3 Frames: \(t=0\) for free
Let \(x_1,\dots ,x_n\in R\) and \(\alpha _1,\dots ,\alpha _n,\alpha _\infty \ge 0\) with \(\sum _P\alpha _P+\alpha _\infty =m\), and let \(H\) be the matrix of the map that sends \(V\in R[t]_{{\lt}m}\) to its \(\alpha _P\)-jets at the \(x_P\) and its \(\alpha _\infty \)-jet at \(\infty \). Then \(\det H=\pm \prod _{P{\lt}Q}(x_Q-x_P)^{\alpha _P\alpha _Q}\). In particular, over a field, \(H\) is invertible if the \(x_P\) with \(\alpha _P\ge 1\) are pairwise distinct, and \(H\) has two equal rows if two of them coincide.
The value is the confluent Vandermonde determinant [ 46 , Sec. 6 ] . The consequences used below also follow by counting zeros: if over a field the points are distinct and all jets of \(V\) vanish, the product \(\prod _P(t-x_P)^{\alpha _P}\) of pairwise coprime factors, of degree \(m-\alpha _\infty \), divides \(V\), while \(\deg V{\lt}m-\alpha _\infty \) because the jet at \(\infty \) vanishes; so \(V=0\). If \(x_P=x_Q\) and \(\alpha _P,\alpha _Q\ge 1\), the rows \(V(x_P)\) and \(V(x_Q)\) coincide. Over a field this zero count is Skriganov’s proof that the hyperderivative (Rosenbloom–Tsfasman Reed–Solomon) codes [ 52 ] are MDS in that metric [ 56 , Prop. 5.1, Thms. 5.1, 5.2 ] .
3.1 The Möbius frame
Let \(P_{ji}=\binom {i}{j}\) be the upper Pascal matrix, \(J\) the reversal, \(D_{\pm }=\operatorname {diag}((-1)^i)\), and \((\Lambda _m)_{ij}=(-1)^{i-j}\binom {m-1-j}{i-j}\) (lower-unitriangular). Then \(\Psi =\Lambda _m P\) and \(\Psi ^2=D_{\pm }JP\). Consequently, for each \(m\), any net with generator matrices \((LT^2,L,LT)\), \(L\) lower-triangular, is (after re-indexing \(v\mapsto D_{\pm }Pv\)) the net with generator matrices
the Hammersley net of the two-dimensional generalized Faure sequence \((A_1,A_2P)\) of Tezuka [ 62 ] with \(A_1=s_kL\Lambda _m D_{\pm }\), \(A_2=LD_{\pm }\), and its Hammersley coordinate scrambled by \(\pm L\). Since \(A_1,A_2\) depend on \(m\), the family is not the set of prefixes of one sequence.
Expand \(V=\sum _j w_j(t-1)^j\) with \(w=Pv\). Then \(\Psi V=\sum _j w_j t^j(1-t)^{m-1-j}\), giving \(\Psi =\Lambda _m P\), and \(\Psi ^2 V=\sum _j w_j(-t)^{m-1-j}\), giving \(\Psi ^2=D_{\pm }JP\). Finally use \(P^{-1}=D_{\pm }PD_{\pm }\) and \(D_{\pm }JD_{\pm }=(-1)^{m-1}J\).
3.2 The frame of \(s\) places
Every box minor of the frame is, up to sign, \(\prod _{\{ c,d\} \subset F}(a_c-a_d)^{d_cd_d}\). In particular, for \(m\ge 2\) and \(q\) coprime to the denominators of the places, the frame is a \((0,m,s)\)-net over \(\mathbb {Z}/q\mathbb {Z}\) if and only if the finite places are pairwise distinct modulo every prime factor of \(q\).
The box matrix for the depths \((d_c)\) is the Hermite matrix of Lemma 9 for the places with multiplicities \(d_c\), which gives the value, a unit modulo a prime \(p\) if the finite places are distinct modulo \(p\). Conversely, if \(a_c\equiv a_d\pmod p\), a box with \(d_c=d_d=1\) (it exists as \(m\ge 2\)) contains the two equal rows \(V(a_c)\), \(V(a_d)\) modulo \(p\). The net statement then follows from Lemma 1.
For every \(k\ge 1\) and \(c\ne d\), row \(0\) of \(\Phi ^F_{cd}\) has the generating function
and \(1/\Pi _d(d+x)\) for \(c=\infty \), \(d\in F\).
The \((t-b)^k\), \(b\in F\), are pairwise comaximal in \(R[t]\), so partial fractions give \(V/\Pi =H+\sum _{b\in F}N_b/(t-b)^k\) with \(\deg H{\lt}k\) (as \(\deg V{\lt}sk=\deg \Pi +k\)) and \(\deg N_b{\lt}k\); that is, \(V=\Pi H+\sum _b\Pi _bN_b\).
Cells. For \(b\in F\), \(V\equiv \Pi _bN_b\pmod{(t-b)^k}\) with \(\Pi _b(b)=Q_b^k\) a unit, so the cell of \(b\) and \(N_b\) determine each other. As \(\deg \sum _b\Pi _bN_b{\lt}m-k\), the cell of \(\infty \) (the top \(k\) coefficients of \(V\)) is that of \(\Pi H\), which determines \(H\) as \(\Pi \) is monic.
Digit \(k\) at \(c\in F\). \(V=\Pi _cN_c+(t-c)^k\Pi _cG_c\) with \(G_c=H+\sum _{b\ne c}N_b/(t-b)^k\), regular at \(c\). So digit \(k\) is a term depending on \(N_c\) only (the own cell) plus \(Q_c^kG_c(c)=Q_c^k\bigl(H(c)+\sum _{b\ne c}\mathrm{pp}_b(c)\bigr)\), where \(\mathrm{pp}_b=N_b/(t-b)^k\) is the principal part at \(b\).
Digit \(k\) at \(\infty \), the coefficient of \(t^{m-1-k}\), is a term from \(\Pi H\) (the own cell) plus, from each \(\Pi _bN_b\), the top coefficient of \(N_b\), i.e. \(\operatorname {Res}_b(V\, dt/\Pi )\).
Column \((d,j)\), \(d\in F\). All other cells vanish, so \(H=0\), \(N_b=0\) for \(b\ne d\), and \(\mathrm{pp}_d\) is the principal part at \(d\) of \((t-d)^{j-k}/\Pi _d\). For a function \(f\) whose only pole near \(d\) has principal part \(\mathrm{pp}\), \(\mathrm{pp}(c)=\operatorname {Res}_{t=d}f(t)/(c-t)\); hence \(\mathrm{pp}_d(c)=[u^{k-1-j}]\, 1/(\Pi _d(d+u)(c-d-u))\) and \(\operatorname {Res}_d(V/\Pi )=[u^{k-1-j}]\, 1/\Pi _d(d+u)\). Summing over \(j\) with weights \(x^{k-1-j}\) gives the first and the third entry.
Column \((\infty ,j)\). The finite cells vanish, so \(V=\Pi H\equiv t^{m-1-j}\) modulo degree \({\lt}m-k\), and \(H\) is the polynomial part of \(t^{k-1-j}\prod _b(1-b/t)^{-k}\), and \(\sum _jx^{k-1-j}H_j(c)=\prod _b(1-bx)^{-k}/(1-cx)\bmod x^k\); multiply by \(Q_c^k\).
Apply the move of Lemma 2(c) with \(L_c=\operatorname {diag}(Q_c^{-k},1,\dots ,1)\), \(U_d=\operatorname {Toep}(\Pi _d(d+x))\) for \(d\in F\) and \(U_\infty =\operatorname {Toep}(\prod _{b\in F}(1-bx)^k)\); their diagonal entries are units of \(R\). Then row \(0\) of every off-diagonal block becomes
independently of \(k\), where \(w_{cd}=y_{cd}=1/(a_c-a_d)\) for \(c,d\in F\); \(w=1\), \(y=0\) for \(c=\infty \); and \(w=1\), \(y=a_c\) for \(d=\infty \).
Substitute into Lemma 12: for instance \(Q_c^{-k}\cdot Q_c^k/(\Pi _d(d+x)(c-d-x))\cdot \Pi _d(d+x)=1/(c-d-x)\).
3.3 Designs that work for every \(k\)
Suppose the residual problem is solvable. For \(k\ge 1\) put \(\lambda _c=\lambda '_cQ_c^{-k}\) and \(U_d=U^0_dU'^{(k)}_d\), where \(U^0_d\) is the Toeplitz column move of Proposition 13, and \(U'^{(k)}_d\) agrees with \(U'_d\) at the carry depths \({\lt}\min (s-1,k)\) and has, for \(s-1\le n{\lt}k\), the column \(e_n-\sum _{i\le s-2}\beta _{n,i}\cdot (\text{column }i\text{ of }U'_d)\), where \(\beta _n\in R^{s-1}\) solves \(\lambda '\odot v_n=\sum _i\beta _{n,i}F_i[\cdot ][d]\) (rows \(c\ne d\)). Then row \(0\) of \(L_c\Phi ^F_{cd}U_d\) equals the design at the carry depths \({\lt}s-1\) (truncated if \(k{\lt}s-1\)) and vanishes at the larger carry depths; all unit factors and all entries lie in \(R\).
For \(n{\lt}s-1\) this is the residual problem (Proposition 13). The \(y_{cd}\), \(c\ne d\), are pairwise distinct with unit differences (\(1/(a_c-a_d)-1/(a_{c'}-a_d)=(a_{c'}-a_c)/((a_c-a_d)(a_{c'}-a_d))\), and \(0\) resp. the \(a_c\) in the other cases), and the \(w_{cd}\) are units, so the matrix \((v_0,\dots ,v_{s-2})\) is invertible over \(R\), and so is \((F_i[\cdot ][d])_{i\le s-2}=\Lambda '(v_0,\dots ,v_{s-2})U'_d\). Hence \(\beta _n\) exists over \(R\), and the new column \(n\) has row-\(0\) entries \(\lambda '_cv_n[c]-\sum _i\beta _{n,i}F_i[c][d]=0\) and diagonal entry \(1\).
Let \((F_n)\) be a design whose residual problem is solvable over \(R'=R[1/D']\), where \(D'\) collects the primes in the units and denominators of \(\lambda '\) and \(U'\), and let \(\gcd (q,DD')=1\). The digital net \(\mathcal P^F_{k,q}\) over \(\mathbb {Z}/q\mathbb {Z}\) is the graph form with depths \((k,\dots ,k)\) and blocks \(\Phi '_{cd}=L_c\Phi ^F_{cd}U_d\) (\(c\ne d\); Proposition 14), \(\Phi '_{cc}\) with row \(0\) equal to \((F_{s-2}[c][c],\dots ,F_0[c][c])\) on the levels \((k-s+1,\dots ,k-1)\) (truncated if \(k{\lt}s-1\)) and all other entries \(0\), all entries reduced modulo \(q\).
Every box minor of \(\mathcal P^F_{k,q}\) over \(R'\) is a unit of \(R'\); so \(\mathcal P^F_{k,q}\) is a digital \((0,sk,s)\)-net over \(\mathbb {Z}/q\mathbb {Z}\) for every \(k\ge 1\), with one point in every cell.
Let \(m=sk\) be odd with \((s,k)\ne (3,1)\), and \(r^*=(m-1)/2\). For any two coordinates of \(\mathcal P^F_{k,q}\) (Definition 15), and of \(\mathcal P_{k,q}\) (Definition 43), the digits with index \(r^*\) agree up to a sign and a linear combination of more significant digits; for \(\mathcal P^F_{k,q}\) the sign is \(+1\) if one of the two places is \(\infty \), and \((-1)^{r^*}\) otherwise. For the places \(\infty \) and \(0\) this is the reversal symmetry of the two-dimensional Hammersley set, which reads digit \(r^*\) as \([t^{m-1-r^*}]V=[t^{r^*}]V\); a Möbius change of variable transports it to every pair of places, and the lower-triangular scrambles of Corollary 16 keep it, since \(r^*\ge k+1\) and their diagonal entries in the rows below the first offset row are \(1\). This digit relation is proved. Its visible effect is computed, not proved: in the computed cases every two-dimensional projection lies on parallel lines, at a dual frequency of length about \(\sqrt{2/q}\, \sqrt N\) (Figure fig:projlines), which the \(t\)-parameter does not see. The phenomenon is akin to the known diagonal and rank-\(1\)-lattice clustering of generalized Faure sequences whose generator matrices coincide up to a scalar [ 38 , 37 ] ; there the coincidence is in the top digits, here in the middle one. The remedy is of the classical kind: a \(t\)-preserving left multiplication by a nonsingular lower-triangular, here diagonal, matrix, as in Tezuka’s generalized Faure sequences [ 61 , 62 ] and Matoušek’s linear scrambling [ 43 ] , with per-coordinate multipliers chosen on two-dimensional projections as in [ 17 , 38 ] . What is specific here is that only row \(r^*\) is rescaled, by units \(\rho _c\). This keeps \(t=0\), as it multiplies every box minor by a unit, and, since Lemma 4 and Theorem 48 use that row only through (H2), it keeps the proved separation bounds (of Lemma 4, Proposition 21, Theorems A(2) and 48); both facts are proved. For a two-dimensional projection with points \(x\) let \(S(\mathbf h)=N^{-1}\sum _xe^{2\pi i\, \mathbf h\cdot x}\), \(\mathbf h\in \mathbb {Z}^2\), and call the least \(|\mathbf h|/\sqrt N\) over the frequencies with \(|S(\mathbf h)|\ge \frac12\) its spectral value (about \(1\) for a good two-dimensional lattice; lines at spacing \(N^{-1/2}\) divided by this value appear when it is small). With \(\rho \) found per \(q\) by an exhaustive search that keeps an estimate of \(|S(\mathbf h)|\) small for \(|\mathbf h|{\lt}0.8\sqrt N\) in every pair of coordinates, the least spectral value over the pairs of coordinates at \(s=5\), \(q=13\)–\(23\) rises from \(0.22\)–\(0.30\) to about \(0.80\) (computed).
3.4 Reachable designs
The off-diagonal part of a design is reachable from the frame by the gauge moves of Lemma 2(c) with row scalars \(\lambda '\) (that is, the residual problem is solvable with these \(\lambda '\)) if and only if, for every column \(d\) and every \(j\le s-2\),
for a polynomial \(g_{d,j}\in R[X]\) of degree \(j\) whose leading coefficient is a unit. For finite \(c,d\) this says \(M^{(j)}[c][d]=\lambda '_c\tilde g_{d,j}(z_{cd})\) with \(z_{cd}=1/(a_c-a_d)\) and \(\tilde g_{d,j}(z)=z\, g_{d,j}(z)\) of degree \(j+1\) without constant term.
A lower-triangular column move with unit diagonal is a Toeplitz move times an upper-triangular matrix in the carry depths, so the residual problem captures all of Lemma 2(c) on row \(0\). Column \(n\) of \(\lambda '^{-1}\odot F_{\cdot d}\) must be \(\sum _{n'\le n}v_{n'}(U'_d)_{n'n}\), i.e. \((w_{cd}f_n(y_{cd}))_c\) with \(\deg f_n\le n\) and the coefficient of \(X^n\) a unit; as the \(y_{cd}\) are distinct with unit differences, interpolation is unique over \(R\), and this is exactly the condition. Partial sums of such polynomials have the same form, and conversely.
3.5 Cauchy identities
Let \(R\) be a commutative ring, \(\alpha ,\beta \in R^s\), \(D_{cd}=\alpha _c\beta _d-\alpha _d\beta _c\), and \(W_{cd}=\lambda _c\mu _d/D_{cd}\) for \(c\ne d\), where all \(\lambda _c\), \(\mu _d\) and \(D_{cd}\) are units of \(R\); let \(\nu _c=\lambda _c\mu _c\) and \(\pi _{cd}=W_{cd}W_{dc}\).
For distinct \(c,d,e\): \(W_{cd}W_{de}W_{ec}=-W_{dc}W_{ed}W_{ce}\). Hence, if \(R\) is an ordered field, \(\pi _{cd}\pi _{de}\pi _{ec}{\lt}0\), and “\(c\sim d\) iff \(\pi _{cd}{\lt}0\)” is an equivalence relation with at most two classes.
For distinct \(a,b,c,d\) let \(\chi =D_{ac}D_{bd}/(D_{ab}D_{cd})\) and \(\psi =D_{ad}D_{bc}/(D_{ab}D_{cd})\). Then \(\chi -\psi =1\), \(\chi ^2=\pi _{ab}\pi _{cd}/(\pi _{ac}\pi _{bd})\) and \(\psi ^2=\pi _{ab}\pi _{cd}/(\pi _{ad}\pi _{bc})\); consequently \((\chi ^2-\psi ^2-1)^2=4\psi ^2\).
If \(R\subset \mathbb {R}\): for distinct \(i,j,k,l\), the largest of \(|\pi _{ij}\pi _{kl}|^{-1/2}\), \(|\pi _{ik}\pi _{jl}|^{-1/2}\), \(|\pi _{il}\pi _{jk}|^{-1/2}\) is the sum of the other two.
For distinct \(i,j,l,d\):
\[ \frac{W_{ji}W_{ld}}{W_{jd}W_{li}}+\frac{W_{ij}W_{ld}}{W_{id}W_{lj}}=1 . \]
Everything follows from \(D_{dc}=-D_{cd}\) and the Plücker relation \(D_{ab}D_{cd}-D_{ac}D_{bd}+D_{ad}D_{bc}=0\). (C) The \(\lambda \), \(\mu \) cancel, and the \(D\) cancel up to \((-1)^3\); then \(\pi _{cd}\pi _{de}\pi _{ec}=-(W_{dc}W_{ed}W_{ce})^2\). If \(c\sim d\) and \(d\sim e\), two of the three factors are negative, so the third is too; if \(c\not\sim d\) and \(d\not\sim e\), then \(c\sim e\). (P) Dividing the Plücker relation by \(D_{ab}D_{cd}\) gives \(\chi -\psi =1\). Since \(\pi _{xy}=-\nu _x\nu _y/D_{xy}^2\), we have \(\pi _{ab}\pi _{cd}=\nu _a\nu _b\nu _c\nu _d/(D_{ab}D_{cd})^2\) and similarly for the other pairings, which gives \(\chi ^2\) and \(\psi ^2\); and \(\chi =1+\psi \) gives \(2\psi =\chi ^2-\psi ^2-1\). (H) By the same formula \(|D_{ij}D_{kl}|\), \(|D_{ik}D_{jl}|\), \(|D_{il}D_{jk}|\) are proportional to the three numbers, and by the Plücker relation the products sum to zero with signs. (X) The \(\lambda \) and \(\mu \) cancel, the two terms are \(D_{jd}D_{il}/(D_{ij}D_{ld})\) and \(-D_{id}D_{jl}/(D_{ij}D_{ld})\), and \(D_{il}D_{jd}-D_{id}D_{jl}=D_{ij}D_{ld}\) is the Plücker relation for \((i,j,l,d)\).