2 Graph forms and the one-digit relaxation
For each \(d\), every elementary box with \(d_i\) digits in coordinate \(i\) contains exactly one point if and only if the box minor for \(d\) is a unit of \(R\). In particular the point set is a \((0,m,s)\)-net in base \(q\) if and only if every box minor is a unit of \(R\), i.e., over a field, nonzero.
The point with index \(v\) lies in a given box of shape \(d\) if and only if the box matrix maps \(v\) to the prescribed leading digits. So the boxes of shape \(d\) correspond bijectively to the values of the linear map \(v\mapsto (\text{box matrix})\, v\) on \(R^m\). There are \(q^m\) points and \(q^m\) boxes, so every box holds exactly one point if and only if this map is bijective, i.e. if and only if the box matrix is invertible, i.e. its determinant is a unit. This is [ 46 , Thm. 6.1 ] .
2.1 Graph forms
The box minor of a graph form for \(d\) equals, up to sign, the determinant of the square matrix formed by the top \(d_c-k_c\) offset rows of the over coordinates \(c\), restricted to the last \(k_e-d_e\) levels of the under coordinates \(e\) (it is \(1\) if \(d=\mathbf k\)). In particular the diagonal blocks \(\Phi _{cc}\) never enter. Consequently the following gauge moves multiply every box minor by a unit of \(R\), and so, by Lemma 1, map \((0,m,s)\)-nets over \(\mathbb {Z}/q\mathbb {Z}\) or \(\mathbb {F}_q\) to \((0,m,s)\)-nets: (a) replacing the diagonal blocks arbitrarily; (b) for \(s=3\) only: \(\Phi _{cd}\mapsto \mu _{cd}\Phi _{cd}\) (\(c\ne d\)) with units \(\mu _{cd}\) of \(R\) (block scalings); (c) \(\Phi _{cd}\mapsto L_c\Phi _{cd}U_d\) (\(c\ne d\)) with \(L_c\), \(U_d\) lower-triangular whose diagonal entries are units of \(R\).
The first \(\min (d_c,k_c)\) rows of \(C_c\) are unit rows selecting the first \(\min (d_c,k_c)\) levels of \(n_c\). Laplace expansion along all these unit rows removes the selected columns; what remains are the top \(d_c-k_c\) offset rows of the over coordinates, on the unselected columns, which are the levels \(\ge d_e\) of the under coordinates. A coordinate is not both over and under, so no diagonal block occurs. (c) Left multiplication by \(L_c\) preserves the span of the top rows of \(\Phi _{c\cdot }\), and right multiplication by \(U_d\) the span of the last columns of \(\Phi _{\cdot d}\); the determinant changes by the product of the corresponding diagonal entries. (b) For \(d\ne \mathbf k\) both over and under coordinates exist; for \(s=3\) there is therefore a single over or a single under coordinate, so the matrix is \([\Phi _{cd}\, |\, \Phi _{cd'}]\) or \([\Phi _{cd};\Phi _{c'd}]\) restricted in this way, and scaling one block multiplies its determinant by a power of \(\mu _{cd}\). For \(s\ge 4\) a box can have two over and two under coordinates, for instance \((k+1,k+1,k-1,k-1)\), and single-block scalings are no longer gauge moves.
Let a graph form have the depths \((k,\dots ,k)\). The first offset digit of a coordinate \(c\) (its digit with index \(k\)) is its geometric digit \(r_c\). The geometric layer is the integer matrix \(\tilde M\) of the centred representatives \(\tilde M_{cd}\in (-q/2,q/2]\) of the level-\((k-1)\) entries \((\Phi _{cd})_{0,k-1}\) of the first offset rows, diagonal included.
2.2 The one-digit relaxation
Let \(q\ge 3\), \(k\ge 1\), and let a digital net over \(\mathbb {Z}/q\mathbb {Z}\) be given in graph form with depths \((k,\dots ,k)\) such that (H1) row \(0\) of every block \(\Phi '_{cd}\), diagonal included, equals a design \((F_0,\dots ,F_{L-1})\), \(1\le L\le k\), on the levels \(k-1,\dots ,k-L\) and vanishes on the levels \(0,\dots ,k-1-L\); (H2) the rows \(\ge 1\) of the diagonal blocks vanish. In cell units write coordinate \(i\) of a point as \(n_i+(r_i+f_i)/q\) with \(r_i\) its first offset digit and \(f_i\in [0,1)\). For a step \(e\in \{ -1,0,1\} ^s\setminus \{ 0\} \) to a toroidally adjacent cell and carry depths \(j_d\) on the moving axes (\(j_d\) is the number of further levels that change, by a carry if \(e_d=1\) and by a borrow if \(e_d=-1\)), put
and \(b_i=\min _t\max (0,|e_iq+t|-\mathrm{jit}_i)\), the minimum over \(t\in \{ \bar\sigma _i,\bar\sigma _i-q\} \) if \(\bar\sigma _i\ne 0\) and over \(t=0\) if \(\bar\sigma _i=0\). Then the two points in cells \(n\) and \(n+e\) satisfy \(q|\Delta _i|\ge b_i\) for every \(i\), where \(\Delta _i\) is coordinate \(i\) of the point in cell \(n+e\) minus that of the point in cell \(n\), in cell units (\(n+e\) taken as the toroidal neighbour), and the toroidal minimum distance satisfies
A carry of depth \(j\) on axis \(d\) changes the levels \(k-1,\dots ,k-1-j\) of \(n_d\) by \(e_d\) modulo \(q\); the torus wrap has \(j=k-1\) (at \(k=1\) every step has \(j=0\)). By (H1), \(r_i\) changes by \(\sigma _i\) modulo \(q\), and capping the carry depth at \(L-1\) is exact because row \(0\) vanishes on the levels \(0,\dots ,k-1-L\). Since \(r'_i-r_i\in (-q,q)\) and \(r_i'-r_i\equiv \sigma _i\), we have \(r'_i-r_i\in \{ \bar\sigma _i,\bar\sigma _i-q\} \), and \(r'_i-r_i=0\) if \(\bar\sigma _i=0\). Always \(|f'_i-f_i|{\lt}1\), and \(f'_i=f_i\) if only axis \(i\) moves: by (H2) the deeper digits of \(i\) do not depend on \(n_i\). So \(q\Delta _i=e_iq+(r'_i-r_i)+(f'_i-f_i)\) gives \(q|\Delta _i|\ge b_i\).
It remains to compare with the toroidal distance; this is the only place where the size of \(q^k\) enters. In cell units the torus has period \(G=q^k\ge 3\). Let \(n\ne n'\) be two cells, let \(E\in \mathbb {Z}^s\) be the representative of \(n'-n\) modulo \(G\) with \(|E_c|\le G/2\), and let \(\theta _c\in (-1,1)\) be coordinate \(c\) of \((r'-r+f'-f)/q\). The toroidal distance in coordinate \(c\) is \(\operatorname {tor}_c=\min _{z\in \mathbb {Z}}|E_c+zG+\theta _c|\). If some \(|E_c|\ge 2\), then \(|E_c+zG|\ge |E_c|\) for every \(z\), so \(\operatorname {tor}_c\ge |E_c|-|\theta _c|{\gt}1\). Otherwise \(E=e\in \{ -1,0,1\} ^s\setminus \{ 0\} \) and \(\Delta =e+\theta \) with \(|\Delta _c|{\lt}2\); if \(|\Delta _c|\le G/2\), then \(\operatorname {tor}_c=|\Delta _c|\ge b_c/q\), and otherwise \(\operatorname {tor}_c\ge G-|\Delta _c|{\gt}G-2\ge 1\). So either some coordinate has toroidal distance at least one cell, or the distance is at least \((\sum _ib_i^2)^{1/2}/q\) cells; as a cell has side \(q^{-k}\), this is the claim.
Call a moving coordinate backward if \(e_i\sigma _i\le 0\) and forward otherwise. For a fixed design and \(q\to \infty \): if some configuration with at least two moving axes has no backward coordinate, or some diagonal partial sum \(M^{(j)}[a][a]\) is positive, then \(\mathrm{LB}=O(1/q)\); otherwise \(\mathrm{LB}=1-K/q+O(q^{-2})\) with \(K\) the maximum of \(|M^{(j)}[a][a]|\) and of \(|\sigma _b|+1\) over the configurations with at least two moving axes and a single backward coordinate \(b\). (Two backward coordinates cost nothing at first order, and the jitter costs the \(+1\).) In dimension three, \(M=-I+S\) with \(S\) skew with entries \(\pm 1\) gives \(K=1\).
2.3 The lattice picture
The permutation \((0,m,2)\)-nets of GrünschloSS and Keller [ 20 , Sec. 3 ] for even \(m\) show this mechanism in the smallest case, base \(2\) and dimension two (Figure fig:arcgk). For even \(m=2r\ge 4\), \(Q=2^r\), \(N=Q^2\) and \(0\le i,j{\lt}Q\) they are the points \((X,Y)/N\) with \(X=\bigl(Q\varrho (i)+\lfloor i/4\rfloor +\tfrac Q4(j+1)\bigr)\bmod N\), \(Y=i+Qj\), where \(\varrho \) reverses \(r\) bits; distances are taken modulo the lattice \(\langle (N,0),(N/4,N)\rangle \) of the row-shifted tiling for which they were designed [ 20 , 53 ] (the tiling metric). (i) Cells and geometric digit. Since \(\lfloor i/4\rfloor {\lt}Q/4\), the binary digits of \(X\) are the fields \([c_x;\, (j+1)\bmod 4;\, \lfloor i/4\rfloor ]\) with \(c_x=\varrho (i)+\lfloor (j+1)/4\rfloor \bmod Q\), and those of \(Y\) are \([j;\, i]\). So the cells of depths \((r,r)\) hold one point each, and the first two offset digits of \(X\), one base-\(4\) geometric digit, are \((j+1)\bmod 4\), a function of the last two cell digits of \(Y\). The occupied cells of depths \((r+2,r)\) are exactly the coset \(\{ (a,b):a\equiv b+1\pmod4\} \) of an index-\(4\) lattice. (ii) Lattice and jitter. With \(e=(Q,0)\), \(v=(Q/4,Q)\) and \(u=\varrho (i)\),
the points lie on the lines of the lattice \(\Lambda =\mathbb {Z}e+\mathbb {Z}v\) in direction \(v\), exactly \(v\) apart along each line, and line \(u\) is slid along itself by \(\varrho (u)/Q\) of one step (the slide of line \(u\), a van der Corput value), up to less than one grid unit. The slides are the jitter. The tiling lattice is \(Q\Lambda \), so the lines close up and there are no slips. (iii) Separation. The functional \(X-Y/4\) separates distinct lines by at least \(Q-\frac34\), so \(\delta \sqrt N\ge (4/\sqrt{17})(1-1/Q)\) for every choice of the slides; the van der Corput slides give exactly \(\delta ^2N=241/256\) in the tiling metric for every even \(m\ge 8\) [ 53 ] . On these lines the slides that keep \(t=0\) are exactly the nested scramblings of the van der Corput slides, and all of them are equally far from lattice slides (both computed for \(m\le 8\)), so only their arrangement is free. Flipping the third binary digit of the slide on the lines \(u\equiv 2,3\pmod4\) is again a nested scrambling, so it gives a \((0,m,2)\)-net, by (iv) and [ 47 ] . In the tiling metric it is worse at \(m=8\) (\(\delta ^2N=229/256\) against \(241/256\)), and for every even \(m\) with \(10\le m\le 20\) it has \(\delta ^2N=977/1024\) against \(241/256=964/1024\), i.e. \(\delta \sqrt N=0.9767\ldots \) against \(0.9702\ldots \) (computed). The jitter is not reduced, only moved off the direction across the lines, as in Section 11. (iv) The net property. The slides carry \(t=0\): the net is a nested (Owen) scrambling of the Hammersley net, the frame of the two places \(\infty \), \(0\), and nested scramblings of a \((t,m,s)\)-net are \((t,m,s)\)-nets [ 47 ] ; this gives a short proof of the net property proved in [ 53 ] . The net is not digital (it does not contain the origin [ 20 , 53 ] ), its scramblings are not linear, and its metric (the row-shifted torus) is adapted to its lattice of lines. The nets of this paper are digital: they use linear scrambles of frames with more places, keep the standard torus, and pay with a slip plane wherever a geometric digit wraps around modulo \(q\); a nonlinear permutation can traverse a sub-cell (a box of side \(1/q\) of a cell) without such a jump, and linear offsets cannot (Section 12).
2.4 Notation
For a point set given by digit vectors \(x_c\in (\mathbb {Z}/q\mathbb {Z})^m\) and \(\tau =(\tau _1,\dots ,\tau _s)\in (\mathbb {Z}/q\mathbb {Z})^{s\times m}\), the digitally shifted point set \(P\oplus \tau \) has the digit vectors \(x_c+\tau _c\), added digit by digit modulo \(q\) without carries. For prime \(q\) this is [ 11 , Def. 2.2 ] . A digital net is unshifted (\(\tau =0\)) unless stated otherwise.