Narrow Results

We study 1-generator quasi-cyclic codes and four-circulant codes, which are also quasi-cyclic but have 2 generators. We state the hull dimensions for both classes of codes in terms of the polynomials in their generating elements. We prove results such as the hull dimension of a four-circulant code is even and one-dimensional hull for double-circulant codes, which are special 1-generator codes, is not possible when the alphabet size $q$ is congruent to 3 mod 4. We also characterize linear complementary pairs among both classes of codes. Computational results on the code families in consideration are provided as well.

Quasi-cyclic codes form a generalization of cyclic codes, and contain a large number of record breaking codes. In this paper, we provide a method for constructing self-orthogonal quasi-cyclic codes, and obtain a large number of new quantum quasi-cyclic codes by CSS construction.

In this paper, we study skew cyclic and skew constacyclic codes over the ring ${𝔽}_p+u_1{𝔽}_p+\cdots +u_{2m}{𝔽}_p$, where $u_i^2=u_i$ and $u_i{u}_j=0$, for $i,j=1,\dots ,2m$ and $i\ne j$. We have introduced some new Gray maps and determined the relation between codes using the ${𝔽}_p$-Gray images of skew constacyclic codes.

In this paper, we study $\lambda$-constacyclic codes over the ring $R={\mathbb{Z}}_4+u{\mathbb{Z}}_4$, where $u^2=3$ for $\lambda =1+2u$ and $3+2u$, respectively. We define some new Gray maps from $R$ to the copies of ${\mathbb{Z}}_4$. It is shown that Gray images of $\lambda$-constacyclic codes over $R$ are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over ${\mathbb{Z}}_4$. Further, we extend and obtain cyclic codes, $\lambda$-constacyclic codes and permutation equivalent to quasi-cyclic codes over ${\mathbb{Z}}_4$, respectively, as Gray images of skew $\lambda$-constacyclic codes over $R$.