We first define a new Gray map from R = Z 4 + uZ 4 to Z 2 4 , where u 2 = 1 and study (1 + 2u)-constacyclic codes over R. Also of interest are some properties of (1 + 2u)-constacyclic codes over R. Considering their Z 4 images, we prove that the Gray images of (1+2u)-constacyclic codes of length n over R are cyclic codes of length 2n over Z 4 . In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic R-codes published byÖzen et al. in Finite Fields and Their Applications, 38, (2016) 27-39.

No abstract

In this paper, new few weights linear codes over the local ring R = F p +uF p +vF p +uvF p , with u 2 = v 2 = 0, uv = vu, are constructed by using the trace function defined over an extension ring of degree m. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gaussian sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to F 4 p , we obtain several families of new p-ary codes from trace codes of dimension 4m. For the first defining set: when m is even, or m is odd and p ≡ 3 (mod 4), we obtain a new family of two-weight codes, which are shown to be optimal by the application of the Griesmer bound; when m is even and under some special conditions, we obtain two new classes of three-weight codes. For the second defining set: we obtain a new class of two-weight codes and prove that it meets the Griesmer bound. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented.

No abstract

<p style='text-indent:20px;'>In this paper, for each of six families of three-valued <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring <inline-formula><tex-math id="M2">\begin{document}$ R = \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ u^2 = 0. $\end{document}</tex-math></inline-formula> The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are <inline-formula><tex-math id="M4">\begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $\end{document}</tex-math></inline-formula> for some explicit <inline-formula><tex-math id="M5">\begin{document}$ r. $\end{document}</tex-math></inline-formula></p>