Embedding of lattices and K3-covers of an enriques surface

In this study, we establish necessary conditions for the embeddings of lattices and apply these conditions to the problem of characterizing algebraic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> <m:mn>3</m:mn> </m:math> K3 surfaces that cover an Enriques surface. By refining existing criteria and providing a more elementary approach, we offer a new perspective on the structure of such surfaces. Our results apply to any lattices, extending beyond specific cases and offering a comprehensive framework for understanding the embedding conditions in terms of Gram matrices.