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Double-circulant and bordered-double-circulant constructions for self-dual codes over R_2
Abstract
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.
... Especially self-dual codes over the rings of order 4, finite chain rings and Frobenius rings have been studied quite extensively. For some of these works we refer to [6], [5], [23], [13], [15]. ...
... Recently, the authors have found extremal binary codes of new weight enumerators by using self-dual codes over a family of rings of characteristic 2. ( [13], [15]). ...
... For example in [2], a construction was given to construct selfdual codes with a particular automorphism. In [13], a construction of self-dual codes over the family of rings R k is given which is then used to construct binary self-dual codes. Very general versions of the building up construction of self-dual codes were given in [7] and [6]. ...
... Also from Equation (2) and (3) for a rectangular 3-class symmetric association scheme we have 1 + a + mn = 0, ac + r + s(m + 1) + t(n + 1) + u(m + 1)(n + 1) = 0. (14) where nm = v. Since (s + u) m = 0 and (t + u) n = 0, Equations (13) and (14) can be reduced to r + s + t + u = 1, ...
3-Class association schemes are used to construct binary self-dual codes. We use the pure and bordered construction to get self-dual codes starting from the adjacency matrices of symmetric and non-symmetric 3-class association schemes. In some specific cases, we also study constructions of self-dual codes over . For symmetric 3-class association schemes, we focus on the rectangular scheme and association schemes derived from symmetric designs.
... Especially self-dual codes over the rings of order 4, finite chain rings and Frobenius rings have been studied quite extensively. For some of these works we refer to [6], [5], [23], [13], [15]. ...
... Recently, the authors have found extremal binary codes of new weight enumerators by using self-dual codes over a family of rings of characteristic 2. ( [13], [15]). ...
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.
... The ring R 2 is introduced by Yildiz and Karadeniz in [16]. Codes over this ring have been studied by them further in [17], [18], [19]. ...
... Both methods have been used extensively in the literature to produce many extremal binary self-dual codes. (See [18], [22], [14] and references therein for some related works.) ...
We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F_2+uF_2, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.
... Instead, generating sets has been used to study such codes. Other studies on R(q, 2) followed soon after ( [1], [5], [6], [9],[10], [12], [14]). 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. ...
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.
... The double circulant constructions for FSD codes were given in [8]. In [1], these constructions were extended to λ-circulant matrices. ...
We shall describe several families of X-rings and construct self-dual and formally self-dual codes over these rings. We then use a Gray map to construct binary formally self-dual codes from these codes. In several cases, we produce binary formally self-dual codes with larger minimum distances than known self-dual codes. We also produce non-linear codes which are better than the best known linear codes.
... Since then, construction of extremal codes with new weight enumerators has generated a great interest. See [3] [5] [11] [18] [15] for some of them. There are two types of weight enumerators for extremal self-dual codes of length 58 as was described in [6]: ...
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.
... Among others, linear codes and cyclic codes over this ring are studied in [15] and [16]. Moreover, using self-dual codes over R 2 , some new extremal binary self-dual codes are obtained in [13]. On the other hand, the class of quasi-cyclic (QC) codes occupies an important place in both codes over finite fields and codes over finite rings. ...
We consider quasi-cyclic codes over the ring F2 + u F 2 + v F2 + u v F2, a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions.
... 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.
... Among the techniques used to construct extremal binary self-dual codes are double-circulant and bordered-double-circulant constructions, using neighbouring codes and automorphism groups. For some of the works done in this direction we refer to [1,3,9,17,13]. ...
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.
In this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F_4+uF_4 together with the well known extension and neighbour methods, we construct new self-dual codes of length 68. In particular, we find 41 new codes of length 68 that were not known in the literature before.
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 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.
This paper studies and classifies all binary self-dual [62,31,12] and [64,32,12] codes having an automorphism of order 7 with 8 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order p. There are exactly 8 inequivalent binary self-dual [62,31,12] codes with an automorphism of type 7-(8,6). As for binary [64,32,12] self-dual codes with an automorphism of type 7-(8,8) there are 44465 doubly-even and 557 singly-even such codes. Some of the constructed singly-even codes for both lengths have weight enumerators for which the existence was not known before.
In this work, we study construction techniques of formally self-dual codes over the infinite family of rings Rk=F2[u1,u2,…,uk]/〈ui2=0,uiuj=ujui〉. These codes give rise to binary formally self-dual codes. Using these constructions, we obtain a number of good formally self-dual binary codes including even formally self-dual binary codes of parameters [72,36,14][72,36,14], [56,28,12][56,28,12], [44,22,10][44,22,10] and odd formally self-dual binary codes of parameters [72,36,13][72,36,13], all of which have better minimum distances than the best known self-dual codes of the same lengths.
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.
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.
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)
It is known that only the prime numbers 2,3,5,7 and 19 can divide the order of the automorphism group of a doubly-even self-dual [40,20,8]-code. The authors construct all binary doubly even self-dual [40,20,8]-codes admitting an automorphism of order 5. Their number equals 45.
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.
Previously, Type IV self dual codes over the ring F2+uF 2 of order 4 have been introduced. In this paper, we construct five optimal Type IV self-dual codes which are the first examples of such codes for that length. We also give several new optimal type IV self-dual codes, and Type IV self-dual codes which improve the upper bounds for minimum weights
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.
In this paper, we construct new binary singly even self-dual codes with larger minimum weights than the previously known singly even self-dual codes for several lengths. Several known construction methods are used to construct the new self-dual codes.
In the present paper we develop a method to determine the coset weight distributions and covering radius of doubly even self-dual extremal binary codes of length 40. The method is algebraic in nature and largely eliminates necessary computations by electronic computers. The method easily applies to longer codes (e.g. self-dual [56,28,12] binary codes) or to non-extremal codes.
We construct new extremal self-dual [40,20,8] codes with covering radius 7. It is also shown that the vectors of a fixed weight in a coset of weight 4n+2 in an extremal doubly even self-dual code of length 24n+16 such that the coset has no vector of weight 4n+4 form a 1-design.
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).
It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y
8+872y
10+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.
All singly-even self-dual [40,20,8] binary codes which have an automorphism of prime order
are obtained up to equivalence. There are two inequivalent codes with an automorphism of order 7 and 37 inequivalent codes with an automorphism of order 5. These codes have highest possible minimal distance and some of them are the first known codes with weight enumerators prescribed by Conway and Sloane.
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.
All distinct extremal double-circulant self-dual codes of length up to 62 have been found by exhaustive search. These codes are classified in this paper using new and previously known methods. Several of these codes are new extremal self-dual codes.
In this note, an extremal doubly-even [40,20,8] code with covering radius 7 is given. This code is the first extremal doubly-even code whose covering radius does not meet the Delsarte bound. Extremal singly-even codes with the smallest covering radius are also given. One of them is an extremal singly-even [44,22,8] code with covering radius 7, which determines t[43,22]=6 by puncturing.
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.
We present a method to determine the complete coset weight
distributions of doubly even binary self-dual extremal [56, 28, 12]
codes. The most important steps are (1) to describe the shape of the
basis for the linear space of rigid Jacobi polynomials associated with
such codes in each index i, (2) to describe the basis polynomials for
the coset weight enumerators of the assigned coset weight i by means of
rigid Jacobi polynomials of index i. The multiplicity of the cosets of
weight i have a connection with the frequency of the rigid reference
binary vectors v of weight i for the Jacobi polynomials. This
information is sufficient to determine the complete coset weight
distributions. Determination of the covering radius of the codes is an
immediate consequence of this method. One important practical advantage
of this method is that it is enough to get information on 8190 codewords
of weight 12 (minimal-weight words) in each such code for computing
every necessary information