{ "description": "Certificate format (SM Section S5.3, Eqs. (S5.14)-(S5.17)). For a pair (W_A, W_B) of the rank-19 lattices and an integer matrix M with det M != 0, q(M) = tr(M^T W_A M W_B) = , G = M^T W_A M. The certificate data are: the multiplier Gram B = W_B; rank-1 terms (y_a in Z^19, weight w_a); pair terms (y_i, y_j in Z^19 independent, l1, l2 >= 0, m) with Lambda = L L^T, L = [[l1,0],[m,l2]]; D = 512 (a lower bound for det G since det W_A >= 512); beta = a rational lower bound for sqrt(12). Define R = B - sum_a w_a y_a y_a^T - sum_p Y_p Lambda_p Y_p^T (Y_p = [y_i y_j]). If R is positive definite then q(M) >= V := 19 t/1000 + 4 sum_a w_a + beta sum_p 2 l1_p l2_p, where t = floor(1000 (D det R)^(1/19)), using (P1) y^T G y >= 4 (G is the Gram matrix of a sublattice of the root-free even lattice W_A), (P2) det G >= D, (P3) every rank-2 section of G has determinant >= 12 (Hermite's constant with minimum 4), and the arithmetic-geometric mean inequality >= 19 (det R det G)^(1/19). Weights w_a >= 0 give an unconditional bound; a negative weight is allowed only on a unit vector y = e_j and then the bound holds on the stratum G_jj = 4 (column j of M of minimal norm), where w_j (G_jj - 4) = 0. Since q is even, V > 46 gives q >= 48 (N_flux = q/2 >= 24) and V > 48 gives q >= 50 (N_flux >= 25). value_frac is the exact rational V.", "files": [ { "file": "table3_C-1024.json", "label": "C-1024", "multiplier": "W1024", "table3_row": 1, "conditional": false, "n_rank1": 195, "n_pairs": 0, "value_frac": "13245254530664589/274877906944000" }, { "file": "table3_C-512u.json", "label": "C-512u", "multiplier": "W512", "table3_row": 2, "conditional": false, "n_rank1": 105, "n_pairs": 0, "value_frac": "41004589043118677/879609302220800" }, { "file": "table3_C-512pu.json", "label": "C-512pu", "multiplier": "W512p", "table3_row": 3, "conditional": false, "n_rank1": 103, "n_pairs": 0, "value_frac": "819922474416771261/17592186044416000" }, { "file": "table3_C-512k0.json", "label": "C-512k0", "multiplier": "W512", "table3_row": 4, "conditional": true, "n_rank1": 485, "n_pairs": 3600, "value_frac": "32367743330235055603013986807893563514994926733/633825300114114700748351602688000000000000000" }, 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