There are exactly 222 \(L\)-types of primitive five-dimensional lattices (Q1612757)

Two positive definite quadratic forms on \(R^n\) are said to be of the same \(L\)-type if their Delaunay partitions (the duals of the Voronoĭ partitions) are affinely equivalent. This notion is due to Voronoĭ, who initiated the study of the problem of enumeration of \(L\)-types, with which this paper is also concerned. For \(n = 5\) dimensional space, \(221\) \(L\)-types were found in the paper by \textit{S. S. Ryšhkov} and \textit{E. P. Baranovskiĭ} [translated in Proceedings of Steklov Institute of Mathematics (1978, issue 4), AMS]; however, in [\textit{P. Engel}, Proc. Inst. Math. Natl. Acad. Sci. Ukr. Math. Appl. 21(2), 22-60 (1998; Zbl 0955.51011)], \(222\) distinct \(L\)-types were listed. This paper disposes of the nontrivial problem of determining which of the \(222\) had been omitted in the earlier enumeration.