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Self-dual Rk lifts of binary self-dual codes
Abstract
In this paper, we give a method to lift binary self-dual codes to the ring . The lifting method requires solving a system of linear equations over . This technique is applied to binary self-dual code to obtain self-dual codes over . As Gray images of these codes, a substantial number of self-dual codes are generated. By using the extension theorem given by Bouyuklieva and Bouyukliev in [2], ten new extremal binary self-dual codes of length 58 with new enumerators are found which were not previously known to exist.
... F 4 + uF 4 -lifts of self-dual quaternary codes were considered in [13]. The ring F 2 + uF 2 + vF 2 + uvF 2 were used in [11,15]. In [10], Karadeniz and Yildiz obtained self-dual codes as lifts of [8,4,4] 2 extended binary Hamming code to the ring R 3 which is of size 2 8 . ...
... Generator matrices in a special form could be used for lifts. For more details about the ring R 2 and lifting a binary code to R 2 we refer to [6,10,20,11,15]. ...
... Lifting quaternary codes to F 4 + uF 4 have been used in [13]. For more research in this direction we refer the reader to [9,10,11,13,15,16]. The binary self-dual codes obtained from facevertex incidence matrix of connected bicubic planar graphs is in good shape which is suitable for lifting. ...
In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring R_2 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the fi?rst time in the literature. The results are tabulated.
... , 58, 63}, β = 2 and γ ∈ {2m | m = 0, 16, . . . , 50, 55} (see [9], [10], [13]). ...
... Proposition 2. There is an extremal singly even self-dual [58, 29, 10] code with weight enumerator W 58,2 for β = 0 and γ ∈ {2m | m = 2, 3, 4, 7, 12}, β = 1 and γ ∈ {2m | m = 8, 9, 10, 11, 12, 15}, β = 2 and γ ∈ {2m | m = 4, 6,7,8,9,10,11,12,13,14,15, 51, 52, 53, 54}. ...
We give a classification of four-circulant singly even self-dual [60,30,d] codes for d=10 and 12. These codes are used to construct extremal singly even self-dual [60,30,12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60,30,12] codes, we also construct extremal singly even self-dual [58,29,10] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight 1.
... In particular, binary self-dual codes have garnered extensive attention, with considerable research effort dedicated to developing techniques for constructing new extremal and optimal binary self-dual codes. These known construction techniques include the doublecirculant and bordered double-circulant constructions [1][2][3][4], the four-circulant construction and its variations [5][6][7][8][9][10][11], group rings and their connection to self-dual codes [12][13][14][15][16][17][18][19][20], bordered matrix constructions [12][13][14]17,19,[21][22][23][24], neighbours of binary self-dual codes [25,26], the widely employed building-up construction [27][28][29], the production of binary self-dual codes as Gray images of self-dual codes over finite commutative Frobenius rings of characteristic 2 [28,[30][31][32], and the well-known lifting method [10,11,[33][34][35][36][37]. ...
In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for k×2k matrices M over R satisfying the conditions that its rows are linearly independent over R and MM⊤=α⊤α for an R-linearly independent vector α∈Rk. Let C be a linear code generated by such a matrix M. We prove that the dual code C⊥ of C is also a free linear code with dimension k, as well as C/Hull(C) and C⊥/Hull(C) are one-dimensional free R-modules, where Hull(C) represents the hull of C. Based on these facts, an isometry from Rx+Ry onto R2 is established, assuming that x+Hull(C) and y+Hull(C) are bases for C/Hull(C) and C⊥/Hull(C) over R, respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix MM⊤ takes the form of α⊤α with α being a linearly independent vector in Rk, a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy MM⊤=0. The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach.
... There are some construction methods in the literature to obtain binary self-dual codes. Some of them are known as the double circulant construction, bordered double circulant construction, constructions with a specific automorphism group and building-up construction [1,[9][10][11][13][14][15][16]. ...
In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring where v2 = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over . Also, we obtain upper bounds on the minimum distance of linear codes for both the Lee distance and the Gray distance. Moreover, we find some free Euclidean and free Hermitian self-dual codes over via some useful construction methods.
... , 58, 63}, β = 2 and γ ∈ {2m | m = 0, 16, . . . , 50, 55} (see [9,10,14]). ...
We give a classification of four-circulant singly even self-dual [60,30,d] codes for d=10 and 12. These codes are used to construct extremal singly even self-dual [60,30,12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60,30,12] codes, we also construct extremal singly even self-dual [58,29,10] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight 1.
In this work, we study linear codes over the ring and their weight enumerators, where . We first give the structure of the ring and investigate linear codes over this ring. We also define two weights called Lee weight and Gray weight for these codes. Then we introduce two Gray maps from to and study the Gray images of linear codes over the ring. Moreover, we prove MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators.
Recently, Karadeniz and Yildiz introduced an efficient method to search for self-dual codes. It is called lifting method and can be applied to some alphabets. In this work, by considering -lifts of binary self-dual codes of length 16 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the first time in the literature. The results are tabulated.
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.
An efficient algorithm for classification of binary self-dual codes is
presented. As an application, a complete classification of the self-dual codes
of length 38 is given.
In this paper, we investigate binary self-dual codes with an
automorphism of order 2 with c cycles and f fixed points. A method for
constructing such codes using self-orthogonal codes of length c and
self-dual codes of length f is presented. We apply this method to
construct extremal self-dual codes of lengths 40, 42, 44, 52, 54, and
58. Some of them have weight enumerators for which self-dual codes were
previously not known to exist. We prove that there do not exist
self-dual [50, 25, 10] and [96, 48, 20] codes with an automorphism of
order 2 with f fixed points for f>0 in their automorphism
groups
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
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)
In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to . Using this method we construct self-dual codes over of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which have weight enumerators that were not previously known to exist. The new codes that we found have and 52 in and they all have automorphism groups of size 8.
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.
Using a method for constructing binary self-dual codes with an automorphism of odd prime order
p
p
, we give a full classification of all optimal binary self-dual
[50+2t,25+t]
[
50
+
2
t
,
25
+
t
]
codes having an automorphism of order 5 for
t
=
0
,
⋯
,
5
. As a consequence, we determine the weight enumerators for which there is an optimal binary self-dual
[
52
,
26
,
10
]
code. Some of the constructed codes for lengths 52, 54, 58, and 60 have new values for the parameter in their weight enumerator. We also construct more than 3,000 new doubly-even
[56,28,12]
[
56
,
28
,
12
]
self-dual codes.
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.
In this correspondence, a lift of the shortened binary [8,4,4][8,4,4] Hamming code to the ring R3 is described. Using this lift, a family of self-dual codes over R3 of length 7 are obtained. By taking the binary images of these codes and applying the extension method given in [15], 11 new extremal binary self-dual [58,29,10][58,29,10] codes with new weight enumerators are found which 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.
All binary [ n , n /2] optimal self-dual codes for length 52 ≤ n ≤ 60 with an automorphism of order 7 or 13 are classified up to equivalence. Two of the constructed [54,27,10] codes have weight enumerators that were not previously known to exist. There are also some [58,29,10] codes with new values of the parameters in their weight enumerator.
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).
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.
We construct extremal self-dual codes with lengths 44, 50, 54, and
58. They have weight enumerators for which extremal codes were
previously not known to exist. Two methods are used for constructing the
codes using self dual codes of same or smaller length. To obtain the
codes we use a combinatorial optimization search
The authors discuss extremal self-dual [60,30,12] codes and then
go on to present new extremal self dual [58,29,10] codes
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)
Some new extremal self-dual codes with lengths 44
- I Bouyukliev
- S Bouyuklieva
I. Bouyukliev, S. Bouyuklieva, Some new extremal self-dual codes with lengths 44, 50, 54 and 58,
IEEE Trans. Inf. Theory 44 (2) (1998) 809–812.