Related papers: On the Kernel of $\mathbb{Z}_{2^s}$-Linear Hadamar…
For the past decades, linear codes with few weights have been widely studied, since they have applications in space communications, data storage and cryptography. In this paper, a class of binary linear codes is constructed and their weight…
By using the logarithmic approach of the classical kernels for the K2 of number fields, we compute the 2-rank of the wild kernel WK2(F) and the 2-rank of the subgroup of infinite heigh elements in K2(F) in terms of positive class groups for…
In this paper, we first generalize the class of linear codes by Ding and Ding (IEEE TIT, 61(11), pp. 5835-5842, 2015). Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums…
In this paper, we study the algebraic structure of Z_2[u]Z_2[u, v]-additive codes which are Z_2[u, v]-submodules where u^2 = v^2 = 0 and uv = vu. In particular, we determine a Gray map from Z_2[u]Z_2 [u, v] to Z_2^{2{\alpha}+8\b{eta}} and…
We propose a framework for 2D shape analysis using positive definite kernels defined on Kendall's shape manifold. Different representations of 2D shapes are known to generate different nonlinear spaces. Due to the nonlinearity of these…
Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings $R_k$.…
We give methods for constructing many self-dual $\mathbb{Z}_m$-codes and Type II $\mathbb{Z}_{2k}$-codes of length $2n$ starting from a given self-dual $\mathbb{Z}_m$-code and Type II $\mathbb{Z}_{2k}$-code of length $2n$, respectively. As…
We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these…
Studying the generalized Hamming weights of linear codes is a significant research area within coding theory, as it provides valuable structural information about the codes and plays a crucial role in determining their performance in…
One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over ${\mathbb Z} _t \times {\mathbb Z}_2^2$. Two types of equivalence relations for classifying cocyclic matrices…
We show that the kernel of the monodromy representation for hypersurfaces of degree d and dimension n is large for d at least three with the exception of the cases (d,n) = (3,0) and (3,1). For these the kernel is finite. By "large" we mean…
We consider the class of linear antipodal two-weight rank metric codes and discuss their properties and characterization in terms of $t$-spreads. It is shown that the dimension of such codes is $2$ and the minimum rank distance is at least…
A characterization of $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices is described, depending on the notions of {\em distributions}, {\em ingredients} and {\em recipes}. In particular, these notions lead to the…
The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over…
Code decompositions (a.k.a code nestings) are used to design good binary polar code kernels. The proposed kernels are in general non-linear and show a better rate of polarization under successive cancelation decoding, than the ones…
In this paper, a recent method to construct complementary sequence sets and complete complementary codes by Hadamard matrices is deeply studied. By taking the algebraic structure of Hadamard matrices into consideration, our main result…
We introduce and examine the notion of principal $\mathbb{Z}_2^n$-bundles, i.e., principal bundles in the category of $\mathbb{Z}_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $\mathbb{Z}_2^n$-grading…
Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full…
We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ…
Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight…