Delsarte Duals: Theory in Coding and Geometry

• Delsarte duals are dual constructions that associate primal extremal objects with complementary dual certificates across Euclidean, finite-field, and coding theory settings.
• They enable the transfer of structural information by linking orthogonality, defect sequences, and linear programming duality to achieve sharp bounds and tiling properties.
• Applications range from optimizing convex body packings and analyzing rank-metric code distributions to unifying diverse constructs in finite geometry and q-matroid theory.

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 (Kolountzakis et al., 11 Oct 2025, Borello et al., 29 Sep 2025, Cruz et al., 2015, Coregliano et al., 8 Jan 2025). 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 $U^{\perp_D}$0	$U^{\perp_D}$1	$U^{\perp_D}$2 and quotient-model dual $U^{\perp_D}$3
Delsarte rank-metric codes	$U^{\perp_D}$4	$U^{\perp_D}$5 under $U^{\perp_D}$6
Higher-order Delsarte LPs	level-$U^{\perp_D}$7 primal feasible $U^{\perp_D}$8	dual feasible family $U^{\perp_D}$9

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 (Kolountzakis et al., 11 Oct 2025, Borello et al., 29 Sep 2025, Cruz et al., 2015, Coregliano et al., 8 Jan 2025).

2. Euclidean-space Delsarte duality

For an open set $U^d$0 of finite Lebesgue measure, symmetric with respect to the origin and containing $U^d$1, the Euclidean Delsarte primal problem seeks real-valued functions $U^d$2 on $U^d$3 such that $U^d$4, $U^d$5 is even, $U^d$6, $U^d$7 for all $U^d$8, and $U^d$9 for all $\mathbb F_q$0. The Delsarte constant is

$\mathbb F_q$1

The same exposition states the universal bound $\mathbb F_q$2 (Kolountzakis et al., 11 Oct 2025).

The dual variable is a tempered distribution of the form

$\mathbb F_q$3

where $\mathbb F_q$4 is a positive Radon measure supported in $\mathbb F_q$5. Its Fourier transform is again a positive measure,

$\mathbb F_q$6

The dual Delsarte constant is

$\mathbb F_q$7

A central point is that weak and strong duality must be proved in the continuous setting. Weak duality gives

$\mathbb F_q$8

while strong duality sharpens this to

$\mathbb F_q$9

The proof of strong duality proceeds by embedding the primal feasible region into a convex cone in

$\mathbb F_{q^m}$0

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 $\mathbb F_{q^m}$1, and the dual supremum is attained by some $\mathbb F_{q^m}$2.

When $\mathbb F_{q^m}$3 and $\mathbb F_{q^m}$4 are extremal, one obtains tiling-type complementarity relations. The support of $\mathbb F_{q^m}$5 lies in $\mathbb F_{q^m}$6, and the support of $\mathbb F_{q^m}$7 lies in $\mathbb F_{q^m}$8. Consequently,

$\mathbb F_{q^m}$9

and also

$C^\perp$0

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 $C^\perp$1 of positive measure, its essential difference set is

$C^\perp$2

The classical Cohn–Elkies argument gives

$C^\perp$3

The trivial volume bound $C^\perp$4 is recovered by the admissible trial function $C^\perp$5. If $C^\perp$6 is a convex body that does not tile $C^\perp$7 by translations, then

$C^\perp$8

The proof uses extremal data for $C^\perp$9 to obtain a measure $f$0 with

$f$1

so $f$2 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 (Kolountzakis et al., 11 Oct 2025).

3. Subspace Delsarte duality, weight, and defect

Let $f$3, viewed also as an $f$4-space of dimension $f$5. Write $f$6 for the lattice of $f$7-subspaces and $f$8 for the lattice of all $f$9-subspaces. Fix a nondegenerate $\mathbb R^d$0-bilinear form $\mathbb R^d$1, and let $\mathbb R^d$2. For $\mathbb R^d$3, the trace-orthogonal Delsarte dual is

$\mathbb R^d$4

This satisfies

$\mathbb R^d$5

so $\mathbb R^d$6 is an order-reversing involution of $\mathbb R^d$7 (Borello et al., 29 Sep 2025).

The same framework introduces weight and defect relative to a fixed $\mathbb R^d$8 of $\mathbb R^d$9-dimension $\Phi=\delta_0+\nu$0 with $\Phi=\delta_0+\nu$1. For any $\Phi=\delta_0+\nu$2,

$\Phi=\delta_0+\nu$3

For each $\Phi=\delta_0+\nu$4,

$\Phi=\delta_0+\nu$5

The sequence $\Phi=\delta_0+\nu$6 is nondecreasing, with $\Phi=\delta_0+\nu$7 and $\Phi=\delta_0+\nu$8. Its positive jumps determine the sequence of maximum nonzero defects

$\Phi=\delta_0+\nu$9

A second notion, denoted $\mathbb F_q$0, is defined in a quotient model originating from the Lunardon–Polverino “translation ovoid” argument. Every $\mathbb F_q$1 of dimension $\mathbb F_q$2 with $\mathbb F_q$3 can be realized as the projection of an $\mathbb F_q$4-dimensional $\mathbb F_q$5-subgeometry $\mathbb F_q$6 onto $\mathbb F_q$7, where $\mathbb F_q$8 is an $\mathbb F_q$9-space disjoint from $U^{\perp_D}$00. If $U^{\perp_D}$01 is the $U^{\perp_D}$02-orthogonal of $U^{\perp_D}$03, then

$U^{\perp_D}$04

This $U^{\perp_D}$05 is called a Delsarte dual of $U^{\perp_D}$06; provided $U^{\perp_D}$07, one has $U^{\perp_D}$08 and $U^{\perp_D}$09.

The quotient-model dual controls defect data exactly. If

$U^{\perp_D}$10

then

$U^{\perp_D}$11

Thus $U^{\perp_D}$12 has the same defect-sequence length $U^{\perp_D}$13, with positive defects occurring at complementary dimensions. The same paper also gives a bijection $U^{\perp_D}$14 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 (Borello et al., 29 Sep 2025).

4. Delsarte duals of rank-metric codes

For a prime power $U^{\perp_D}$15, let $U^{\perp_D}$16 be endowed with the nondegenerate symmetric bilinear form

$U^{\perp_D}$17

If $U^{\perp_D}$18 is an $U^{\perp_D}$19-linear rank-metric code, its Delsarte dual is

$U^{\perp_D}$20

Nondegeneracy gives

$U^{\perp_D}$21

The associated rank distribution is

$U^{\perp_D}$22

and if $U^{\perp_D}$23, the Delsarte analogue of the Singleton bound is

$U^{\perp_D}$24

The rank defect is

$U^{\perp_D}$25

with $U^{\perp_D}$26 (Cruz et al., 2015).

MacWilliams identities relate the rank distributions of $U^{\perp_D}$27 and $U^{\perp_D}$28. In the form quoted in the paper, for $U^{\perp_D}$29,

$U^{\perp_D}$30

Hence the full rank distribution of $U^{\perp_D}$31 determines that of $U^{\perp_D}$32, and conversely. If $U^{\perp_D}$33 is MRD or, more generally, dually-QMRD, then for each $U^{\perp_D}$34 with $U^{\perp_D}$35,

$U^{\perp_D}$36

so the entire distribution depends only on $U^{\perp_D}$37. In the general positive-defect case, the high-rank frequencies are determined by the parameters together with the “small-rank” numbers. When $U^{\perp_D}$38, $U^{\perp_D}$39, and $U^{\perp_D}$40, one has

$U^{\perp_D}$41

A complementary geometric description uses $U^{\perp_D}$42 rank-metric codes. If $U^{\perp_D}$43 is a generator matrix of a nondegenerate $U^{\perp_D}$44-linear code $U^{\perp_D}$45, and $U^{\perp_D}$46 is the $U^{\perp_D}$47-span of its column vectors in $U^{\perp_D}$48, then

$U^{\perp_D}$49

and the minimum rank distance is

$U^{\perp_D}$50

Moreover, $U^{\perp_D}$51 is, up to $U^{\perp_D}$52-equivalence, the Delsarte dual of $U^{\perp_D}$53, and the defect-sequence rule recovers the Wei-type duality

$U^{\perp_D}$54

Within this framework, MRD, near-MRD, quasi-MRD, and $U^{\perp_D}$55-MRD codes are analyzed via defect sequences, and several of these classes are shown to be closed under duality (Borello et al., 29 Sep 2025).

5. Dual formulations of higher-order Delsarte LPs

For binary codes of block length $U^{\perp_D}$56 and minimum distance $U^{\perp_D}$57, Delsarte’s original level-1 primal LP uses a function $U^{\perp_D}$58 supported on

$U^{\perp_D}$59

normalized by $U^{\perp_D}$60, with $U^{\perp_D}$61, $U^{\perp_D}$62, and $U^{\perp_D}$63. Its dual can be written in terms of a function $U^{\perp_D}$64 as

$U^{\perp_D}$65

subject to

$U^{\perp_D}$66

A feasible $U^{\perp_D}$67 certifies an upper bound on code size. Classically, choosing $U^{\perp_D}$68 in terms of Krawtchouk polynomials recovers the MRRW bound (Coregliano et al., 8 Jan 2025).

The higher-order hierarchy replaces single codewords by matrices $U^{\perp_D}$69 and defines $U^{\perp_D}$70 through the linear span of the rows of $U^{\perp_D}$71. The primal formulation imposes positivity of suitable partial Fourier transforms after applying elements of $U^{\perp_D}$72. The symmetrized dual uses functions

$U^{\perp_D}$73

with objective

$U^{\perp_D}$74

subject to a validity inequality on $U^{\perp_D}$75 and the positivity constraints $U^{\perp_D}$76. When $U^{\perp_D}$77, this reduces to the classical level-1 dual; at level $U^{\perp_D}$78, a feasible dual bounds $U^{\perp_D}$79.

A central structural result is the lifting theorem. If $U^{\perp_D}$80 and $U^{\perp_D}$81 is feasible for the level-$U^{\perp_D}$82 dual with objective value

$U^{\perp_D}$83

then one can explicitly construct a feasible level-$U^{\perp_D}$84 dual $U^{\perp_D}$85 with objective

$U^{\perp_D}$86

The paper interprets this as showing that any level-$U^{\perp_D}$87 dual bound automatically lifts to level $U^{\perp_D}$88 without loss in the natural exponent. It also gives a dual-based completeness proof: if $U^{\perp_D}$89 is closed under taking subspaces and

$U^{\perp_D}$90

then for every $U^{\perp_D}$91 the optimum of the subspace-symmetric dual is exactly

$U^{\perp_D}$92

The same work further states that the spectral construction via the polynomial $U^{\perp_D}$93 and combinatorial “cylinder” arguments is the first from-scratch dual at level $U^{\perp_D}$94, and that it recovers the MRRW rate

$U^{\perp_D}$95

up to lower-order terms for small relative distance $U^{\perp_D}$96 (Coregliano et al., 8 Jan 2025).

6. Applications, closure properties, and recurring distinctions

Delsarte duality has direct consequences beyond coding bounds. In the q-matroid setting, a q-matroid $U^{\perp_D}$97 on $U^{\perp_D}$98 is specified by a rank function $U^{\perp_D}$99 satisfying the axioms $U^d$00 $U^d$01, $U^d$02 monotonicity, and $U^d$03 submodularity. If $U^d$04 is a $U^d$05 matrix over $U^d$06 of rank $U^d$07, then

$U^d$08

defines an $U^d$09-representable q-matroid $U^d$10, depending only on the $U^d$11-system $U^d$12 spanned by the columns, and $U^d$13 is the dual q-matroid $U^d$14. For direct sums of uniform q-matroids, Delsarte duality yields the equivalence

$U^d$15

if and only if

$U^d$16

The same paper also defines the rank-generating function

$U^d$17

and states that when $U^d$18, the weight enumerator $U^d$19 is recovered from $U^d$20 by a simple substitution (Borello et al., 29 Sep 2025).

A recurring source of ambiguity is that the same expression refers to different dual mechanisms. In the subspace theory, $U^d$21 is the trace-orthogonal complement inside $U^d$22, whereas $U^d$23 is a quotient-model dual living in a different ambient dimension. In Euclidean extremal theory, the dual is a tempered distribution $U^d$24, not an orthogonal complement. In higher-order coding LPs, the dual consists of feasible certificates $U^d$25. 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 (Kolountzakis et al., 11 Oct 2025). 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.