On the Extremality of an 80-Dimensional Lattice

Abstract

Abstract. We show that a specific even unimodular lattice of dimension 80, first investigated by Schulze-Pillot and others, is extremal (i.e., the minimal nonzero norm is 8). This is the third known extremal lattice in this dimension. The known part of its automorphism group is isomorphic to SL2(F79), which is smaller (in cardinality) than the two previous examples. The technique to show extremality involves using the positivity of the Θ-series, along with fast vector enumeration techniques including pruning, while also using the automorphisms of the lattice.

Another 80-dimensional extremal lattice

“…Similar to the proof in [34], we show extremality by enumerating all the vectors of norm 10 and using the positivity of the Θ-series. We indicate various improvements over the methods used in [34], in particular those which allowed us to work with an automorphism group that does not have such a nice representation as with SL 2 (F 79 ).…”

“…Bachoc and Nebe [3] had previously constructed two extremal lattices in dimension 80, proving these were extremal using coding theory; and more recently Stehlé and the author [34] used a more computationally intensive method to show that the lattice associated to the (binary) extended quadratic residue code of length 80 is a third example in this dimension. We use techniques similar to those exposed in [34] to prove the extremality of a fourth lattice in this dimension. This lattice corresponds to the rank 20 quaternionic matrix group SL 2 (F 41 )⊗S 3 as constructed by Nebe in [22, §3], and in [22,Lemma 4.3(i)] it is noted that there is an additional 2-extension in the automorphism group.…”

Algorithms for the Shortest and Closest Lattice Vector Problems

“…The main heuristic used for accelerating Enum consists in pruning the corresponding enumeration tree, by cutting off branches with low ratio between the estimated branch size and the likeliness of containing the desired solution [75,76,80,25]. Pruning consists in replacing the sets L (i) ∩ B(t (i) , A) by subsets L (i) ∩ B(t (i) , p i · A), for some pruning coefficients 0 < p n ≤ .…”

“…Several heuristics are known for the solvers of the saturation and enumeration families (at the moment, the Micciancio-Voulgaris Voronoi-based algorithm seems uncompetitive, and would require further practical investigation). However, the heuristic implementations of the enumeration families, relying on tree pruning strategies [75,76,80,25] seem to outperform the heuristic implementations of the saturation families [65,58]. This observation has led to hardware implementations of the enumeration [37,16].…”

“…The Theta series is a formal series whose coefficient of degree k is the number of vectors of squared norm equal to k (for all k ≥ 0). The first coefficients of the Theta series of a lattice are typically computed with enumerationbased algorithms (see, e.g., [80]). …”