How the nets are built

Everything starts from a frame: a digital net whose coordinates read one polynomial at different places. In two dimensions it is the Hammersley net. Write the index i in base q; the coordinate x reads its digits from the most significant one down, and y reads them in reverse order, which is the van der Corput sequence. So the point with index i is (i, rev(i))/N.

An elementary box of shape (d_x, d_y) with d_x + d_y = m fixes the first d_x digits of x and the first d_y digits of y. For the Hammersley net these are the top d_x and the bottom d_y digits of i, which together are all of them: each box holds exactly one point, for every shape, including the long thin ones. The cells of the later sections are just the shape (k, k). In matrix terms, the box matrix stacks the first d_x rows of C_x and the first d_y rows of C_y; every box holds one point exactly when its determinant is a unit modulo q. For a frame every such determinant is ±1, in every base.

The frame fills every box, but some of its points nearly touch. Its geometric digit, the digit after the cell digits, is r = M̃ℓ mod q with the layer M̃ = [0 1; 1 0]: the digit of x is the last cell digit of y and vice versa. Away from carries this is harmless: the local lattice (I + M̃/q)ℤ² has its shortest vector close to one cell (see the table). The trouble sits at the carries.

Step diagonally across a carry: from a cell whose last digits are both q − 1 to the next cell in both directions, where both last digits roll over to 0. Both geometric digits wrap from q − 1 to 0, and each wrap pulls the second point back by almost a whole cell, nearly cancelling the step: the two points end up about 1/q of a cell apart in each coordinate. A coordinate of the frame does not see its own step (the diagonal of M̃ is 0), so nothing stops the wraps from cancelling it; in the paper's words this step has two moving axes and no backward coordinate (rem:K). With one cell digit the only such carry is the corner of the torus. With more cell digits it happens at every corner of a block of q × q cells, and the separation falls with every added digit (in base 2 the closest difference on the torus is one unit in each coordinate at every size shown). These values are computed; the paper proves no separation value for the unscrambled frame.

A lower-triangular scramble of a coordinate adds to each of its digits a combination of that coordinate's earlier digits, and may multiply the digit by a unit. Two points that share the first d digits of x still share them afterwards, so every box goes to a box of the same shape, and one point per box survives for free. In matrices: the first d rows of LC are the leading block of L times the first d rows of C, so every box minor is multiplied by a unit. This is why linear scrambling (Matoušek) and Tezuka's generalized Faure sequences keep t; Owen's nested scrambles are a larger, nonlinear family that also keeps it.

The construction uses this freedom on one digit only. It rewrites the geometric digit of each coordinate by subtracting the coordinate's own last cell digit (and, for x, flipping its sign), so that the layer becomes the turned layer of section 4. The deeper digits, the jitter, are borrowed from the frame unchanged up to a sign; they are what keeps every thin box filled. The table compares the two nets matrix by matrix.

The repaired net keeps the frame's cells and its deeper digits (up to a sign) and changes only the geometric digit, to r = M̃ℓ mod q with the turned layer M̃ = −I + quarter turn = [−1 −1; 1 −1]; ℓ holds the last cell digit of each coordinate. Along its own axis each coordinate's digit now moves backward (the diagonal of M̃ is −1), so on a step to a neighbouring cell, carries included, a wrap can only lengthen the coordinates that move. In higher dimensions the frame's geometric digit also depends on how far a carry reaches (lem:Rs), and the construction chooses it through a design that behaves well at every carry depth (prop:Gs); in two dimensions the last cell digits suffice. Between the dashed slip lines the points sit on the lattice (I + M̃/q)ℤ², a copy of the square grid turned by arctan(1/(q−1)) and shrunk slightly. Click a cell to read its digits; switch the layer to compare with the plain frame.

Now turn the deeper digits off and show the boxes of a long thin shape. The points drop to their lattice images, and those boxes are no longer filled one by one: some are empty, some hold several points. The borrowed jitter is exactly what fills them. This layer is the xy-block of the three-dimensional layer in section 6.

This is the implementation view of the repaired net. A digital net is computed one digit at a time. Write the index i of a point in base q, multiply its digit vector by one integer matrix per coordinate, reduce modulo q, and read the result as the base-q digits of the coordinate. The steps below do this for one point, then split the digits into cell, geometric digit and jitter, and show where the lattice comes from.

In three dimensions the frame reads one polynomial at the places ∞, 0 and 1; it is the Hammersley net of the two-dimensional Faure sequence, and every box minor is ±1. Its closest pairs are again far too close. The net 𝒫_2,q has N = q⁶ points in q² × q² × q² cubes. One integer lower-triangular scramble of the frame of the places ∞, 0, 1 makes digit 3 of each coordinate equal to Mℓ + Jh mod q, with ℓ and h the last and second-last cell digits of all three coordinates. Its layer M turns ℤ³ slightly about (1,1,1) and shrinks it. A one-layer slab seen from above shows the lattice patches and the staircase slips, and the readout compares the exact minimum distance with the bound of the theorem.

The column net 𝒞_q gives all three coordinates the same two digits after the cell digit. Each point then sits on the body diagonal of its cell, and the points form q² lines parallel to (1,1,1). Seen down the diagonal, these columns stand over a triangular lattice.

In a column net, a step to the neighbouring column changes the common height of the points. The ideal arrangement, the cube ℤ³ turned half a turn about (1,1,1), climbs q²/3 per step. After q steps in one direction you are back at the same column, so the heights must agree modulo q², but q · q²/3 is not a multiple of q² when 3 does not divide q.

The leftover, a third or two thirds of a period, has to be spread over the q steps of the cycle. The chart shows the realized heights against the ideal along the three cycles through one column. The largest excess η of a single step sets the separation, δN^1/3 = 1 − η/q² + …, and among all column nets 𝒞_q spreads the leftover as evenly as possible.

The column nets 𝒞_q of section 6 and the nets 𝒯_p and 𝒲_p of page 3 do not come from a frame. They are graph forms whose second and third digits are chosen directly, and their net property is proved the plain way: every box minor is checked to be a unit (thm:col(1), prop:twocoset(1), prop:wnet(1); prop:half5(1) does the same in dimension five with 126 minors). For 𝒯_p one minor vanishes exactly at p = 23, which is why that prime is excluded.

The table lists every box shape of the chosen net with its determinant modulo the base, and the number of points per box counted directly. Set the third digit to 0 to see which shapes need it: the third digit plays the part that the borrowed jitter plays for the frame nets.