# Inefficient covering by translates

While trying to answer this question, I arrived at another question:

How many translates of $$\{0,1\}^n$$ does it take to cover $$\mathbb F_3^n$$?

The broader context is: consider a set $$S$$ and a collection $$\mathcal P$$ of subsets of $$S$$; each subset of the same size, $$t$$, and each element of $$S$$ belonging to the same number, $$k$$, of elements of $$\mathcal P$$. Write $$N(S,\mathcal P)$$ for the minimal number of elements of $$\mathcal P$$ to cover $$S$$.

By analogy with definitions about covering and packing of subsets of $$\mathbb R^d$$, define the covering density of $$\mathcal P$$ to be $$t\times N(S,\mathcal P)/|S|$$, that is the average number of times than an element of $$S$$ is covered by a minimal covering from $$\mathcal P$$.

By random methods, I can show that one can find $$n\times (\frac 32)^n$$ translates of $$\{0,1\}^n$$ that cover $$\mathbb F_3^n$$ for a covering density of at most $$n$$.

• Are there any known lower bounds on the covering density?
• Are there well known places to look for good examples with large covering density?
• Are random methods the standard approach to questions of this type?