Coset Weight Distribution Problem

Updated 2 February 2026

Coset Weight Distribution Problem is the study of enumerating Hamming weight spectra in cosets of linear codes, determining the count of vectors at each weight.
It leverages combinatorial and algebraic methods, with explicit case studies in MDS and Reed–Solomon codes demonstrating structured weight distributions via geometric and binomial frameworks.
Algorithmic approaches like Gröbner representations and dynamic programming for polar codes enable efficient computation of coset spectra, informing error correction and pseudorandomness analyses.

A coset weight distribution problem concerns the enumeration and characterization of the Hamming weight spectra of all cosets of a linear code, providing deep insight into combinatorial, probabilistic, and algebraic structure. For a code $C\subseteq\mathbb{F}_q^n$ , the coset $C+v$ consists of all vectors $c+v$ with $c\in C$ , and its coset weight distribution (or coset weight enumerator) records, for each $w$ , the number $N_w(C+v)$ of vectors of Hamming weight $w$ in the coset. Key questions include the dependency of these spectra on the code and parameter regime, the algorithmic computability of coset distributions, and the relation between the coset spectra and the code's dual or geometric properties.

1. Core Definitions and Problem Formulation

For any $[n, k, d]_q$ linear code $C$ over $\mathbb{F}_q$ , a coset $C+v$ (for $v\in \mathbb{F}_q^n$ ) has:

Coset weight: $W = \min\{ \mathrm{wt}(w): w \in C+v \}$ .
Coset weight enumerator: $W_{C+v}(x) = \sum_{w=0}^n N_w(C+v)x^w$ , with $N_w(C+v) = |\{ z \in C+v : \mathrm{wt}(z) = w \}|$ .
Coset-leader weight distribution (Editor's term): The count $W_j = |\{ \text{cosets } C+v : \min_{w\in C+v} \mathrm{wt}(w) = j \}|$ for each $j$ .

The general Coset Weight Distribution Problem is to determine, for each coset of $C$ , the sequence $\{N_w(C+v)\}_{w=0}^n$ , classify cosets by these sequences, and analyze their aggregate or average behavior relative to combinatorial baselines (e.g., binomial distributions for random codes) or to code parameters.

2. Structural Results for MDS and Small Dual Distance Codes

For Maximum Distance Separable (MDS) codes, explicit characterizations of coset spectra are possible due to their combinatorial extremality.

For an MDS $[n, k, d]_q$ code, the covering radius $R=d-1$ , and every coset has weight at most $R$ . The small-weight coset spectra (for coset weights $W=1,2,3$ ) admit closed-form aggregate (“integral weight spectrum”) formulas in terms of code parameters, binomial coefficients, and simple correction terms (Davydov et al., 2020), with the coset structure tightly controlled by the code's symmetry.
For cosets of weight $1$ (i.e., those containing a minimal-weight nonzero vector of weight $1$), the spectrum is identical across all such cosets and depends only on $(n, d, q)$ and the codeword weight distribution $A_w(C)$ . For larger coset weights, e.g., $W=2$ or $3$, aggregate spectra likewise admit structured combinatorial expressions but can have several distinct spectral types within the same weight class, exhibiting symmetries in the difference of distributions (Davydov et al., 2021, Davydov et al., 2020).
In the particularly rich case of generalized doubly-extended Reed-Solomon codes ( $[q+1, q-3, 5]_q$ ), coset spectra and the symmetries between coset classes are completely classified via projective geometry and incidence combinatorics, with profound implications for higher codimension MDS codes (Davydov et al., 2020).

An important general principle—both for MDS codes and more broadly—is that for every coset $C+v$ , one only needs the number of vectors in $C+v$ of weights $0\leq t\leq d-2$ ; the remainder of the distribution for $w\ge d-1$ is then determined by universal combinatorial translation formulas (Bonneau-type relations) exploiting the standard weight distribution of $C$ (Davydov et al., 2021).

3. Average and Asymptotic Behavior: Binomiality and Dual Distance

A central quantitative theme is the proximity of coset weight distributions to the ambient (unstructured) binomial law in the Boolean cube (for binary codes), especially as measured by $L_\infty$ and $L_1$ (total variation) distances.

For a binary code $Q\subseteq \mathbb{F}_2^n$ , the average $L_\infty$ -distance between the normalized coset weight distribution $m_{Q+u}$ and the binomial law $B_n(w) = 2^{-n} \binom{n}{w}$ decays rapidly as the bilateral minimum distance $d$ of the dual $Q^\perp$ increases (Bazzi, 2014).
Formally, if $d=2t+1$ , $t\geq 1$ and $Q^\perp$ has bilateral minimum distance at least $d$ ,

$\mathbb{E}_{u} \|m_{Q+u} - B_n\|_\infty \leq \min\{ \big(e\ln (n/2t)\big)^t (2t/n)^{t/2}, \sqrt{2}e^{-t/10}\}$

$\mathbb{E}_{u} \|m_{Q+u} - B_n\|_1 \leq \min\{ (2t+1)\big(e\ln (n/2t)\big)^t (2t/n)^{t/2-1}, \sqrt{2}(n+1)e^{-t/10} \}$

This reveals two asymptotic regimes:
- For $d=O(1)$ , decay is polynomial in $n$ , capturing the slow convergence regime.
- For $d=\Theta(n)$ , decay is exponential, and almost all cosets are exponentially binomial.

This theoretical framework is realized in major code families, e.g., extended Hadamard and extended dual BCH codes, where the average coset spectrum is sharply controlled by the dual code's bilateral minimum distance. Thus, it is the dual code's "spread" around $n/2$ (not the code size) that determines how pseudorandom the coset spectrum is (Bazzi, 2014).

4. Algorithmic Approaches for Coset Weight Distribution

Several algorithmic approaches solve the coset weight distribution problem:

For binary codes, the Gröbner representation $(\mathcal{N},\varphi)$ enumerates all coset leaders and, hence, provides the coset-leader-weight distribution $W_j$ for all $j$ . The main algorithm proceeds via an ordered list construction, traversing minimal-weight representatives in weight-compatible order, and, for each coset, constructing the full set of coset leaders as well as the mapping table $\phi$ (Borges-Quintana et al., 2012).
For polar codes, a deterministic dynamic-programming recursion computes the exact weight enumerator of any polar coset $C_n(u^i)$ in $O(n^2)$ time. The reduction of total coset enumeration complexity leverages algebraic automorphisms of decreasing monomial codes and block lower-triangular affine groups, drastically cutting the search space for large $n$ (e.g., from $2^{34}$ to $2^{25.2}$ cosets at $n=128$ ) and enabling tractability for practically relevant codes (Yao et al., 2021).
For MDS cosets, Bonneau-type formulas recursively determine the coset spectrum for all $w\ge d-1$ using only the initial values $\{N_j(C+v) : 0\leq j\leq d-2\}$ and explicit code-parameter dependent correction factors (Davydov et al., 2021). This is especially efficient for small coset weights.

These schemes achieve both exhaustive enumeration and aggregate-statistical analysis of coset spectra, with practical instantiations illustrated in canonical codes such as Hamming, Hadamard, Reed-Solomon, and polar codes.

5. Symmetries, Classification, and Geometric Connections

For MDS codes, cosets of equal weight but distinct weight distributions often exhibit reflection or mirror symmetries in their spectra; for instance, in doubly-extended Reed–Solomon codes, the difference between the $w$ th components of two coset distributions depends only on the difference between their weight-3 components, governed by orbit structure in projective geometry (Davydov et al., 2020).
The combinatorial translation from small-weight coset content to full weight enumerator reduces, for MDS codes, to geometric incidence calculations (e.g., points and planes in projective space), as exemplified by cosets associated with points on the twisted cubic curve in $\mathrm{PG}(3,q)$ (Davydov et al., 2020).
In projective-geometric MDS codes arising from $n$ -arcs, the coset structure and small-weight spectrum relate to bisecant counts and geometric configurations, revealing further algebraic and geometric symmetries (Davydov et al., 2021).
This algebraic–geometric connection extends to the classification of deep holes (maximal-weight cosets), covering properties, and the existence of optimal saturating sets (Davydov et al., 2021).

6. Applications, Scope, and Limitations

Coset weight distribution results underpin analysis of distance properties, pseudorandomness, and error behavior in both classical and modern coding paradigms:

In normed metrics, the tail probabilities and error rates in bounded-distance and maximum-likelihood decoding are directly governed by the coset-leader-weight distributions (Davydov et al., 2020).
Explicit distribution formulas for cosets of small weights provide tight control over error floors and reliability metrics in both random and geometric code families.
For codes with small dual bilateral minimum distance, the coset spectrum may deviate significantly from binomial, providing worst-case scenarios for cryptography and pseudorandomness applications.
Existential limitations arise for non-MDS codes and higher coset weights, where the simple closed-form or two-term correction formulas break down, necessitating more involved combinatorial or computational approaches (Davydov et al., 2020).

For polar codes and decreasing monomial codes, symmetry reduction and automorphism group actions yield polynomial and often feasible exponential-time algorithms, with direct computational validation for $n$ in excess of $100$ (Yao et al., 2021).

The coset weight distribution problem remains central to the combinatorial theory of codes, providing fundamental connections between algebraic invariants (such as dual distance), probabilistic limits (via binomial approximations), computational methods (via Gröbner and recursive algorithms), and geometric or group-theoretic symmetries. Current research continues to extend analytical tractability to broader code classes and higher parameter regimes.