---
title: Symmetrized Weight Enumerator
url: https://www.emergentmind.com/topics/symmetrized-weight-enumerator
type: topic
---

# Symmetrized Weight Enumerator

Searching arXiv for recent and foundational papers on symmetrized weight enumerators and MacWilliams identities.
The **symmetrized weight enumerator** is a weight enumerator obtained by grouping alphabet symbols into equivalence classes and then counting codeword coordinates by class rather than by individual symbol. In the most general formulation now available, it is a special case of a weight enumerator attached to an equivalence relation on the alphabet, with the classical Hamming and complete weight enumerators appearing as extreme cases of coarse and fine partitioning [2510.27358]. Over finite abelian groups and finite commutative Frobenius rings, the standard symmetrization identifies symbols related by additive inversion, so that \(a\) and \(-a\) are treated as equivalent; over module alphabets, more general orbit partitions under automorphism groups yield symmetrized weight compositions [2510.27358], [1412.6083]. The subject is closely tied to MacWilliams identities, invariant theory, and the algebraic structure of Frobenius rings, and it has been extended to tuple enumerators, support-sensitive versions, and quantum analogues [2509.20794], [2409.03576].

## 1. Definition and basic constructions

Let \(A=\{a_0,\dots,a_r\}\) be an alphabet with additive identity \(0=a_0\), and let \(C\subseteq A^n\) be a code. The complete weight enumerator is
\[
\operatorname{cwe}_C(x_0,x_1,\dots,x_r) = \sum_{c=(c_1,\dots,c_n)\in C} \prod_{i=1}^n x_{\iota(c_i)},
\]
where \(\iota(a_j)=j\), while the Hamming weight enumerator is
\[
W_C(x,y) = \sum_{c\in C} x^{\,n-wt(c)} y^{\,wt(c)}.
\]
The latter is obtained from the former by collapsing all nonzero symbols:
\[
W_C(x,y)=\operatorname{cwe}_C(x,y,y,\dots,y).
\]
This establishes the basic pattern: a weight enumerator may be refined or coarsened by deciding which alphabet symbols are distinguished and which are identified [2510.27358].

In the equivalence-relation framework, one fixes a partition
\[
\mathcal{P}=\{A_1,\dots,A_s\}
\]
of the alphabet, induced by an equivalence relation \(\equiv\). For a codeword \(c=(c_1,\dots,c_n)\in C\), define
\[
N_i(c)=\bigl|\{\,j\in\{1,\dots,n\}\mid c_j\in A_i\,\}\bigr|,\qquad i=1,\dots,s.
\]
The associated weight enumerator is
\[
EW_C(x_1,\dots,x_s)=\sum_{c\in C}\prod_{i=1}^s x_i^{\,N_i(c)}.
\]
The symmetrized weight enumerator is obtained when the partition is chosen from a natural symmetry of the alphabet, rather than from mere support/non-support distinction [2510.27358].

For a finite abelian group \(G\), the paper "Weight Enumerators From Equivalence Relations and MacWilliams Identities" defines the **symmetric weight enumerator** by the equivalence relation
\[
a\sim_S b \iff a=\pm b.
\]
If \(A_1,\dots,A_s\) are the equivalence classes of \(\sim_S\), then
\[
SW_C(x_1,\dots,x_s)=\sum_{c\in C}\prod_{i=1}^s x_i^{N_i(c)}.
\]
This is the standard symmetrization by additive inversion and matches what is commonly called the symmetrized weight enumerator in the literature, especially over rings such as \(\mathbb{Z}_4\) [2510.27358].

## 2. Algebraic meaning of the symmetrization

The defining equivalence \(a\sim_S b\iff a=\pm b\) identifies symbols lying in the same orbit under the involution \(a\mapsto -a\). In this sense, the symmetrized weight enumerator is the orbit enumerator for the \(\{\pm1\}\)-action on the alphabet. Compared with the complete weight enumerator, it forgets the distinction between \(a\) and \(-a\) while retaining the distinction between inequivalent orbits such as \(\{0\}\), \(\{a,-a\}\), and singleton order-two elements [2510.27358].

This orbit interpretation is part of a broader pattern. Over a finite left \(R\)-module \(A\), with a subgroup \(G\le \operatorname{Aut}_R(A)\), one defines an equivalence relation by
\[
a\sim b \quad \text{if and only if} \quad a=bT \text{ for some } T\in G.
\]
The corresponding **symmetrized weight composition** is
\[
\operatorname{swc}_a(x)=\left|\{i\in\{1,\dots,n\}:x_i\sim a\}\right|,
\]
and the associated symmetrized weight enumerator is
\[
W_C^{\text{swc}}(Z_1,\dots,Z_t)=\sum_{x\in C}\prod_{j=1}^t Z_j^{\operatorname{swc}_{a_j}(x)},
\]
where \(A/G=\{a_1,\dots,a_t\}\) denotes the orbit set [1412.6083]. This formulation makes clear that symmetrization is not inherently tied to inversion; it is a general procedure of passing from symbol-by-symbol data to orbit-by-orbit data.

A particularly important ring-theoretic formulation, developed in "A multiset approach to MacWilliams identities" [2509.20794], uses a finite commutative Frobenius ring \(R\) and the equivalence relation
\[
r\approx s \iff r=us \text{ for some unit } u\in R^\times,
\]
equivalently \(rR=sR\). Thus the equivalence classes are precisely the principal ideals up to unit multiples. If \(\{a_0,\dots,a_t\}\) is a set of representatives of these classes and
\[
S_i(c)=\{\ell\in[n]:c_\ell\approx a_i\},
\qquad
\mathrm{swc}_i(c)=|S_i(c)|,
\]
then
\[
swe_C(x_0,\dots,x_t)=\sum_{c\in C}x_0^{\mathrm{swc}_0(c)}x_1^{\mathrm{swc}_1(c)}\cdots x_t^{\mathrm{swc}_t(c)}.
\]
This version refines the Hamming weight enumerator by separating coordinates according to the principal ideal generated by each entry [2509.20794].

## 3. Canonical examples

The simplest nontrivial example occurs over \(\mathbb{Z}_4\). Under \(a\sim_S b\iff a=\pm b\), the equivalence classes are
\[
A_1=\{0\},\qquad A_2=\{1,3\},\qquad A_3=\{2\}.
\]
Thus the symmetrized weight enumerator has three variables rather than four, and distinguishes the coordinate types \(0\), odd, and even-nonzero [2510.27358]. For the explicit code
\[
C=\{(0,0,0),(0,1,2),(0,2,0),(0,3,2),(2,0,0),(2,1,2),(2,2,0),(2,3,2)\},
\]
the enumerator is
\[
SW_C(x_1,x_2,x_3)=x_1^3+2x_1x_2x_3+2x_1^2x_2+2x_1x_3^3+x_2x_3^2.
\]
The exponent of \(x_1\) counts zeros, the exponent of \(x_2\) counts \(1\) or \(3\), and the exponent of \(x_3\) counts coordinates equal to \(2\) [2510.27358].

Characteristic \(2\) provides the degenerate case. If the alphabet has characteristic \(2\), then \(-a=a\) for all \(a\), so every orbit under \(a\mapsto -a\) is a singleton. In that case,
\[
SW_C(x_1,\dots,x_{r+1})=\operatorname{cwe}_C(x_1,\dots,x_{r+1}),
\]
and no nontrivial symmetrization occurs [2510.27358]. This observation is important for interpreting binary and \(\mathbb{F}_2^m\)-based codes: the symmetrized and complete enumerators coincide.

The same orbit idea appears in more elaborate settings. For \(R=\mathbb{Z}_{2k_1}\times\cdots\times\mathbb{Z}_{2k_g}\), one may define
\[
a\sim b \Longleftrightarrow a=b \ \text{or} \ a=-b,
\]
and form symmetrized weight enumerators of tuples of codes over the factors. In the case \(R=\mathbb{F}_2\times\mathbb{Z}_4\), pairs of Type II codes over \(\mathbb{F}_2\) and \(\mathbb{Z}_4\) give multivariate symmetrized weight enumerators that are invariant under a finite group \(H\) and, in degrees \(8\) and \(16\), span the invariant subspaces of the corresponding invariant ring [2407.19815].

## 4. MacWilliams identities and the condition for their validity

The classical complete and Hamming weight enumerators satisfy MacWilliams identities. In the unified group/ring setting of [2510.27358], one has
\[
\operatorname{cwe}_{C^*}(x_0,\dots,x_r)=\frac{1}{|C|}\,\operatorname{cwe}_C\bigl(M\cdot(x_0,\dots,x_r)\bigr),
\]
and
\[
W_{C^*}(x,y)=\frac{1}{|C|}\,W_C(x+ry,x-y),
\]
where \(M\) is the appropriate character table or generating-character matrix and \(r=|A|-1\) [2510.27358].

For an equivalence-relation-based enumerator \(EW_C\), MacWilliams identities do not hold automatically. The decisive notion is that of an **autodual partition**. If \(\mathcal{P}=\{A_1,\dots,A_s\}\) is the alphabet partition induced by the equivalence relation, and if \(\mathcal{P}\) is autodual, then the coarse transform matrix
\[
\overline{M}_{A_i,A_j}=\sum_{b\in A_j}M_{a,b},\qquad a\in A_i,
\]
is well defined and
\[
EW_{C^*}(x_1,\dots,x_s)=\frac{1}{|C|}\,EW_C\bigl(\overline{M}\cdot(x_1,\dots,x_s)\bigr).
\]
The paper identifies autoduality as the criterion under which a partition-based weight enumerator inherits MacWilliams reciprocity from the complete weight enumerator [2510.27358].

The symmetrized partition \(a\sim_S b\iff a=\pm b\) satisfies this condition over finite abelian groups. Consequently,
\[
SW_{C^M}(x_1,\dots,x_s)=\frac{1}{|C|}\,SW_C\bigl(\overline{M}\cdot(x_1,\dots,x_s)\bigr),
\]
where \(\overline{M}\) is obtained by summing the character-table entries over \(\pm\)-orbits [2510.27358]. This places the symmetrized weight enumerator on the same structural footing as the Hamming and complete enumerators.

A later development provides a different proof strategy. The paper "A multiset approach to MacWilliams identities" interprets the symmetrized weight enumerator over finite commutative Frobenius rings as a sum over multisets and proves the MacWilliams identity combinatorially, without generating characters [2509.20794]. In that setting the transform matrix factors as
\[
\mathbf{S}=\mathbf{QDA}^{-1},
\]
where \(\mathbf{A}\) is the adjacency matrix of the poset of principal ideals under inclusion, \(\mathbf{D}\) is the diagonal matrix of ideal sizes, and \(\mathbf{Q}\) encodes containment in annihilator ideals. The resulting identity is
\[
swe_{C^\perp}(x)=\frac{1}{|C|}\,swe_C(\mathbf{QDA}^{-1}x).
\]
This factorization shows that the transform is governed by the incidence combinatorics of principal ideals rather than only by character theory [2509.20794].

## 5. Variants, refinements, and failures of overspecialization

Several important generalizations of the symmetrized weight enumerator follow the same pattern of grouping alphabet symbols by a structural equivalence.

For a finite commutative Frobenius ring and a unit \(\lambda\) with \(\lambda^2=1\), one may define
\[
a\sim_\lambda b \iff a=\lambda b \text{ or } a=b,
\]
obtaining the \(\lambda\)-weight enumerator. The case \(\lambda=-1\) recovers the symmetric weight equivalence on rings, and the corresponding MacWilliams identity follows from the same partition framework [2510.27358]. For finite chain rings, symbols may instead be grouped by order or ideal layer, yielding the chain ideal weight enumerator
\[
CRW_C(x_0,\dots,x_e)=\sum_{c\in C}\prod_{i=0}^e x_i^{N_i(c)},
\]
which also satisfies a MacWilliams identity with a coarsened transform matrix [2510.27358].

In module-theoretic language, symmetrized weight compositions are central to the **Extension Property**. A finite module alphabet \(A\) over a finite ring \(R\) has the Extension Property with respect to symmetrized weight compositions if every linear isomorphism between codes preserving the symmetrized composition extends to a \(G\)-monomial transformation of the ambient module [1412.6083]. The characterization proved in "On Symmetrized Weight Compositions" is that this holds if and only if \(A\) can be embedded in the character group \(\widehat{R}\), equivalently if \(A\) has a cyclic socle [1412.6083]. This result clarifies when symmetrized weight enumerators behave as robust invariants of code equivalence.

A crucial caveat is that passing from a multivariate symmetrized enumerator to a more compressed two-variable weight enumerator may destroy the MacWilliams property. "Weights with Maximal Symmetry and Failures of the MacWilliams Identities" studies weights \(w\) that are constant on maximal symmetry classes over finite chain rings and matrix rings, and shows that in many cases, including the homogeneous weight, MacWilliams identities for the induced \(w\)-weight enumerators fail [2404.07154]. The paper emphasizes that the orbit or rank partition enumerators still satisfy MacWilliams identities, but specialized two-variable \(w\)-weight enumerators often do not. This suggests that symmetrized enumerators are frequently the correct level of resolution at which duality is preserved.

## 6. Invariant theory, quantum analogues, and broader significance

The symmetrized weight enumerator is also an invariant-theoretic object. In the setting of pairs of Type II codes over \(\mathbb{F}_2\) and \(\mathbb{Z}_4\), symmetrized weight enumerators are invariant under a finite group \(H\), and in degrees \(8\) and \(16\) they form bases of the graded invariant spaces determined by the Molien series [2407.19815]. This places them in the same tradition as Gleason-type theorems for classical self-dual codes.

A different invariant-theoretic viewpoint appears for formally self-dual quantum codes. There, three enumerators are studied: the single weight enumerator \(B(x,y)\), the double weight enumerator \(C(x,y,z,w)\), and the complete weight enumerator \(D(M)\). The complete enumerator is the most refined object, while the double and single enumerators are symmetrized versions obtained by aggregating Pauli error types [2409.03576]. For example, the single enumerator becomes invariant under the transform
\[
B(x,y)=B(x+(q^2-1)y,x-y),
\]
and its invariant ring is generated by two explicit polynomials; the double enumerator lies in a hypersurface generated by five invariants [2409.03576]. Although the terminology differs, these are symmetrized weight enumerators in the same structural sense: they arise by retaining only the orbit data relevant to the symmetry of the problem.

The multiset formulation of [2509.20794] further extends the classical theory by defining symmetrized support enumerators and \(\lambda\)-tuple symmetrized weight enumerators. For \(\lambda\)-tuples, the transform becomes
\[
swe^{[\lambda]}_{C^\perp}(x)=\frac{1}{|C|^\lambda}\,swe^{[\lambda]}_C\big(\mathbf{QD^\lambda A}^{-1}x\big),
\]
which generalizes the \(1\)-tuple MacWilliams identity and shows that the same ideal-poset combinatorics governs more refined joint distributions [2509.20794].

A plausible implication is that the enduring significance of the symmetrized weight enumerator lies in its position between complete and Hamming enumerators. It is coarse enough to reflect genuine alphabet symmetries, yet fine enough to retain the duality information that can be lost under further specialization. The recent equivalence-relation formulation makes this explicit: symmetrized weight enumerators are not ad hoc constructions, but instances of a general mechanism in which partitions of the alphabet determine both the enumerator and the possibility of a MacWilliams identity [2510.27358].

Source: https://www.emergentmind.com/topics/symmetrized-weight-enumerator