Extremal problems on sum-free sets and coverings in tridimensional spaces (Q621366)

Let \(q\) be a prime power, and let \(\mathbb F_q\) be the field of order \(q\). Define \(c(q)\) to be the minimum cardinality of a subset \(H\) of \(\mathbb F_q\) such that every vector in this space differs in at most 1 coordinate from a multiple of a vector in \(H\). The authors prove that \(c(5)=4.\) Then they use a construction of short coverings from sum-free sets to prove the following. If \(q\) is odd, then \(c(q) \leq 6 \lceil {{q-1}\over{12}} \rceil + 6 \lceil \log_4({{q-1}\over{4}})\rceil +3\) and if \(q\) is even, then \(c(q) \leq 6 \lceil {{q-1}\over{9}} \rceil + 6 \lceil \log_4({{q-1}\over{3}})\rceil +3\).