Designs, Codes and Cryptography

In this paper, we determine the weight distribution of several classes of double cyclic codes over Galois rings by Gauss sums.

In this paper, we first generalize the polycyclic codes over finite fields to polycyclic codes over the mixed alphabet ${\mathbb{Z}}_2{\mathbb{Z}}_4$, and we show that these codes can be identified as ${\mathbb{Z}}_4[x]$-submodules of ${\mathcal{R}}_{\alpha ,\beta }$ with ${\mathcal{R}}_{\alpha ,\beta }={\mathbb{Z}}_2[x]/⟨t_1(x)⟩\times {\mathbb{Z}}_4[x]/⟨t_2(x)⟩$, where $t_1(...$${}_{\mathrm{}}$${\mathbb{}}_{\mathrm{}}$${\mathbb{}}_{\mathrm{}}$${\mathbb{}}_{\mathrm{}}{\mathbb{}}_{\mathrm{}}$${\mathbb{}}_{\mathrm{}}{\mathbb{}}_{\mathrm{}}$${\mathbb{}}_{\mathrm{}}{\mathbb{}}_{\mathrm{}}$

Given k points in $S^1$ satisfying certain conditions which are determined through their symmetric functions, we introduce a method for constructing spherical t-designs in ${\mathbb{R}}^2$ with $2t+k$ elements. This approach points toward a better understanding of ...

In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length n over ${\mathbb{F}}_q$ and ${\mathbb{Z}}_k$. We give the MacWilliams type identity for the complete joint Jacobi polynomials of codes. We also introduce the concepts ...${\mathbb{}}_{}$${\mathbb{}}_{}$${\mathbb{}}_{}$${\mathbb{}}_{}$$\mathrm{}$${\mathbb{}}_{}$${\mathbb{}}_{}$

The automorphism group of the Zetterberg code Z of length 17 (also a quadratic residue code) is a rank three group whose orbits on the coordinate pairs determine two strongly regular graphs equivalent to the Paley graph attached to the prime 17. ...$\mathrm{}\mathrm{}$

In a study of multilength variable-weight optical orthogonal codes (MLVWOOCs), compatible (N, M, W, 1, Q; 2) difference packing (briefly (N, M, W, 1, Q; 2)-CDP) set systems play an important role. In this paper, a new consequence of Weil’s theorem ...$\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}$

The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in Jitman and Ling (Des Codes Cryptogr 74:511–531, 2015), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling and ...

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull ...

The first author recently proved the extension theorem for linear codes over integer residue rings equipped with the Lee or the Euclidean weight by introducing a determinant criterion that is dual to earlier approaches. In this paper we generalize ...

Let ${\mathbb{F}}_q$ be a finite field with $q=p^e$ elements, where p is a prime and e is a positive integer. In 2017, Fan and Zhang introduced $\ell$-Galois inner products on the n-dimensional vector space ${\mathbb{F}}_q^n$ for $0\le \ell <e$, which generalized the Euclidean inner product ...$$$$${\mathbb{}}_{}$${\mathbb{}}_{}$$$$$$$${\mathbb{}}_{}$$$$$${\mathbb{}}_{}$${\mathbb{}}_{}$$$$$${\mathbb{}}_{}$

This paper gives new methods of constructing symmetric self-dual codes over a finite field GF(q) where q is a power of an odd prime. These methods are motivated by the well-known Pless symmetry codes and quadratic double circulant codes. Using ...

It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design D(q) associated with a Paley–Hadamard matrix of type II. ...$\mathrm{}\mathrm{}\mathrm{}\mathrm{}$${}^{}$${}^{}$

Let ${\mathbb{F}}_q$ be the finite field of order $q=p^m$, where p is a prime, m is a positive integer, and $\mathcal{R}={\mathbb{F}}_q[u,v]/⟨u^2-u,v^2-v,uv-vu⟩$. Thus $\mathcal{R}[x;\theta ,{\delta }_{\theta }]$ is a noncommutative ring, known as skew polynomial ring, where $\theta$ is an automorphism of $\mathcal{R}$ and ${\delta }_{\theta }$ is a $\theta$-...$\mathcal{}$${}_{}$$\mathcal{}$$\mathcal{}{}_{}$${}_{}$$\mathcal{}$${}_{}$${}_{}$$\mathcal{}$