There has been a lot of work recently on constructing line packings that achieve equality in either the Welch bound or the orthoplex bound. It has proven much harder to pack in regimes where these bounds are not tight. To help fill this void, about a month ago, I posted a new paper with Matt Fickus and John Jasper on the arXiv. We provide a few results to approach this case, which I outline below.

**1. The minimal coherence of 6 unit vectors in is 1/3.**

The Welch bound is known to not be tight whenever lies strictly between and (see the next section for a proof sketch). As such, new techniques are required to prove optimality in this range. We leverage ideas from real algebraic geometry to show how to solve the case of vectors in for all sufficiently small . For example, our method provides a new proof of the optimality of 5 non-antipodal vertices of the icosahedron in , as well as the optimality of Sloane’s packing of 6 lines in .