Harmonic Higher Weight Enumerators

• Harmonic higher weight enumerators are harmonic generalizations of traditional higher weight enumerators that refine subcode support distributions using discrete harmonic functions.
• They preserve compatibility with MacWilliams-type and Greene-type transforms, linking refined support analysis with design-theoretic criteria in coding theory.
• This framework provides a unified approach that parallels extended and m-tuple weight enumerators, connecting discrete harmonic analysis, invariant theory, and coding invariants.

Harmonic higher weight enumerators are harmonic generalizations of higher weight enumerators for linear codes over finite fields. They replace the unweighted counting of supports of $r$-dimensional subcodes by counts weighted with discrete harmonic functions on the coordinate set. In this form, they refine the distribution of subcode supports while remaining compatible with MacWilliams-type transforms, Greene-type evaluations through harmonic Tutte polynomials, and design-theoretic criteria for subcode supports (Britz et al., 2024, Chakraborty et al., 2021).

1. Discrete harmonic framework

The harmonic input is a discrete harmonic function on subsets of the coordinate set $E=\{1,2,\ldots,n\}$. For $d$ between $0$ and $n$, with $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$, the space of discrete harmonic functions of degree $d$ is

$\operatorname{Harm}_d(n)=\ker(\gamma),$

where $$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$ is the discrete differentiation operator. This is the discrete harmonic formalism used in the coding-theoretic developments of Delsarte and Bachoc, and later in harmonic Tutte theory (Chakraborty et al., 2021, Britz et al., 2022).

For a linear code $C$ of length $E=\{1,2,\ldots,n\}$0 over $E=\{1,2,\ldots,n\}$1, the harmonic $E=\{1,2,\ldots,n\}$2-th higher weight enumerator associated to $E=\{1,2,\ldots,n\}$3 is defined by

$E=\{1,2,\ldots,n\}$4

where

$E=\{1,2,\ldots,n\}$5

Here $E=\{1,2,\ldots,n\}$6 is the set of all $E=\{1,2,\ldots,n\}$7-dimensional subcodes of $E=\{1,2,\ldots,n\}$8, and $E=\{1,2,\ldots,n\}$9 is the union of supports of all codewords in $d$0 (Britz et al., 2024).

The special case in which $d$1 is constant recovers the usual $d$2-th higher weight enumerator. For degree $d$3, the harmonic weighting produces a refined invariant that detects symmetries and nonuniformities in the support structure that are invisible to the unweighted higher support distribution (Britz et al., 2024).

A related formulation appears in the study of higher Jacobi polynomials. There, the harmonic function is extended from $d$4-subsets to arbitrary subsets by

$d$5

and the harmonic higher weight enumerator is written using $d$6 rather than $d$7 directly (Chakraborty et al., 16 Aug 2025). This is a change of presentation rather than a change of subject: the common structure is the harmonic weighting of subcode supports.

2. Relation to classical higher support weights and harmonic weight enumerators

Classically, the $d$8-th support weight polynomial of a code is

$d$9

where $0$0 is the number of $0$1-dimensional subcodes of weight $0$2. Britz’s theorem expresses this polynomial in terms of the Tutte polynomial of the associated matroid, and the generalized Hamming weights $0$3 arise from the minimal support sizes among $0$4-dimensional subcodes (Lalitha et al., 2015). Harmonic higher weight enumerators retain the same subcode-support perspective, but replace the raw counts $0$5 by harmonic sums over supports (Britz et al., 2024).

They also extend the earlier harmonic weight enumerators attached to codewords rather than subcodes. For a linear code $0$6 and $0$7, the harmonic weight enumerator is

$0$8

where $0$9 means $n$0 (Chakraborty et al., 2021, Miezaki et al., 2023). In this sense, harmonic higher weight enumerators belong to the same family of weighted generating functions, but the objects being counted are subcodes rather than individual codewords.

This distinction matters conceptually. Ordinary harmonic weight enumerators probe the geometry of codeword shells, whereas harmonic higher weight enumerators probe the geometry of the support poset of the Grassmannian of subcodes. A plausible implication is that higher harmonic enumerators can detect design-theoretic regularity at the level of subcode supports even when the single-codeword shells do not exhibit comparable regularity.

3. Correspondence with harmonic extended weight enumerators

The 2024 theory develops harmonic higher and harmonic extended weight enumerators simultaneously. For a positive integer $n$1, if $n$2 denotes the $n$3-extension of $n$4, the harmonic $n$5-extended weight enumerator is

$n$6

where

$n$7

Thus the higher and extended theories are parallel harmonic refinements of two classical enumerative constructions (Britz et al., 2024).

A central structural result is that these two families determine one another. Theorem 3.5 gives an explicit expression of the harmonic extended enumerator in terms of the harmonic higher enumerators: $n$8 where $n$9 is the Gaussian binomial coefficient (Britz et al., 2024). Theorem 3.6 supplies the reverse relation, expressing the higher harmonic invariants in terms of the extended ones (Britz et al., 2024).

These correspondences place harmonic higher weight enumerators inside a larger interpolation theory: changing the extension degree $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$0 and changing the subcode dimension $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$1 produce equivalent packages of information. This suggests that the higher and extended harmonic invariants are best viewed as two coordinate systems on the same underlying support-distribution data.

4. MacWilliams-type identities and Greene-type interpretations

The harmonic higher theory retains the characteristic factorization familiar from harmonic weight enumerators. For an $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$2 code $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$3 over $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$4 and $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$5,

$\mathcal{E}_d=\{X\subset E\mid |X|=d\}$6

where $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$7 is a homogeneous polynomial of degree $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$8 (Britz et al., 2024). The same paper gives a MacWilliams-type identity relating these $\mathcal{E}_d=\{X\subset E\mid |X|=d\}$9-polynomials to corresponding polynomials for the dual code $d$0 (Britz et al., 2024).

For harmonic extended weight enumerators, the duality statement is explicit: $d$1 with

$d$2

This is the direct harmonic analogue of the classical MacWilliams transform in the extended setting (Britz et al., 2024).

The Greene-type side of the theory uses harmonic Tutte polynomials. For a matroid $d$3 with rank function $d$4 and harmonic $d$5, the harmonic Tutte polynomial is

$d$6

In the ordinary harmonic case, the generalized Greene theorem expresses the harmonic $d$7-polynomial as

$d$8

where $d$9 is the matroid associated with the code (Chakraborty et al., 2021). The 2024 paper extends this Greene-type correspondence to the higher and extended harmonic enumerators (Britz et al., 2024).

This places harmonic higher weight enumerators in the same matroidal framework as classical higher support weights. The unweighted Tutte polynomial controls ordinary higher support distributions; the harmonic Tutte polynomial controls their harmonic refinements.

5. Design-theoretic role

One of the explicit applications of harmonic higher weight enumerators is a new proof of the Assmus-Mattson Theorem for subcode supports of linear codes over finite fields (Britz et al., 2024). In this use, the harmonic higher invariants serve as the vehicle through which vanishing conditions are turned into $\operatorname{Harm}_d(n)=\ker(\gamma),$0-design conclusions.

The later work on higher Jacobi polynomials sharpens this perspective. It states that harmonic higher weight enumerators encode the distribution of supports of all $\operatorname{Harm}_d(n)=\ker(\gamma),$1-dimensional subcodes, and that by vanishing and symmetry properties, as in Delsarte’s theorem, they characterize when the subcode supports form $\operatorname{Harm}_d(n)=\ker(\gamma),$2-designs (Chakraborty et al., 16 Aug 2025). The same paper also states that higher Jacobi polynomials for linear codes whose subcode supports form $\operatorname{Harm}_d(n)=\ker(\gamma),$3-designs can be uniquely determined from the higher weight enumerators of the codes via polarization technique, and that higher Jacobi polynomials can be computed from harmonic higher weight enumerators with the help of Hahn polynomials (Chakraborty et al., 16 Aug 2025).

The broader harmonic-enumerator literature shows how such criteria operate in concrete families. For ordinary harmonic weight enumerators and Jacobi polynomials, explicit formulas for first-order Reed–Muller codes and extended Hamming codes lead to the corollary that no shell of $\operatorname{Harm}_d(n)=\ker(\gamma),$4 or $\operatorname{Harm}_d(n)=\ker(\gamma),$5 supports a combinatorial $\operatorname{Harm}_d(n)=\ker(\gamma),$6-design for any weight (Miezaki et al., 2023). This is a codeword-shell statement rather than a higher-support statement, but it clarifies the general mechanism: harmonic coefficients detect the failure or presence of design uniformity.

6. Parallel generalizations and algebraic structure

Harmonic higher weight enumerators are part of a wider harmonic extension of coding-theoretic enumerators. A parallel development introduces harmonic $\operatorname{Harm}_d(n)=\ker(\gamma),$7-tuple weight enumerators for codes over finite Frobenius rings: $\operatorname{Harm}_d(n)=\ker(\gamma),$8 where $\operatorname{Harm}_d(n)=\ker(\gamma),$9 counts $$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$0-tuples of codewords whose joint support is $$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$1 (Britz et al., 2022). That work proves a harmonic MacWilliams-type identity, defines harmonic Tutte and harmonic coboundary polynomials for demi-matroids, and proves a Greene-type identity connecting these polynomials to the harmonic $$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$2-tuple weight enumerators (Britz et al., 2022). The ring-theoretic and demi-matroidal setting is distinct from the finite-field higher-support setting, but the conceptual pattern is the same: harmonic weighting, Tutte-type control, and duality.

The algebraic background of the subject comes from the structure theory of discrete harmonic polynomials on Hamming space. Using finite-dimensional representation theory of $$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$3, the space of degree-$$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$4 discrete polynomials admits the decomposition

$$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$5

with

$$\gamma:\mathbb{R}^{\mathcal{E}_d}\to \mathbb{R}^{\mathcal{E}_{d-1}}$$6

The same framework yields a generalized MacWilliams identity for harmonic weight enumerators of binary linear codes (Elkies et al., 2011). This structural theory explains why harmonic enumerators naturally interact with duality, invariant theory, and design criteria.

In that broader landscape, harmonic higher weight enumerators occupy the subcode-support corner of a unified program linking discrete harmonics, coding invariants, matroid or demi-matroid polynomials, and design theory. The available results indicate that they are not merely weighted variants of classical higher weight enumerators, but a framework in which support distributions, duality, and design regularity can be studied simultaneously (Britz et al., 2024, Chakraborty et al., 16 Aug 2025).