Unimodular lattices in dimension 29 and other lattices

Found here. To use the functions defined in the GP scripts it is necessary to compile the two C files (eqfminim.c and orbmod2.c) first. How to do this is platform-dependent: look for the line modules_build in the file pari.cfg provided with your distribution of PARI/GP, and run this command where %s is replaced by eqfminim or orbmod2. This generates two libraries which can then be loaded in GP (see the top of the files eqfminim.gp and orbmod2.gp for how this is done). Loading all.gp (e.g. by running gp all.gp) loads all our functions in an interpreter.

• tools.gp defines general-purpose functions, in particular the functions gram_to_vec and vec_to_gram which translate between symmetric matrices and vectors (the vector associated to a symmetric matrix gram is composed of the diagonal coefficients gram[i,i], then the coefficients gram[i,i+1], etc). It also defines the function nei([d,x,t]) which computes a Gram matrix of the d-neighbor of \(\mathrm{I}_n\) associated to

  • an integer \(d \geq 2\)
  • an n by 1 matrix x = [x_1;...;x_n] where the coefficients \(x_i\) are non-decreasing and satisfy \(x_1=1\), \(x_n \leq d/2\) and \(x \cdot x \equiv 0 \pmod d\) (resp. \(x \cdot x \equiv 0 \pmod{2d}\)) if \(d\) is odd (resp. even),
  • t is an extra parameter equal to 0 or 1, only needed for d even.

  (For more details see Chenevier, Gaëtan, Unimodular hunting, Algebraic Geometry 12, No. 6, pp. 769–812)

• rs.gp defines functions related to root systems, in particular root_system(gram) which computes the isomorphism class of the root system of a lattice with Gram matrix gram (a string as explained in 2.3.2) and order_weyl(rs) which computes the order of the Weyl group given such a string rs.
• eqfminim.gp (relying on eqfminim.c) defines
  • An exact variant of the Fincke-Pohst algorithm, eqfminim(gram, M), where gram is a positive definite Gram matrix with integral coefficients and M is a bound (an integer). It returns [L_1,...,L_M] where L_i is a matrix whose columns are all vectors v of length i (i.e. qfeval(gram,v) is equal to i) whose first non-zero coefficient is positive.
  • A variant of the Plesken-Souvignier algorithm, qfautors(gram), computing the reduced isometry group of the Gram matrix gram, returning a vector whose first two components are its order and a vector of generators. (The other components give additional orbit information, or information useful for profiling.) For qfautors(gram) to be efficient, gram should be the Gram matrix in a basis of short vectors (see good_bases.gp below).
  • A (fast) implementation fast_marked_HBV(gram, mark) of the marked BV invariant (Definition 1.14)
• invariants.gp uses this function fast_marked_HBV(gram, mark) to implement the invariant \(\mathrm{BV}_{n,p}\) (as defined just before Corollary 7.23 in the paper), as BVhdiscp(gram).
• good_bases.gp (relying on eqfminim.c) defines
  • good_basis_rand(gram, B), which tries to find (randomly) a basis of vectors of norm (for gram) at most B, prioritizing vectors of length less than B (see Section 4 of Allombert, Bill; Chenevier, Gaëtan, Unimodular hunting. II, Forum Math. Sigma 13, Paper No. e136, 18 p. (2025)).
  • good_basis_hybrid(gram, B), which takes as input a Gram matrix gram which is already LLL-reduced and returns a basis of short vectors (of norm at most B) using a hybrid method (random and deterministic). If such a basis does not exist it will return a basis composed of n-1 vectors of norm at most B and another vector of norm as small as possible, or fall back to good_basis_rand() if such a basis is not found either. This method is generally faster than good_basis_rand().
• orbmod2.gp (relying on orbmod2.c) defines functions computing orbits in \(\mathbb{F}_2^n\) for a subgroup of \(\operatorname{GL}_n(\mathbb{F}_2)\) (acting on column vectors) given by generators:
  • orbmod2(gens) takes as input a Vec of generators (more precisely, lifts with integral coefficients) and returns a Vec of Vecsmall([len, x]) where len is the cardinality of the orbit and x a representative, given in binary format (so int2binvec(x, n) recovers a Vec with coefficients 0 or 1).
  • fororbmod2(gens, V, block) evaluates block once for each orbit with V equal to (the Vec) [len, v] where len is the cardinality of the orbit and v is a representative (Vec with coefficients 0 or 1).
  • fororbmod2aff(gens, V, block) is a variant for subgroups of the affine group \(\mathbb{F}_2^n \rtimes \operatorname{GL}_n(\mathbb{F}_2)\): gens is now a Vec of pairs [g,t] where g is an n by n matrix with integral coefficients and t is a Vec (or Col) with integral coefficients, representing \(T_t \circ g\) where \(T_t\) is translation by \(t \in \mathbb{F}_2^n\).

The file check.gp contains functions allowing one to check that the lists are correct:

• that each line is correct: gram represents a lattice in the genus and rs, roa and inv are correct,
• and that each isomorphism class occurs exactly once (i.e. the invariants are distinct and the total mass is as predicted by the Smith-Minkowski-Siegel mass formula).

Assuming that the directories src and lattices are placed in the same directory, run gp check.gp from within src and the following commands (then wait a few days!):

• check_all_odd_uni(1,1)
• check_all_even(1,1)

to check all lists except for odd_uni/X29_no1_*.gz, even_hdiscp/5/X28_*.gz and even_hdiscp/5/X27_*.gz. For these lists the checks do not terminate in reasonable time (on a current computer), so to check them it is necessary to split the lists and run the checks in parallel (ideally, on many computers) using the functions check_rs_inv() and check_roa() defined in check.gp. How to do so is platform-dependent; to check the orders of automorphism groups we recommend mixing the lines during splitting as the lattices for which qfautors takes more time are not uniformly distributed in our lists.