 68,2 are obtained in [16,13,14] together with these, codes exist for By considering R 2,1 -extensions of codes in Table 2 with respect to Theorem 5.2 we were able to obtain 14 new extremal binary self-dual codes, which are listed in Table 5. L i X c γ β L 4 (1313uu0133130u11) 1 + u 2 60 L 4 (1131uu011133u011) 1 2 62 L 4 (0001u11uu3110300) 1 2 64 L 4 (00u1u130u111u1u0) 1 + u 2 66 L 4 (uuu30330013101uu) 1 + u 2 70 L 4 (u0u1u13uu333u3u0) 1 + u 2 72 L 3 (u3000uu33u31u031) 1 2 166 L 3 (u1u0u0u11u31uu13) 1 + u 2 170 L 3 (03u0u00330310u31) 1 + u 2 172 L 3 (u1uuu0u11u31u013) 1 + u 2 174 L 3 (01000u0110310013) 1 + u 2 176 L 3 (011300u031111313) 1 3 156 L 3 (3u131011301u0u10) 1 + u 3 172 L 3 (103130333010u010) 1 + u 3 180 Example 5.3. ...
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.
... In the classical study of the structure of linear cyclic codes over rings, most of the rings are assumed to be commutative. For example, linear cyclic codes over different types of finite commutative chain rings have been studied in [1, 3, 4, 6-8, 10, 12, 14, 15, 18, 20, 22, 23], and those over some finite commutative non-chain rings have been studied in [13,19,21,[24][25][26]. ...
In this study, we consider the finite (not necessary commutative) chain ring , where θ is an automorphism of , and completely explore the structure of left and right cyclic codes of any length N over , that is, left and right ideals of the ring . For a left (right) cyclic code, we determine the structure of its right (left) dual. Using the fact that self-dual codes are bimodules, we discuss on self-dual cyclic codes over . Finally, we study Gray images of cyclic codes over and as some examples, three linear codes over with the parameters of the best known ones, but with different weight distributions, are obtained as the Gray images of cyclic codes over .
... 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].
... Recently, several new families of rings have been studied in connection with coding theory, rings that are not finite chain but are Frobenius. These rings have rich algebraic structures and they lead to binary codes with large automorphism groups and in some cases new binary self-dual codes [22,6,14,15]. F 2 + uF 2 is a size 4 ring that also has generated a lot of interest among coding theorists starting with [7]. There is an interesting connection between Z 4 and F 2 + uF 2 . ...
Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of
Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams
identities for the complete, symmetrized and Lee weight enumerators are proved.
Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and
self-dual codes over Z_4+uZ_4 are studied in connection with these projections.
Finally three constructions are given for formally self-dual codes over
Z_4+uZ_4 and their Z_4-images together with some good examples of formally
self-dual Z_4-codes obtained through these constructions.
In this work, we define a modification of a bordered construction for self-dual codes which utilises \begin{document}\end{document}-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.
In this work, we define a modification of a bordered construction for self-dual codes which utilises -circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.
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].
In this work, we propose a modified four circulant construction for self-dual
codes and a bordered version of the construction using the properties of
\lambda-circulant and \lambda-reverse circulant matrices. By using the
constructions on , we obtain new binary codes of lengths 64 and 68. We
also apply the constructions to the ring and considering the and
-extensions, we obtain new singly-even extremal binary self-dual codes of
lengths 66 and 68. More precisely, we find 3 new codes of length 64, 15 new
codes of length 66 and 22 new codes of length 68. These codes all have weight
enumerators with parameters that were not known to exist in the literature.
We study self-dual codes over chain rings. We describe a technique for constructing new self-dual codes from existing codes and we prove that for chain rings containing an element c with c 2 = −1 all self-dual codes can be constructed by this technique. We extend this construction to self-dual codes over principal ideal rings via the Chinese Remainder Theorem. We use torsion codes to describe the structure of self-dual codes over chain rings and to set bounds on their minimum Hamming weight. Interestingly, we find the first examples of MDS self-dual codes of lengths 6 and 8 and near-MDS self-dual codes of length 10 over a certain chain ring which is not a Galois ring.
We prove that self-dual codes exist over all finite commutative Frobenius rings, via their decomposition by the Chinese Remainder Theorem into local rings. We construct non-free self-dual codes under some conditions, using self-dual codes over finite fields, and we also construct free self-dual codes by lifting elements from the base finite field. We generalize the building-up construction for finite commutative Frobenius rings, showing that all self-dual codes with minimum weight greater than 2 can be obtained in this manner in cases where the construction applies.
A method to design binary self-dual codes with an automorphism of
order two without fixed points is presented. Extremal self-dual codes
with lengths 40, 42, 44, 54, 58, 68 are constructed. Many of them have
weight enumerators for which extremal codes were previously not known to
exist
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)
. This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from finite fields to finite Frobenius rings. It is over Frobenius rings that certain key identifications can be made between the ring and its complex characters. Introduction Since the appearance of [8] and [19], using linear codes over Z=4Z to explain the duality between the non-linear binary Kerdock and Preparata codes, there has been a revival of interest in codes defined over finite rings. This paper examines the foundations of algebraic coding theory over finite rings and singles out the finite Frobenius rings as the most appropriate rings for coding theory. Why are Frobenius rings appropriate for coding theory? Because two classical theorems of MacWilliams---the extension theorem and th...
We develop a construction method for finding self-dual codes with an automorphism of order p with c independent p-cycles. In more detail, we construct a self-dual code with an automorphism of type p-(c,f+2) and length n+2 from a self-dual code with an automorphism of type p-(c,f) and length n, where an automorphism of type p-(c,f) is that of order p with c independent cycles and f fixed points. Using this construction, we find three new inequivalent extremal self-dual [54,27,10] codes with an automorphism of type 7-(7,5) and two new inequivalent extremal self-dual [58,29,10] codes with an automorphism of of type 7-(8,2). We also obtain an extremal self-dual [40,20,8] code with an automorphism of type 3-(10,10), which is constructed from an extremal self-dual [38,19,8] code of type 3-(10,8), and at least 482 inequivalent extremal self-dual [58,29,10] codes with an automorphism of type 3-(18,4), which is constructed from an extremal self-dual [54,27,10] code of type 3-(18,0); we note that the extremality is preserved.
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.
We give a general experimental method generalizing the codes of Carlach and Vervoux (Proceedings of the 13th Applicable Algebra in Engineering Communication and Computing (AAECC 13), Hawaii, USA, 14–19 November 1999, p. 15) to construct self-dual codes. We consider the particular fields GF(2), GF(3), GF(4) (both Euclidean and Hermitian cases), GF(5) and GF(7). We give numerical tables of the best known self-dual codes over these alphabets up to lengths where minimum distances are computable. These tables regularly fill gaps between known codes with good parameters such as the quadratic residue codes, the Pless symmetry codes (J. Combin. Theory Ser. A A12 (1972) 119) or the quadratic double circulant codes (J. Combin. Theory Ser. A 97 (2002) 85). Many new codes with better parameters are constructed: in particular the first extremal ternary self-dual [52,26,15] and [78,39,18] codes and the first binary self-dual [114,57,18] and [116,58,18] codes are constructed. We also give the minimum weight of the Pless symmetry codes of length 84. We also update tables for quadratic residue codes over GF(4),GF(5) and GF(7) and we obtain in particular a [62,31,20] code over GF(5).
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).
The purpose of this paper is to complete the classification of binary
self-dual [48,24,10] codes with an automorphism of odd prime order. We prove
that if there is a self-dual [48, 24, 10] code with an automorphism of type
p-(c,f) with p being an odd prime, then p=3, c=16, f=0. By considering only an
automorphism of type 3-(16,0), we prove that there are exactly 264 inequivalent
self-dual [48, 24, 10] codes with an automorphism of odd prime order,
equivalently, there are exactly 264 inequivalent cubic self-dual [48, 24, 10]
codes.
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, we construct an extremal singly-even self-dual [64,32,12] code with a weight enumerator which was not previously known to exist.
Conway and Sloane (1990) have previously given an upper bound on
the minimum distance of a singly-even self-dual binary code, using the
concept of the shadow of a self-dual code. We improve their bound,
finding that the minimum distance of a self-dual binary code of length n
is at most 4[n/24]+4, except when n mod 24=22, when the bound is
4[n/24]+6. We also show that a code of length a multiple of 24 meeting
the bound cannot be singly-even. The same technique gives similar
results for additive codes over GF(4) (relevant to quantum coding
theory)
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
<em>Self-dual codes over commutative Frobenius rings</em>,
- S. T. Dougherty
<em>$R_2$-generator matrices for extremal self-dual codes of length 68</em>,
- S. Karadeniz
<em>Database of self-dual codes</em>,
- A. Munemasa