Construction Of Self-dual Codes Research Articles

Block designs from self-dual codes obtained from Paley designs and Paley graphs

In 2002, P. Gaborit introduced two constructions of self-dual codes using quadratic residues, so-called pure and bordered construction, as a generalization of the Pless symmetry codes. In this paper, we further study conditions under which the pure and the bordered construction using Paley designs and Paley graphs yield self-dual codes. Special attention is given to the binary and ternary codes. Further, we construct \(t\)-designs from supports of the codewords of a particular weight in the binary and ternary codes obtained.

New constructions of self-dual codes via twisted generalized Reed-Solomon codes

New constructions of self-dual codes via twisted generalized Reed-Solomon codes

Construction of self dual codes from graphs

Construction of self dual codes from graphs

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

Group matrix ring codes and constructions of self-dual codes

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring M_k(R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring M_k(R) are one sided ideals in the group matrix ring M_k(R)G and the corresponding codes over the ring R are G^k-codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes.

Double bordered constructions of self-dual codes from group rings over Frobenius rings

In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings mathbb {F}_{2}+umathbb {F}_{2} and mathbb {F}_{4}+umathbb {F}_{4}. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables.

Constructions of self-dual codes and formally self-dual codes over rings

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.

Construction of self-dual codes over ℤ 2 m $\mathbb {Z}_{2^{m}}$

Construction of self-dual codes over ℤ 2 m $\mathbb {Z}_{2^{m}}$

Euclidean self-dual codes over non-commutative Frobenius rings

We study self-dual codes over non-commutative Frobenius rings. It is shown that a code is equal to its left orthogonal if and only if it is equal to its right orthogonal. Constructions of self-dual codes are given over Frobenius rings that arise from self-dual codes over the center of the ring. These constructions are used to show for which lengths self-dual codes exist over various rings.

Construction of extremal self-dual codes over $${\mathbb {Z}}_{8}$$ Z 8 and $${\mathbb {Z}}_{16}$$ Z 16

We present a method of constructing free self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16 which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over $${\mathbb {Z}}_{2^m}$$Z2m can be found from a binary (extremal or optimal) Type II code for any positive integer $$m \ge 2$$mź2. We find explicit algorithms for construction of self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16. Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over $${\mathbb {Z}}_{2^m}$$Z2m. By using our explicit algorithms, we construct extremal free self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16 up to lengths 40.

Fourth Power Residue Double Circulant Self-Dual Codes

Quadratic residue codes are a well-known class of codes. In this paper, we consider the constructions of self-dual codes by higher power residues, especially fourth power residues. New infinite families of self-dual codes over GF(2), GF(3), GF(4), GF(8), and GF(9) are introduced. Some of them have better minimum weight than previously known codes. We also give general results related to the automorphism group of some of these codes.

Constructing self-dual codes over $$\mathbb{F }_q[u]/(u^t)$$ F q [ u ] / ( u t )

Given a self-dual code over $$\mathbb{F }_q[u]/(u^t)$$ F q [ u ] / ( u t ) we present a method to obtain explicitly new self-dual codes of larger length. Conversely, we also prove that, with the appropriate assumptions on length and number of generators, every self-dual code over $$\mathbb{F }_q[u]/(u^t)$$ F q [ u ] / ( u t ) can be obtained in this manner. We use this construction to produce several optimal self-dual codes over the base field in a manner that generalizes the Lee weight. This construction is based on ideas presented by Han et al. (Bull Korean Math Soc, 49:135---143, 2012) and also by Lee and Kim (An efficient construction of self dual codes, 2012), not only generalizing it, but joining the two different cases from the original paper as special cases of one general construction.

Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$

In this work, the double-circulant, bordered-double-circulant and stripped bordered-double-circulant constructionsfor self-dual codes over the non-chain ring $R_2 = \mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$ 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 $7$.

Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$

Self-dual codes over $\mathbb Z_2\times\mathbb Z_4$ are subgroups of $\mathbb Z_2^\alpha\times\mathbb Z_4^\beta$ that are equal to their orthogonal under an inner-product that relates these codes to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $\alpha,\beta$ such that there exist a self-dual code $\mathcal C\subseteq \mathbb Z_2^\alpha \times\mathbb Z_4^\beta$ are established. Moreover, the construction of such a code for each type and possible pair $(\alpha,\beta)$ is given. The standard techniques of invariant theory are applied to describe the weight enumerators for each type. Finally, we give a construction of self-dual codes from existing self-dual codes.

Self-dual codes from circulant matrices

In this paper, two general methods for constructing self-dual codes are presented. These methods use circulant matrices in circulant or bordered circulant structures to construct the suitable generator matrices. The necessary and sufficient conditions, for the generated codes to be self-dual, are provided. Special cases of the proposed methods include the well known "Pure Double Circulant" construction and the "Bordered Double circulant" construction of self-dual codes. As an example, the methods were applied to search for self-dual codes in GF(5). Many new inequivalent self-dual codes with best known distance are found.

Construction of self-dual codes with an automorphism of order $p$

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 <em>automorphism of type</em> $p-(c, f)$ 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.

Constructions of self-dual codes over finite commutative chain rings

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 c² = −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.

Construction of Self-Dual Codes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Construction methods for self-dual codes are given. By using these methods some new extremal self-dual <emphasis><formula formulatype="inline"><tex>$[{66, 33, 12}]$</tex></formula></emphasis> and <emphasis><formula formulatype="inline"> <tex>$[{68, 34, 12}]$</tex></formula></emphasis> codes are obtained. </para>

Construction of self-dual codes over finite rings [formula omitted

We present an efficient method for constructing self-dual or self-orthogonal codes over finite rings Z p m (or Z m ) with p an odd prime and m a positive integer. This is an extension of the previous work [J.-L. Kim, Y. Lee, Euclidean and Hermitian self-dual MDS codes over large finite fields, J. Combin. Theory Ser. A 105 (2004) 79–95] over large finite fields GF ( p m ) to finite rings Z p m (or Z m ). Using this method we construct self-dual or self-orthogonal codes of length at least up to 10 over various finite rings Z p m or Z p q with q an odd prime, where p m = 25 , 125, 169, 289 and p q = 65 , 85. All the self-dual codes we obtained are MDS, MDR, near MDS, or near MDR codes.

Combinatorial designs and codes over some prime fields

Combinatorial designs are widely used in the construction of self-dual codes. Recently new methods for constructing self-dual codes are established using orthogonal designs, generalized orthogonal designs and Diophantine equations over GF ( p ) . These methods have led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we propose some methods to generate self-dual codes, over GF ( p ) . Moreover, we apply shortening and padding to obtain self-orthogonal codes over GF ( p ) , for some primes p.