---
title: 'Delsarte Duals: Theory in Coding and Geometry'
url: https://www.emergentmind.com/topics/delsarte-duals
type: topic
---

# Delsarte Duals: Theory in Coding and Geometry

Searching arXiv for the cited papers to ground the article in current literature.
Delsarte duals are dual objects associated with Delsarte-type extremal problems and orthogonality constructions. In current usage, the term includes the dual tempered distributions \(\Phi=\delta_0+\nu\) for the Euclidean Delsarte problem, the trace-orthogonal complement \(U^{\perp_D}\) and the quotient-model dual \(U^d\) for \(\mathbb F_q\)-subspaces of \(\mathbb F_{q^m}\)-vector spaces, the trace-orthogonal dual \(C^\perp\) of Delsarte rank-metric codes, and dual feasible solutions for higher-order Delsarte linear programs for binary codes [2510.10172], [2509.24409], [1510.01008], [2501.04854]. Across these settings, duality transfers structural information between primal objects and dual certificates, complements, or quotient constructions.

## 1. Principal meanings of the term

The literature does not use “Delsarte dual” for a single uniform construction. Instead, the name appears in several technically distinct frameworks.

| Setting | Primal object | Dual object |
|---|---|---|
| Euclidean Delsarte problem | admissible positive-definite \(f\) on \(\mathbb R^d\) | tempered distribution \(\Phi=\delta_0+\nu\) |
| \(\mathbb F_q\)-subspaces of \(\mathbb F_{q^m}^k\) | \(U\in\mathcal L_q(V)\) | \(U^{\perp_D}\) and quotient-model dual \(U^d\) |
| Delsarte rank-metric codes | \(C\subseteq\mathbb F_q^{\,n\times m}\) | \(C^\perp\) under \(\Tr(MN^T)\) |
| Higher-order Delsarte LPs | level-\(\ell\) primal feasible \(f\) | dual feasible family \((g_1,\dots,g_\ell)\) |

In the finite-field settings, duality is induced by nondegenerate bilinear forms and dimension-complement formulas. In the Euclidean and coding-theoretic LP settings, duality is linear-programming duality, where the dual object is a certificate for an extremal bound rather than an orthogonal complement. The recent literature treats these viewpoints in parallel rather than reducing them to a single formalism [2510.10172], [2509.24409], [1510.01008], [2501.04854].

## 2. Euclidean-space Delsarte duality

For an open set \(U\subset\mathbb R^d\) of finite Lebesgue measure, symmetric with respect to the origin and containing \(0\), the Euclidean Delsarte primal problem seeks real-valued functions \(f\) on \(\mathbb R^d\) such that \(f\in C(\mathbb R^d)\cap L^1(\mathbb R^d)\), \(f\) is even, \(f(0)=1\), \(f(t)\le 0\) for all \(t\in U^c\), and \(\widehat f(\xi)\ge 0\) for all \(\xi\). The Delsarte constant is
\[
D(U)=\sup\Bigl\{\int_{\mathbb R^d} f(x)\,dx:\;f\text{ satisfies the admissibility conditions}\Bigr\}.
\]
The same exposition states the universal bound \(D(U)\le m(U)\) [2510.10172].

The dual variable is a tempered distribution of the form
\[
\Phi=\delta_0+\nu,
\]
where \(\nu\ge 0\) is a positive Radon measure supported in \(U^c\). Its Fourier transform is again a positive measure,
\[
\widehat\Phi=\widehat\Phi(\{0\})\,\delta_0+\mu,\qquad \mu\ge 0.
\]
The dual Delsarte constant is
\[
D'(U)=\sup\{\widehat\Phi(\{0\}):\;\Phi=\delta_0+\nu,\;\supp\nu\subset U^c,\;\widehat\Phi\ge 0\}.
\]

A central point is that weak and strong duality must be proved in the continuous setting. Weak duality gives
\[
D(U)D'(U)\ge 1,
\]
while strong duality sharpens this to
\[
D(U)D'(U)=1.
\]
The proof of strong duality proceeds by embedding the primal feasible region into a convex cone in
\[
X=L^1(\mathbb R^d)\times L^1(U^c)\times\mathbb R\times\mathbb R
\]
and applying a Hahn–Banach separation argument. The same work also proves existence of extremizers for both the primal and dual problems: the primal supremum is attained by a continuous admissible \(f\), and the dual supremum is attained by some \(\Phi=\delta_0+\nu\).

When \(f\) and \(\Phi\) are extremal, one obtains tiling-type complementarity relations. The support of \(\nu\) lies in \(\{x:f(x)=0\}\cup\{0\}\), and the support of \(\mu=\widehat\Phi-\widehat\Phi(\{0\})\delta_0\) lies in \(\{\xi:\widehat f(\xi)=0\}\cup\{0\}\). Consequently,
\[
f(x)\,(\delta_0+\nu)(x)=\delta_0(x),\qquad \widehat f\ast \widehat\Phi(\xi)=1\quad\text{a.e.},
\]
and also
\[
\widehat f(\xi)\,\widehat\Phi(\xi)=\delta_0(\xi),\qquad f\ast\Phi(x)=1\quad\text{a.e.}
\]
The paper describes this by saying that the primal extremal function and the dual extremal measure “tile” one another by convolution in space and in frequency.

The principal application in the same work concerns convex bodies and packings. For a bounded measurable set \(A\subset\mathbb R^d\) of positive measure, its essential difference set is
\[
\Delta(A)=\{t:\;m(A\cap(A+t))>0\}.
\]
The classical Cohn–Elkies argument gives
\[
\text{packing density}\le \frac1{D(\Delta(A))}.
\]
The trivial volume bound \(m(A)^{-1}\) is recovered by the admissible trial function \((1/m(A))\,1_A\ast 1_{-A}\). If \(A\) is a convex body that does not tile \(\mathbb R^d\) by translations, then
\[
D(\Delta(A))>m(A),
\qquad
\frac1{D(\Delta(A))}<\frac1{m(A)}.
\]
The proof uses extremal data for \(U=\Delta(A)\) to obtain a measure \(\nu\ge 0\) with
\[
1_A\ast \nu=1_{A^c}\quad\text{a.e.},
\]
so \(A\) weakly tiles its complement; the argument then invokes the structure theorem of Kolountzakis–Lev–Matolcsi 2023 that a convex body which weakly tiles its complement must tile properly [2510.10172].

## 3. Subspace Delsarte duality, weight, and defect

Let \(V=\mathbb F_{q^m}^k\), viewed also as an \(\mathbb F_q\)-space of dimension \(mk\). Write \(\mathcal L(V)\) for the lattice of \(\mathbb F_{q^m}\)-subspaces and \(\mathcal L_q(V)\) for the lattice of all \(\mathbb F_q\)-subspaces. Fix a nondegenerate \(\mathbb F_{q^m}\)-bilinear form \(\langle\cdot,\cdot\rangle:V\times V\to\mathbb F_{q^m}\), and let \(\Tr=\Tr_{\mathbb F_{q^m}/\mathbb F_q}\). For \(U\in\mathcal L_q(V)\), the trace-orthogonal Delsarte dual is
\[
U^{\perp_D}=\{v\in V:\Tr(\langle u,v\rangle)=0\ \forall\,u\in U\}\in\mathcal L_q(V).
\]
This satisfies
\[
\dim_{\mathbb F_q}(U)+\dim_{\mathbb F_q}(U^{\perp_D})=mk,
\qquad
(U^{\perp_D})^{\perp_D}=U,
\]
so \(U\mapsto U^{\perp_D}\) is an order-reversing involution of \(\mathcal L_q(V)\) [2509.24409].

The same framework introduces weight and defect relative to a fixed \(U\in\mathcal L_q(V)\) of \(\mathbb F_q\)-dimension \(n\) with \(\langle U\rangle_{\mathbb F_{q^m}}=V\). For any \(T\in\mathcal L(V)\),
\[
w_U(T)=\dim_{\mathbb F_q}(U\cap T),
\qquad
\epsilon_U(T)=w_U(T)-\dim_{\mathbb F_q}(T).
\]
For each \(r=0,1,\dots,k\),
\[
\epsilon_U(r)=\max\{\epsilon_U(T):\;T\in\mathcal L(V),\ \dim_{\mathbb F_{q^m}}T=r\}.
\]
The sequence \(r\mapsto\epsilon_U(r)\) is nondecreasing, with \(\epsilon_U(0)=0\) and \(\epsilon_U(k)=n-k\). Its positive jumps determine the sequence of maximum nonzero defects
\[
0=t_0<t_1<\cdots<t_s\le k,
\qquad
0<\epsilon_1<\cdots<\epsilon_s=n-k.
\]

A second notion, denoted \(U^d\), is defined in a quotient model originating from the Lunardon–Polverino “translation ovoid” argument. Every \(U\) of dimension \(n>k\) with \(\langle U\rangle=V\) can be realized as the projection of an \(n\)-dimensional \(\mathbb F_q\)-subgeometry \(W\subset\mathbb F_{q^m}^n\) onto \(V\simeq\mathbb F_{q^m}^n/\Gamma\), where \(\Gamma\) is an \((n-k)\)-space disjoint from \(W\). If \(\Gamma^\perp\) is the \(\langle\cdot,\cdot\rangle\)-orthogonal of \(\Gamma\), then
\[
U=(W+\Gamma)/\Gamma\subset \mathbb F_{q^m}^n/\Gamma,
\qquad
U^d=(W+\Gamma^\perp)/\Gamma^\perp\subset \mathbb F_{q^m}^n/\Gamma^\perp\simeq\mathbb F_{q^m}^{\,n-k}.
\]
This \(U^d\) is called a Delsarte dual of \(U\); provided \(\Gamma^\perp\cap W=0\), one has \(\dim U^d=n-k\) and \((U^d)^d=U\).

The quotient-model dual controls defect data exactly. If
\[
0<\epsilon_1<\cdots<\epsilon_s=n-k
\quad\text{at dimensions}\quad
t_1<\cdots<t_s=k,
\]
then
\[
\epsilon_{U^d}(n-k-\epsilon_i)=k-t_i,\qquad i=1,\dots,s.
\]
Thus \(U^d\) has the same defect-sequence length \(s\), with positive defects occurring at complementary dimensions. The same paper also gives a bijection \(\mathcal E_U\leftrightarrow\mathcal E_{U^d}\) between minimal subspaces of positive defect, reversing inclusion. This framework is used there to unify classes of subspaces studied in finite geometry and to give a new geometric interpretation of code duality [2509.24409].

## 4. Delsarte duals of rank-metric codes

For a prime power \(q\), let \(\mathbb F_q^{\,n\times m}\) be endowed with the nondegenerate symmetric bilinear form
\[
\langle M,N\rangle=\Tr(MN^T)\in\mathbb F_q.
\]
If \(C\subseteq\mathbb F_q^{\,n\times m}\) is an \(\mathbb F_q\)-linear rank-metric code, its Delsarte dual is
\[
C^\perp=\{M\in\mathbb F_q^{\,n\times m}:\Tr(MN^T)=0\ \forall\,N\in C\}.
\]
Nondegeneracy gives
\[
\dim_{\mathbb F_q}C^\perp=nm-\dim_{\mathbb F_q}C,
\qquad
(C^\perp)^\perp=C.
\]
The associated rank distribution is
\[
A_i(C)=|\{M\in C:\rk(M)=i\}|,
\]
and if \(\dim C=t\), the Delsarte analogue of the Singleton bound is
\[
t\le m(n-d(C)+1),
\qquad
d(C)\le n-\Bigl\lceil \frac tm\Bigr\rceil+1.
\]
The rank defect is
\[
\Rdef(C)=n-\Bigl\lceil \frac tm\Bigr\rceil-d(C)+1,
\]
with \(\Rdef(C)\ge 0\) [1510.01008].

MacWilliams identities relate the rank distributions of \(C\) and \(C^\perp\). In the form quoted in the paper, for \(0\le r\le n\),
\[
\sum_{i=0}^{n-r} A_i(C)\binom{n-i}{r}_q
=
q^{mr}|C|
\sum_{j=0}^r A_j(C^\perp)\binom{n-j}{r-j}_q.
\]
Hence the full rank distribution of \(C\) determines that of \(C^\perp\), and conversely. If \(C\) is MRD or, more generally, dually-QMRD, then for each \(r\) with \(d(C)\le r\le n\),
\[
A_r(C)=\binom{n}{r}_q\sum_{i=0}^{r-d(C)}(-1)^i\binom{r}{i}_q\bigl(q^{m(r-d(C)+1-i)}-1\bigr),
\]
so the entire distribution depends only on \((n,m,d(C))\). In the general positive-defect case, the high-rank frequencies are determined by the parameters together with the “small-rank” numbers. When \(\Rdef(C)=\Rdef(C^\perp)=1\), \(d+d^\perp=n\), and \(m\mid\dim C\), one has
\[
A_d(C)=A_{d^\perp}(C^\perp).
\]

A complementary geometric description uses \([n,k]_{q^m/q}\) rank-metric codes. If \(G\) is a generator matrix of a nondegenerate \(\mathbb F_{q^m}\)-linear code \(\mathcal C\subset\mathbb F_{q^m}^n\), and \(U_\mathcal C\) is the \(\mathbb F_q\)-span of its column vectors in \(\mathbb F_{q^m}^k\), then
\[
w(xG)=n-\dim_q(U_\mathcal C\cap x^\perp),
\]
and the minimum rank distance is
\[
d=n-\max\{\dim_q(U_\mathcal C\cap H): H\text{ an }\mathbb F_{q^m}\text{-hyperplane}\}.
\]
Moreover, \(U_{\mathcal C^\perp}\) is, up to \(\mathbb F_{q^m}\)-equivalence, the Delsarte dual of \(U_\mathcal C\), and the defect-sequence rule recovers the Wei-type duality
\[
\{d_s(\mathcal C^\perp):1\le s\le n-k\}
=
\{1,\dots,n\}\setminus\{n+1-d_r(\mathcal C):1\le r\le k\}.
\]
Within this framework, MRD, near-MRD, quasi-MRD, and \((\mathbf n,\mathbf k)\)-MRD codes are analyzed via defect sequences, and several of these classes are shown to be closed under duality [2509.24409].

## 5. Dual formulations of higher-order Delsarte LPs

For binary codes of block length \(n\) and minimum distance \(d\), Delsarte’s original level-1 primal LP uses a function \(f:\mathbb F_2^n\to\mathbb R\) supported on
\[
\Valid_{n,1}=\{x\in\mathbb F_2^n:|x|\ge d\}\cup\{0\},
\]
normalized by \(f(0)=1\), with \(\widehat f\ge 0\), \(f\ge 0\), and \(f(x)=f(-x)\). Its dual can be written in terms of a function \(g:\mathbb F_2^n\to\mathbb R\) as
\[
\min g(0)
\]
subject to
\[
\sum_{y\in\mathbb F_2^n} g(y)=1,\qquad
g(x)\le 0\ \forall x\in \Valid_{n,1}\setminus\{0\},\qquad
\widehat g(y)\ge 0\ \forall y\in\mathbb F_2^n.
\]
A feasible \(g\) certifies an upper bound on code size. Classically, choosing \(g(x)=Q_k(|x|)\) in terms of Krawtchouk polynomials recovers the MRRW bound [2501.04854].

The higher-order hierarchy replaces single codewords by matrices \(X\in\mathbb F_2^{\ell\times n}\) and defines \(\Valid_{n,\ell}\) through the linear span of the rows of \(X\). The primal formulation imposes positivity of suitable partial Fourier transforms after applying elements of \(\GL_\ell(\mathbb F_2)\). The symmetrized dual uses functions
\[
g_1,\dots,g_\ell:\mathbb F_2^{\ell\times n}\to\mathbb R
\]
with objective
\[
1+\sum_{k=1}^\ell g_k(0),
\]
subject to a validity inequality on \(\Valid_{n,\ell}\setminus\{0\}\) and the positivity constraints \({}_k g_k\ge 0\). When \(\ell=1\), this reduces to the classical level-1 dual; at level \(\ell\), a feasible dual bounds \(|C|^\ell\).

A central structural result is the lifting theorem. If \(k\mid \ell\) and \((h_1,\dots,h_k)\) is feasible for the level-\(k\) dual with objective value
\[
V=1+\sum_{i=1}^k h_i(0),
\]
then one can explicitly construct a feasible level-\(\ell\) dual \((g_1,\dots,g_\ell)\) with objective
\[
1+\sum_{i=1}^\ell g_i(0)=V^{\ell/k}.
\]
The paper interprets this as showing that any level-\(k\) dual bound automatically lifts to level \(\ell\) without loss in the natural exponent. It also gives a dual-based completeness proof: if \(\Valid_n\) is closed under taking subspaces and
\[
k=\max\{\dim(S):S\in\Valid_n\},
\]
then for every \(\ell\ge n\) the optimum of the subspace-symmetric dual is exactly
\[
2^{k\ell}.
\]
The same work further states that the spectral construction via the polynomial \(\Phi_m\) and combinatorial “cylinder” arguments is the first from-scratch dual at level \(\ell\ge 2\), and that it recovers the MRRW rate
\[
\frac14\,\epsilon^2\log_2\!\frac1\epsilon
\]
up to lower-order terms for small relative distance \(\delta=\frac{1-\epsilon}{2}\) [2501.04854].

## 6. Applications, closure properties, and recurring distinctions

Delsarte duality has direct consequences beyond coding bounds. In the q-matroid setting, a q-matroid \(M\) on \(E\simeq\mathbb F_q^n\) is specified by a rank function \(\rho:\mathcal L(E)\to\mathbb N\) satisfying the axioms \((R1)\) \(0\le \rho(A)\le \dim A\), \((R2)\) monotonicity, and \((R3)\) submodularity. If \(G\) is a \(k\times n\) matrix over \(\mathbb F_{q^m}\) of rank \(k\), then
\[
\rho_G(A)=\rank_{\mathbb F_{q^m}}(G\cdot A^U)
\]
defines an \(\mathbb F_{q^m}\)-representable q-matroid \(M_G\), depending only on the \(\mathbb F_q\)-system \(U_\mathcal C\) spanned by the columns, and \(M_{G^\perp}\) is the dual q-matroid \(M_G^*\). For direct sums of uniform q-matroids, Delsarte duality yields the equivalence
\[
U_{k_1,n_1}\oplus\cdots\oplus U_{k_t,n_t}\text{ is representable over }\mathbb F_{q^m}
\]
if and only if
\[
U_{n_1-k_1,n_1}\oplus\cdots\oplus U_{n_t-k_t,n_t}\text{ is representable over }\mathbb F_{q^m}.
\]
The same paper also defines the rank-generating function
\[
R_M(X_1,X_2,X_3,X_4)
=
\sum_{D\in\mathcal L(E)}
X_1^{\rho(E)-\rho(D)}
X_2^{\dim D-\rho(D)}
\prod_{i=0}^{\dim D-1}(X_3-q^iX_4),
\]
and states that when \(M=M_G\), the weight enumerator \(W_\mathcal C(X,Y)\) is recovered from \(R_M\) by a simple substitution [2509.24409].

A recurring source of ambiguity is that the same expression refers to different dual mechanisms. In the subspace theory, \(U^{\perp_D}\) is the trace-orthogonal complement inside \(\mathcal L_q(V)\), whereas \(U^d\) is a quotient-model dual living in a different ambient dimension. In Euclidean extremal theory, the dual is a tempered distribution \(\Phi=\delta_0+\nu\), not an orthogonal complement. In higher-order coding LPs, the dual consists of feasible certificates \((g_1,\dots,g_\ell)\). Another common simplification is to regard weak and strong linear duality as automatic in every setting; the Euclidean theory explicitly notes that this is automatic in the finite group setting, but requires proof in the continuous setting [2510.10172]. These distinctions are structural rather than terminological: each version of Delsarte duality is tailored to the ambient category in which the extremal problem is posed.

Source: https://www.emergentmind.com/topics/delsarte-duals