History

This page lists the extremal even unimodular lattices known today, dimension by dimension, together with who found them and when. How they were constructed is the subject of the next page.

Up to dimension $24$ the even unimodular lattices are completely classified, so all the extremal ones are known. They are $E_8$ in dimension $8$, the two lattices $E_8\perp E_8$ and $D_{16}^{+}$ in dimension $16$ (the bound is only $2$ there, so both are extremal), and the Leech lattice in dimension $24$.

In dimensions $32$ and $40$ there are so many extremal lattices that they cannot be listed; only lower bounds for their number are known. From dimension $48$ on, every known example comes from an explicit construction. The table names individual lattices where only a few are known.

The list below gives the constructions of extremal lattices in dimensions $24$ and above, together with two results that limit the dimensions in which extremal lattices can exist. Each item is placed in the year in which the result first became public, either as a preprint (a paper made public before peer review) or in a journal.

• 1967. Leech publishes the Leech lattice $\Lambda_{24}$ [Lee67]. Two years later Conway proves that it is the only even unimodular lattice of dimension $24$ without vectors of norm $2$ [Con69].
• 1971. Leech and Sloane construct the first two extremal lattices of dimension $48$, $P_{48p}$ and $P_{48q}$, from ternary self-dual codes (codes over the field with three elements; see the next page) [LS71].
• 1975. Mallows, Odlyzko and Sloane prove the bound $\min\Lambda\le 2\lfloor n/24\rfloor+2$ and show that extremal lattices cannot exist in sufficiently large dimensions [MOS75].
• 1984. Quebbemann introduces a general construction and obtains an extremal lattice of dimension $64$ [Que84].
• 1989. Ozeki obtains an extremal lattice of dimension $56$ from ternary codes [Oze89].
• 1995. Batut, Quebbemann and Scharlau find another extremal lattice of dimension $56$, using cyclotomic fields, the fields obtained by adjoining roots of unity to $\mathbb{Q}$ [BQS95].
• 1998. Nebe constructs a third extremal lattice of dimension $48$, $P_{48n}$. The same paper also constructs extremal lattices of dimension $56$ on which the group $\mathrm{SL}_2(29)$ acts [Neb98]. In the same year Bachoc and Nebe construct two extremal lattices of dimension $80$ [BN98].
• 2010. Stehlé and Watkins prove that a third lattice of dimension $80$ is extremal [SW10]. Nebe constructs $\Gamma_{72}$, the first extremal lattice of dimension $72$ (preprint 2010, published 2012) [Neb12].
• 2011. Jenkins and Rouse show that extremal lattices do not exist in any dimension greater than $163\,264$ [JR11].
• 2012. Watkins finds a fourth extremal lattice of dimension $80$ [Wat12].
• 2013. Nebe finds a fourth extremal lattice of dimension $48$, $P_{48m}$ (preprint 2013, published 2014) [Neb14].
• 2018, 2022. Shimada gives a new construction in dimension $64$ [Shi18] and shows that Quebbemann’s construction produces many extremal lattices of dimension $64$ [Shi22].
• 2026. Sakamoto constructs two extremal lattices of dimension $88$, with computer-assisted proofs (preprint) [Sak26]. In October, an extremal lattice of dimension $112$, $L_{112}$, is likewise announced with a computer-assisted proof (preprint) [L112]; the page on $L_{112}$ describes it.

The figure plots, for each dimension from $24$ on, the year in which its first extremal lattice became public. Dimensions were not settled in increasing order. An extremal lattice of dimension $80$ was known twelve years before one of dimension $72$, and an example in dimension $112$ was found while $96$ and $104$ remain open.

So far, the longest delays have come where the bound $2\lfloor n/24\rfloor+2$ has just risen. Dimension $72$ is the first that requires minimum $8$, and it came after dimension $80$. Dimension $96$ is the first that requires minimum $10$, and no example is known. Dimension $48$, the first that requires minimum $6$, is the exception: it was settled early, by the construction from ternary codes. Jenkins and Rouse remark that dimensions divisible by $24$ are particularly challenging [JR11, §1].

The page on dimension $112$ gives a heuristic explanation, based on the mass formula, of why a larger minimum is harder to reach. The same page explains why an example was more likely to be found in dimension $112$ than in dimensions $96$ and $104$, although all three require minimum $10$.