Self-dual Codes Research Articles

Let p be a prime number and m be a positive integer. In this paper, we investigate cyclic codes of length n over the local non-Frobenius ring R=GR(p2,m)[u], where u2=0 and pu=0. We first determine the algebraic structure of cyclic codes of arbitrary length n. For the case gcd(n,p)=1, we explicitly describe the generators of cyclic codes over R. Moreover, we establish necessary and sufficient conditions for the existence of self-dual and LCD codes, together with their enumeration. Several illustrative examples and tables are presented, highlighting the mass formula for cyclic self-orthogonal codes, cyclic LCD codes, and families of new cyclic codes that arise from our results.

A persymmetric matrix is a square matrix that is symmetric concerning its antidiagonal. This article discusses some characteristics of a persymmetric matrix and its application in coding theory. A persymmetric matrix is used to form a generator matrix of binary reversible self-dual codes. A binary reversible self-dual code is a self-dual code whose reverse element is contained in the code. The methodology involves the implementation of flip transpose and column reversal to ensure the generator matrix satisfies both self-duality and reversibility. We begin with small-sized persymmetric matrices (e.g., 2×2 and 3×3) to extend the construction of the larger matrices. Combining a self-dual code and a reversible self-dual code of shorter length, and embedding persymmetric blocks, we construct new binary reversible self-dual codes of longer length. The novelty of this research lies in developing a new construction method for binary reversible self-dual codes derived directly from self-dual codes in the standard form, which has not been explicitly addressed in previous studies.

Formally self-dual codes with desirable hull dimensions

<p>We consider a ring <span class="math inline">\(R_{u^3} = \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2+u^3\mathbb{F}_2, u^4=0\)</span>. We discuss the structure of self-dual codes, Type I codes and Type II codes over the ring <span class="math inline">\(R_{u^3}\)</span> in terms of the structure of their Torsion and Residue codes. We construct Type I and Type II optimal codes over the ring <span class="math inline">\(R_{u^3}\)</span> for different lengths.</p>

Free double cyclic self-dual codes over $$\mathbb {Z}_4$$

We introduce a code construction for Wess-Zumino-Witten (WZW) models associated with simply-laced affine Lie algebras at level 1. The chiral primary fields of these rational CFTs can be parametrized by the elements of the outer automorphism group of the affine algebra, which is isomorphic to the discriminant group G of the root lattice. We show that the classification of even, self-dual codes over the alphabet G is equivalent to the classification of modular-invariant CFTs. Each individual CFT is dual to a Chern-Simons theory, after gauging the maximal, non-anomalous subgroup of its 1-form symmetry group specified by the code. We calculate the ensemble average of these CFTs, which is holographically dual to “CS gravity” — where the bulk theory is summed over topologies. When the alphabet G consists only of elements of square-free order, we explicitly show that this ensemble average reproduces the Poincaré series of the vacuum character, which can be interpreted as the CS path-integral summed only over handlebody topologies. However, when G contains elements of non-square-free order, additional contributions from singular topologies arise.

In this work we focus on minors of binary codes, especially in relation to self-dual binary codes with good error-correcting capabilities as measured by their distance. We show that every optimal binary code with distance at least 4, up to length 32, has an extended binary Hamming code minor. Furthermore, when restricted to the case of Type II codes, we show that all optimal codes beyond a certain length have the binary Hamming code and its dual as a minor. Finally, we cast the code minor problem in the setting of double-circulant codes as a substring problem, demonstrate how it can be used, and discuss the computational challenges involved.

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order p2, where p is a prime satisfying p≡1mod4 or p≡3mod4. The work expands the traditional scope of coding theory by developing and adapting a generalized recursive approach to produce quasi-self-dual and self-dual codes within this algebraic setting. While the method for code generation is rooted in the classical build-up technique, the primary focus is on the duality properties of the resulting codes—especially how these properties manifest under different congruence conditions on p. Computational examples are provided to illustrate the effectiveness of the proposed methods.

Unbiased Hadamard Matrices and Ternary Self-dual Codes

The distance function on Coxeter-like graphs and self-dual codes

In this paper, we consider degree p self-dual codes over the ring Zpm +iZpm (i2 = 1), and discuss their properties over the ring. We also consider the invariant polynomial ring with the complete weight enumerator to describe the relationship between Clifford-Weil group and Jacobi forms.

New binary $$\left[ 72,36,12\right] $$ self-dual codes from group rings and skew group rings

Cyclic codes having length ptq have been studied using the primitive binary idempotent generators when ( ) 2 (2) = pt pt O f and (2) = 1 qO q − with ( ( ) ) 2 , 1 = 1. f pt q − Using the expressions of these idempotent generators and the theory of cyclotomy, expressions of idempotent generators of ℤ4 – cyclic codes are obtained. In this case, all the self-dual codes and LCD codes of length ptq over ℤ4 are defined in terms of the idempotent generators. Clearly, these self-dual cyclic codes of length ptq are Type-I codes. Also, the permutational equivalence of these cyclic codes has been discussed.

Reversible (bisymmetric) self-dual codes over finite fields with characteristic two and their applications to DNA codes

New self-dual codes from TGRS codes with general $ \ell $ twists

New NMDS self-dual codes over finite fields

On the existence of MRD self-dual codes

Reversible self-dual code is a linear code which combine the properties from self-dual code and reversible code. Previous research shows that reversible self-dual codes have only been developed over field of order 2 and order 4. In this article, we construct reversible self-dual code over any finite field of order F_q , with natural number q=2. We first examine and prove some of fundamental properties of reversible self-dual code over . After a thorough analysis these, we obtain a new generator matrix of reversible self-dual code. A new generator matrix is derived from existing self-dual and reversible self-dual code over . It will be shown that a new reversible self-dual over can be constructs from one and more existing code by specific algebraic methods. Furthermore, using this construction, we determine the minimum distance of reversible self-dual code and ensuring its optimal performance in various applications.

Abstract Weighing matrices with entries in the complex cubic and sextic roots of unity are employed to construct Hermitian self-dual codes and Hermitian linear complementary dual codes over the finite field $\mathrm {GF}(4).$ The parameters of these codes are explored for small matrix orders and weights.

In this paper, we introduce several new construction techniques of linear complimentary dual (LCD) codes. First, we show that if a LCD code is fixed by a transitive automorphism group, then the punctured code is also LCD. We show that several important families such as anti-primitive BCH codes and extended Gabidulin codes are LCD. Finally relationships among self-dual codes, [Formula: see text]-orthogonal matrices and LCD codes are studied. As an application we give LCD codes from Paley-type bipartite graph and from punctured MacDonalds codes. Further, we give a bound on some LCD codes.