Five-Lee-weight linear codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q$

In this study, linear codes having their Lee-weight distributions over the semi-local ring ${\mathbb{F}}_q+u{\mathbb{F}}_q$ with $u^2=1$ are constructed using the defining set and Gauss sums for an odd prime $q$. Moreover, we derive complete Hamming-weight enumerators for the images of the constructed linear codes under the Gray map. We finally show an application to secret sharing schemes.