Extremal Binary Self-dual Codes Research Articles

A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes

<p style='text-indent:20px;'>In this paper, a genetic algorithm, one of the evolutionary algorithm optimization methods, is used for the first time for the problem of computing extremal binary self-dual codes. We present a comparison of the computational times between the genetic algorithm and a linear search for different size search spaces and show that the genetic algorithm is capable of computing binary self-dual codes significantly faster than the linear search. Moreover, by employing a known matrix construction together with the genetic algorithm, we are able to obtain new binary self-dual codes of lengths 68 and 72 in a significantly short time. In particular, we obtain 11 new binary self-dual codes of length 68 and 17 new binary self-dual codes of length 72.</p>

Extremal binary self-dual codes from a bordered four circulant construction

In this paper, we present a new bordered construction for self-dual codes which employs λ-circulant matrices. We give the necessary conditions for our construction to produce self-dual codes over a finite commutative Frobenius ring of characteristic 2. Moreover, using our bordered construction together with the well-known building-up and neighbour methods, we construct many binary self-dual codes of lengths 56, 62, 78, 92 and 94 with parameters in their weight enumerators that were not known in the literature before.

New extremal binary self-dual codes from block circulant matrices and block quadratic residue circulant matrices

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.

Composite matrices from group rings, composite G-codes and constructions of self-dual codes

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma =7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction

New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings

<p style='text-indent:20px;'>Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form <inline-formula><tex-math id="M1">\begin{document}$ G = (I_n \ | \ A), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ I_n $\end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id="M3">\begin{document}$ n \times n $\end{document}</tex-math></inline-formula> identity matrix and <inline-formula><tex-math id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id="M5">\begin{document}$ n \times n $\end{document}</tex-math></inline-formula> matrix fully determined by the first row. In this work, we define a generator matrix in which <inline-formula><tex-math id="M6">\begin{document}$ A $\end{document}</tex-math></inline-formula> is a block matrix, where the blocks come from group rings and also, <inline-formula><tex-math id="M7">\begin{document}$ A $\end{document}</tex-math></inline-formula> is not fully determined by the elements appearing in the first row. By applying our construction over <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula> and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual <inline-formula><tex-math id="M9">\begin{document}$ [68, 34, 12] $\end{document}</tex-math></inline-formula>-codes with the rare parameters <inline-formula><tex-math id="M10">\begin{document}$ \gamma = 7, 8 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ 9 $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M12">\begin{document}$ W_{68, 2}. $\end{document}</tex-math></inline-formula> In particular, we find 92 new binary self-dual <inline-formula><tex-math id="M13">\begin{document}$ [68, 34, 12] $\end{document}</tex-math></inline-formula>-codes.</p>

New extremal binary self-dual codes of length 68 from generalized neighbors

In this work, we use the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k-neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with γ=8 in their W68,2 and 40 with γ=9 in their W68,2. These examples are the first in the literature for these γ values. This completes the theoretical list of possible values for γ in W68,2.

A modified bordered construction for self-dual codes from group rings

We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78.

Constructing self-dual codes from group rings and reverse circulant matrices

<p style='text-indent:20px;'>In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

Modified quadratic residue constructions and new extremal binary self-dual codes of lengths 64, 66 and 68

In this work we consider modified versions of quadratic double circulant and quadratic bordered double circulant constructions over the binary field and the rings F2+uF2 and F4+uF4 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].

New extremal self-dual binary codes of length 68 via composite construction, &amp;Fopf;&lt;SUB align="right"&gt;2 + &lt;i&gt;u&lt;/i&gt;&amp;Fopf;&lt;SUB align="right"&gt;2 lifts, extensions and neighbours

We describe a composite construction from group rings where the groups have orders 16 and 8. This construction is then applied to find the extremal binary self-dual codes with parameters [32, 16, 8] or [32, 16, 6]. We also extend this composite construction by expanding the search field which enables us to find more extremal binary self-dual codes with the above parameters and with different orders of automorphism groups. These codes are then lifted to &#x1d53d;2 + u&#x1d53d;2, to obtain extremal binary images of codes of length 64. Finally, we use the extension method and neighbour construction to obtain new extremal binary self-dual codes of length 68. As a result, we obtain 28 new codes of length 68 which were not known in the literature before.

Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68

<p style='text-indent:20px;'>We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_4 $\end{document}</tex-math></inline-formula>. These codes have binary images with parameters <inline-formula><tex-math id="M2">\begin{document}$ [32,16,8] $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M3">\begin{document}$ [32,16,6] $\end{document}</tex-math></inline-formula>. These are lifted to codes over <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_4+u\mathbb{F}_4 $\end{document}</tex-math></inline-formula>, to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula> to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.

On the classification of binary self-dual codes admitting imprimitive rank 3 permutation groups

One of the questions of current interest in coding theory is the following: given a finite non-solvable permutation group G acting transitively on a set $$\Omega $$ , under what conditions on G are self-dual codes invariant under G existent or nonexistent? In this paper, this problem is investigated under the hypothesis that the group G is an imprimitive rank 3 permutation group. It is proven that if G is an imprimitive rank 3 permutation group acting transitively on the coordinate positions of a self-dual binary code C then G is one of $${\mathrm{M}}_{11}$$ of degree 22; $${\mathrm{Aut}}({\mathrm{M}}_{12})$$ of degree 24; $$\mathrm{PSL}(2,q)$$ of degree $$2(q +1)$$ for $$q {\equiv 1}{({\mathrm{mod}}\,4)};$$ $$\mathrm{PSL}(m, q)$$ of degree $$2\times \frac{q^m-1}{q-1}$$ for $$m \ge 3$$ odd and q an odd prime; $$\mathrm{PSL}(m, q)$$ of degree $$2\times \frac{q^m-1}{q-1}$$ for $$m \ge 4$$ even and q an odd prime, and $$\mathrm{PSL}(3,2)$$ of degree 14. When combined with a result on the classification of binary self-dual codes invariant under primitive rank 3 permutation groups of almost simple type this yields a result on the non-existence of extremal binary self-dual codes invariant under quasiprimitive rank 3 permutation groups of almost simple type.

New extremal binary self-dual codes from a Baumert–Hall array

In this work, we introduce new construction methods for self-dual codes using a Baumert–Hall array. We apply the constructions over the alphabets F2 and F4+uF4 and combine them with extension theorems and neighboring constructions. As a result, we construct 46 new extremal binary self-dual codes of length 68, 26 new best known Type II codes of length 72 and 8 new extremal Type II codes of length 80 that lead to new 3−(80,16,665) designs. Among the new codes of length 68 are the examples of codes with the rare γ=5 parameter in W68,2. All these new codes are tabulated in the paper.

New self-dual and formally self-dual codes from group ring constructions

In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings $ &nbsp;\mathbb{F}_2+u \mathbb{F}_2 $ and $ &nbsp;\mathbb{F}_4+u \mathbb{F}_4 $, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters $ [72,36,14] $.

An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F2+uF2 and F4+uF4; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with γ=5 in W68,2, which is the first instance of such a γ value in the literature.

Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes

We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F2+uF2, 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.

Constructions for self-dual codes induced from group rings

In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68.

Binary extremal self-dual codes of length 60 and related codes

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 $1$.

New extremal binary self-dual codes of lengths 64 and 66 from R2-lifts

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 R2-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.

NEW EXTREMAL BINARY SELF-DUAL CODES OF LENGTHS 66 AND 68 FROM CODES OVER Rk,m

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.