The Lee complete ρ weight enumerator and the exact complete ρ weight enumerator over Mn×s(Fl+vFl+v2Fl)(v3=v) are defined, and the MacWilliams identities with respect to RT metric for these two weight enumerators of linear codes over Mn×s(Fl+vFl+v2Fl) are obtained, respectively. Finally, we give two examples to illustrate the obtained results.

We investigate linear complementary dual (LCD) rank-metric codes in this paper. We construct a class of LCD generalized Gabidulin codes by a self-dual basis of an extension field over the base field. Moreover, a class of LCD MRD codes, which are obtained by Cartesian products of a generalized Gabidulin code, is constructed.

Let $R$ = $mathbb{F}_{p}+umathbb{F}_{p}+vmathbb{F}_{p}+uvmathbb{F}_{p}$, where u2=u, v2 and uv=vu. A relation between the support weight distribution of a linear code $mathscr{C}$ of type p4k over R and its dual code $mathscr{C}^{ot}$ is established.

The stability theory of stream ciphers plays an important role in designing good stream cipher systems. Two algorithms are presented, to determine the optimal shift and the minimum linear complexity of the sequence, that differs from a given sequence over Fq with period qn-1 by one digit. We also describe how the linear complexity changes with respect to one digit differing from a given sequence.

In this article, we investigate the depth distribution and the depth spectra of linear codes over the ring R=F2+uF2+u2F2, where u3=1. By using homomorphism of abelian groups from R to F2 and the generator matrices of linear codes over R, the depth spectra of linear codes of type 8k14k22k3 are obtained. We also give the depth distribution of a linear code C over R. Finally, some examples are presented to illustrate our obtained results.

The definitions of the Lee complete ρ weight enumerator and the exact complete ρ weight enumerator over Mn×s(F2[u,v]/) are introduced, and the MacWilliams identities with respect to the RT metric for these two weight enumerators of linear codes over Mn×s(F2[u,v]/) are obtained. Finally, we give two examples to illustrate the results we obtained.

In this article, we study skew cyclic codes over $R=mathbb{F}_{q}+vmathbb{F}_{q}+v^{2}mathbb{F}_{q}$, where $q=p^{m}$, $p$ is an odd prime and v3=v. We describe the generator polynomials of skew cyclic codes over this ring and investigate the structural properties of skew cyclic codes over R by a decomposition theorem. We also describe the generator polynomial of the dual of a skew cyclic code over R. Moreover, the idempotent generators of skew cyclic codes over $mathbb{F}_{q}$ and R are considered.

Linear complexity and the k-error linear complexity of periodic sequences are the important security indices of stream cipher systems. This paper focuses on the distribution of p-error linear complexity of p-ary sequences with period pn. For p-ary sequences of period pn with linear complexity pn-p+1, n≥1, we present all possible values of the p-error linear complexity, and derive the exact formulas to count the number of the sequences with any given p-error linear complexity.