Jacobi Analogue of MacWilliams Identity

• The Jacobi analogue of MacWilliams identity refines classical duality by incorporating additional variables to expose enhanced combinatorial and geometric data in codes and lattices.
• It employs finite Fourier transforms or character-sum substitutions to generalize weight enumerators into multidimensional Jacobi polynomials and split variants.
• The framework bridges coding and lattice theories, connecting design theory, invariant analysis, and higher-genus and higher-weight extensions through refined enumeration techniques.

The Jacobi analogue of the MacWilliams identity denotes a family of duality transformations that refine the classical MacWilliams identity for linear codes by introducing additional variables attached to reference coordinates, reference vectors, coordinate blocks, genus, subcodes, extension fields, or lattice data. In the code-theoretic setting, these identities replace the two-variable Hamming weight enumerator by Jacobi polynomials or split Jacobi polynomials and show that passage from a code $C$ to its dual $C^\perp$ is governed by a finite Fourier transform or character-sum substitution. In the lattice-theoretic setting, the same theme appears through analogies with the Jacobi–Poisson summation formula and, for Construction A lattices, through the $\nu$-function identity that answers a conjecture of P. Solé from 1995 (Chakraborty et al., 2021, Chakraborty et al., 2022, Chakraborty et al., 2023, Chakraborty et al., 2024, Chakraborty et al., 16 Aug 2025, Zheng et al., 2024, Chakraborty et al., 2023, Chakraborty et al., 2021).

1. Classical MacWilliams identity as the prototype

For an $[n,k]$ linear code $C\subseteq \mathbb{F}_q^n$, with Hamming weight $w(x)=|\{i:x_i\neq 0\}|$, the homogeneous weight enumerator is

$$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$

If $C^\perp$ denotes the dual code, then the MacWilliams identity reads

$$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$

or equivalently

$W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$

In coefficient form this becomes

$C^\perp$0

where $C^\perp$1 is the Krawtchouk polynomial of degree $C^\perp$2 in $C^\perp$3 (Zheng et al., 2024, Chakraborty et al., 16 Aug 2025).

Modern treatments in the supplied sources describe this identity as a Fourier-transform statement. One formulation says that $C^\perp$4 is an “eigenfunction” up to a simple factor of the finite Fourier (Hadamard) transform on $C^\perp$5 (Zheng et al., 2024). Another formulation writes the transform kernel multiplicatively in coordinates by

$C^\perp$6

and extends this principle to Jacobi-type refinements (Chakraborty et al., 16 Aug 2025).

A recurrent point in later work is that the Jacobi analogue does not replace the classical identity; it refines it. Several papers state explicitly that the ordinary MacWilliams identity is recovered as a special case by collapsing Jacobi variables, taking $C^\perp$7, taking $C^\perp$8, or setting $C^\perp$9 and using one block (Chakraborty et al., 2022, Chakraborty et al., 2023, Chakraborty et al., 2024, Chakraborty et al., 2021).

2. Jacobi polynomials for codes and the basic duality transform

One genus-$\nu$0 formulation fixes a subset $\nu$1 and, for each $\nu$2, decomposes the support into the reference part $\nu$3 and its complement $\nu$4. Writing

$\nu$5

$\nu$6

the Jacobi polynomial attached to $\nu$7 is

$\nu$8

Equivalently,

$\nu$9

where

$[n,k]$0

The extended-field version replaces $[n,k]$1 by $[n,k]$2 and defines $[n,k]$3 analogously (Chakraborty et al., 16 Aug 2025).

A different but equivalent line of development fixes one or several reference vectors. For $[n,k]$4 reference vectors $[n,k]$5, with $[n,k]$6 and $[n,k]$7 for $[n,k]$8, one defines

$[n,k]$9

for $C\subseteq \mathbb{F}_q^n$0, and then

$C\subseteq \mathbb{F}_q^n$1

When $C\subseteq \mathbb{F}_q^n$2 this reduces to $C\subseteq \mathbb{F}_q^n$3, and when $C\subseteq \mathbb{F}_q^n$4 one recovers the one-reference-vector Jacobi polynomial (Chakraborty et al., 2022).

The corresponding MacWilliams-type transforms are finite-Fourier substitutions. In the one-subset genus-$C\subseteq \mathbb{F}_q^n$5 formalism, the extended-field identity is

$C\subseteq \mathbb{F}_q^n$6

In the $C\subseteq \mathbb{F}_q^n$7-reference-vector formalism, if $C\subseteq \mathbb{F}_q^n$8 is a fixed nontrivial additive character and

$C\subseteq \mathbb{F}_q^n$9

then

$w(x)=|\{i:x_i\neq 0\}|$0

The sources explicitly present this as the Jacobi–MacWilliams theorem and as a coordinate-wise tensor-power Fourier transform (Chakraborty et al., 16 Aug 2025, Chakraborty et al., 2022).

3. Split, complete, and genus-$w(x)=|\{i:x_i\neq 0\}|$1 Jacobi analogues

The literature extends the Jacobi analogue in several orthogonal directions. One is the split complete Jacobi polynomial attached to pairwise disjoint blocks $w(x)=|\{i:x_i\neq 0\}|$2 with $w(x)=|\{i:x_i\neq 0\}|$3, together with distinguished subsets $w(x)=|\{i:x_i\neq 0\}|$4. Using variables $w(x)=|\{i:x_i\neq 0\}|$5 and $w(x)=|\{i:x_i\neq 0\}|$6, the split complete Jacobi polynomial is

$w(x)=|\{i:x_i\neq 0\}|$7

Its MacWilliams identity is

$w(x)=|\{i:x_i\neq 0\}|$8

where $w(x)=|\{i:x_i\neq 0\}|$9 is a fixed nontrivial additive character (Chakraborty et al., 2023).

Another direction is higher genus. For a binary linear code $$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$0 and $$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$1, the genus-$$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$2 weight enumerator is

$$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$3

If $$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$4, the genus-$$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$5 Jacobi polynomial is

$$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$6

In the split genus-$$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$7 setting with blocks $$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$8 and $$W_C(z_1,z_2)=\sum_{x\in C} z_1^{\,n-w(x)}z_2^{\,w(x)}.$$9, the formal identity is

$C^\perp$0

where

$C^\perp$1

The transformation is therefore blockwise by the Fourier matrix $C^\perp$2 with entries $C^\perp$3 (Chakraborty et al., 2024).

A related genus-$C^\perp$4 formulation over $C^\perp$5 or $C^\perp$6 expresses the homogeneous Jacobi polynomial $C^\perp$7 as a $C^\perp$8-fold Fourier transform: $C^\perp$9 where

$$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$0

The same source gives explicit genus-$$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$1 and genus-$$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$2 specializations (Chakraborty et al., 2021).

4. Higher and extended Jacobi polynomials

The higher-weight framework replaces individual codewords by $$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$3-dimensional subcodes. For an $$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$4 linear code $$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$5,

$$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$6

$$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$7

and the $$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$8th higher weight enumerator is

$$W_{C^\perp}(z_1,z_2)=\frac{1}{|C|}\,W_C(z_1+(q-1)z_2,\;z_1-z_2),$$9

The associated higher Jacobi polynomial is

$W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$0

with

$W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$1

The same paper defines extended Jacobi polynomials by passing to $W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$2 and counting vectors in the extension code (Chakraborty et al., 16 Aug 2025).

The higher-genus Jacobi analogue of MacWilliams is stated explicitly as Theorem 3.4: $W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$3 The extended-field version is

$W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$4

The same source emphasizes that in each case the “kernel” is the same shift in the four variables $W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$5, multiplied by explicit $W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$6-powers and Gaussian-binomial factors in the higher-genus case (Chakraborty et al., 16 Aug 2025).

The proof strategy proceeds through subcode-enumeration sums

$W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$7

their block-summed versions $W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$8, an interpolation formula for $W_{C^\perp}(x,y)=q^{-k}\,W_C(x+(q-1)y,\;x-y).$9, and then application of the classical or extension-field MacWilliams identity together with Möbius inversion. This places the higher Jacobi analogue within the same transform calculus as ordinary weight enumerators, but at the level of subcode support distributions (Chakraborty et al., 16 Aug 2025).

5. Design theory, polarization, and invariant-theoretic structure

Several papers relate Jacobi polynomials to design-theoretic regularity conditions. One source states that if a code is $C^\perp$00-homogeneous, that is, the codewords of the code for every given weight hold a $C^\perp$01-design, then its Jacobi polynomial in genus $C^\perp$02 with composition $C^\perp$03 and $C^\perp$04 can be obtained from its weight enumerator in genus $C^\perp$05 using the polarization operator (Chakraborty et al., 2024). In the genus-$C^\perp$06 binary setting, the operator is

$C^\perp$07

Repeated application “moves” coordinates from $C^\perp$08 into $C^\perp$09 and recovers $C^\perp$10 from $C^\perp$11 (Chakraborty et al., 2024).

In the higher-weight setting, the polarization operator is

$C^\perp$12

If the family of $C^\perp$13-dimensional subcode-supports of weight $C^\perp$14 forms a $C^\perp$15-design, then

$C^\perp$16

and more generally, if all subcode-supports of dimension $C^\perp$17 and weight $C^\perp$18 form a $C^\perp$19-design, then for any $C^\perp$20 of size $C^\perp$21,

$C^\perp$22

The same paper states that under the standard Assmus–Mattson-type hypotheses this recovers every Jacobi polynomial from the single higher weight enumerator (Chakraborty et al., 16 Aug 2025).

The split complete Jacobi formalism also gives an exact characterization of generalized $C^\perp$23-colored $C^\perp$24-designs. For split-composition data $C^\perp$25, the corresponding incidence blocks form a generalized $C^\perp$26-colored $C^\perp$27-design if and only if

$C^\perp$28

is independent of which choice of subsets $C^\perp$29 of sizes $C^\perp$30 one makes (Chakraborty et al., 2023).

Invariant-theoretic consequences also appear. One paper states that the generators of the invariant ring appearing for $C^\perp$31 are obtained in the binary case, and that the homogeneous Jacobi polynomials of binary codes in genus $C^\perp$32 are tied to this invariant ring through the polarization formalism (Chakraborty et al., 2024). Another source states that for Type III and Type IV codes the relation between Jacobi polynomials and designs can be interpreted through explicit small examples (Chakraborty et al., 2022). A plausible implication is that the Jacobi analogue serves simultaneously as a duality formula and as a mechanism for extracting design data from refined weight distributions.

6. Lattice analogue: Jacobi–Poisson summation, the $C^\perp$33-function, and Solé’s conjecture

The code–lattice analogy is made explicit in the comparison between the finite-field MacWilliams identity and the Jacobi–Poisson theta-function formula. For a full-rank lattice $C^\perp$34, the theta-series

$C^\perp$35

satisfies

$C^\perp$36

by Poisson summation. The source states that, for purely imaginary $C^\perp$37, the Gaussian measure on $C^\perp$38 is “self-reciprocal” under Fourier transform—exactly as the MacWilliams weight-enumerator is under the finite transform on $C^\perp$39 (Zheng et al., 2024).

The 2024 paper introduces the lattice $C^\perp$40-function

$C^\perp$41

described as counting lattice points in $C^\perp$42-balls rather than Euclidean spheres. In P. Solé’s 1995 paper “Counting lattice points in pyramids” it was observed that although no direct Poisson formula is known for $C^\perp$43, when $C^\perp$44 arises from a binary code by Construction A,

$C^\perp$45

one expects a perfect analogue of MacWilliams (Zheng et al., 2024).

That conjecture is answered positively. If $C^\perp$46 is any binary code, $C^\perp$47, and $C^\perp$48 satisfy

$C^\perp$49

then Theorem 3 of the paper states

$C^\perp$50

The paper further states that this identity exactly parallels

$C^\perp$51

once one observes that the weight enumerator is the finite-field “$C^\perp$52-series” for $C^\perp$53 and that Construction A intertwines dual codes with dual lattices (Zheng et al., 2024).

The proof is organized through six lemmas: Poisson summation on $C^\perp$54, the identity $C^\perp$55, the fact $C^\perp$56, the relation

$C^\perp$57

the volume/count correspondence

$C^\perp$58

and an elementary hyperbolic identity matching the normalizations. The source also states the scope precisely: $C^\perp$59 must be a binary linear code of length $C^\perp$60; no further integrality or evenness of $C^\perp$61 is required beyond its coming from Construction A; and self-duality of $C^\perp$62 yields a fixed-point property (Zheng et al., 2024).

A distinct but related lattice-theoretic formulation appears for $C^\perp$63-codes over finite Frobenius rings. There the Jacobi weight enumerator $C^\perp$64 is defined by substituting theta functions into a complete weight enumerator, and the main theorem gives

$C^\perp$65

which the source identifies as the Jacobi analogue of MacWilliams identity in a modular-inversion form (Chakraborty et al., 2023).

7. Scope, variants, and common points of interpretation

The supplied literature shows that “Jacobi analogue of MacWilliams identity” is not a single formula but a class of closely related Fourier-duality statements. The variants include Jacobi polynomials with one or several reference vectors (Chakraborty et al., 2022), split complete Jacobi polynomials attached to coordinate blocks (Chakraborty et al., 2023), genus-$C^\perp$66 Jacobi polynomials and split genus-$C^\perp$67 analogues (Chakraborty et al., 2021, Chakraborty et al., 2024), higher and extended Jacobi polynomials associated with subcodes and extension fields (Chakraborty et al., 16 Aug 2025), complete joint Jacobi polynomials for pairs or $C^\perp$68-tuples of codes over $C^\perp$69 and $C^\perp$70 (Chakraborty et al., 2021), equivariant forms for $C^\perp$71-codes over finite Frobenius rings (Chakraborty et al., 2023), and lattice $C^\perp$72-function analogues obtained through Construction A (Zheng et al., 2024).

A common misconception would be to identify the Jacobi analogue solely with the genus-$C^\perp$73 four-variable polynomial $C^\perp$74. The sources do not support such a restriction. They present Jacobi analogues in multiple reference-vector, split, higher-weight, higher-genus, averaged, joint, and lattice settings (Chakraborty et al., 2022, Chakraborty et al., 2023, Chakraborty et al., 16 Aug 2025, Zheng et al., 2024, Chakraborty et al., 2021). Another misconception would be to regard the Jacobi analogue as unrelated to the classical MacWilliams identity. On the contrary, the sources repeatedly state that the ordinary MacWilliams identity is recovered by specialization, collapsing variables, taking $C^\perp$75, or reducing genus (Chakraborty et al., 2023, Chakraborty et al., 2024, Chakraborty et al., 2021).

What remains uniform across these variants is the mechanism: duality is implemented by a character-theoretic or Fourier-theoretic transform, often coordinatewise, sometimes blockwise, and in the lattice setting through Poisson summation. This suggests a single structural principle behind the code and lattice theories: the Jacobi analogue refines enumerators so that additional geometric or combinatorial data are visible, while preserving the duality transform that is characteristic of MacWilliams-type theorems.