Minimum Weight Codewords in Coding Theory

Updated 17 November 2025

Minimum weight codewords are the nonzero codewords with the smallest Hamming weight, defining a code's error-detection and correction limits.
They often exhibit deep combinatorial, geometric, or algebraic structures, enabling explicit construction and accurate enumeration in diverse code families.
Their study drives advances in coding theory by informing robust code design and revealing computational complexities in NP-hard cases.

Minimum weight codewords—codewords achieving the minimum nonzero Hamming weight in a linear code—play a central role in coding theory as they directly determine the code’s minimum distance and hence its error-detection and correction capabilities. In both classical and modern code families—such as permutation codes, cyclic codes, Reed–Muller varieties, geometric codes, and polar or BCH codes—minimum-weight codewords often admit deep combinatorial, geometric, or algebraic characterizations, with enumeration and explicit construction at the forefront of current research.

1. Minimum Weight Codewords: Definitions and General Principles

Let $\mathcal{C}$ be a linear code of length $n$ over a finite field $\mathbb{F}_q$ . The minimum (Hamming) weight $d_{\min}$ is defined as

$d_{\min} = \min\{\, w(\mathbf{c}) : \mathbf{c}\in\mathcal{C},\; \mathbf{c}\neq 0 \},$

where $w(\mathbf{c})$ denotes the number of nonzero components in $\mathbf{c}$ . A minimum-weight codeword is any $\mathbf{c}\in\mathcal{C}$ with $w(\mathbf{c})=d_{\min}$ . The collection and enumeration of such codewords, along with understanding their support and algebraic structure, are crucial for assessing code performance under both random and worst-case error events.

Minimum-weight codewords are also geometrically or algebraically significant—forming links to blocking sets, generalized polygons, or algebraic subvarieties in geometric codes, and to specific orbits or group actions in group-theoretic codes (e.g., permutation or monomial codes). In several code classes, all nonzero codewords of minimum weight are equivalent under the action of the automorphism group of the code.

2. Classical Geometric Codes: Minimum Weight Structure

Many projective-geometric codes, including those arising from projective spaces, planes, Grassmannians, and generalized polygons, enjoy precise characterizations of their minimum-weight words.

Projective Geometry Codes: For the code $C_{n-1}(n,q)$ arising from point–hyperplane incidences in $\mathrm{PG}(n, q)$ , the minimum weight is $\theta_{n-1}=1+q+\cdots+q^{n-1}$ , realized exactly by the incidence vector of a single hyperplane. The only codewords of weight $2q^{n-1}$ are (up to scalar) the differences of incidence vectors of two hyperplanes. Extended results show that for $q>17$ , all codewords of substantially higher, but still “small,” weight must also be supported on the union of hyperplanes through a fixed $(n-3)$ -space (Adriaensen et al., 2019).
Codes from Generalized Polygons: In codes defined from incidence matrices of points and lines in generalized $2m$-gons of order $(s,t)$ , the minimum weight is $s+1$ , and all minimum-weight codewords are (up to scalar) incidence vectors of so-called “distance-traces” (generalizing lines in planes): $T_{d,x,y}=P_d(x)\cap P_{2m-d}(y)$ . For thick generalized polygons, this characterization is uniform and holds in all characteristics under mild restrictions (Petit et al., 10 Nov 2025).
Block Designs and Duals: For the duals of codes from projective planes of order $p^2$ , with $p\geq 5$ , every nonzero codeword has weight at least $2p^2-2p+5$ . Equality is only potentially achieved by differences of incidence vectors associated with disjoint embedded subplanes or antipodal planes, but for Desarguesian planes these configurations do not occur, yielding a strict lower bound for the minimum dual codeword weight (Boeck et al., 2022).
Intersecting Line Codes: In $\mathrm{PG}(3,q)$ the minimum weight of the code of intersecting lines is $q^3+q^2+q+1$ (even $q$ ), achieved by the symplectic polar space lines; for odd $q$ , the minimum is $q^3+2q^2+q+1$ , with supports corresponding to all lines meeting a fixed line (Adriaensen et al., 12 Mar 2024).
Geometric AG Codes (e.g. Giulietti–Korchmáros): For one-point AG-codes associated with maximal curves, minimum-weight codewords of the dual are supported on geometrically special configurations (collinear subsets of the curve, conics, or cubics), with exact counts derived using orbit decompositions and affine variety techniques (Bartoli et al., 2018).

3. Algebraic and Group-Theoretic Codes

Permutation Codes

For subgroup permutation codes under the Chebyshev (ℓ∞) metric, the minimum-weight problem is NP-complete, and inapproximable within a factor of $7/6-\epsilon$ unless $P=NP$ . Explicit construction is possible only when additional structure is imposed. For codes with particular block or generator structures, minimum-weight elements can be realized using explicit gadget designs, while the general case remains computationally intractable (Shieh et al., 2010).

Cyclic Codes

The minimum nonzero weight in a cyclic code $C_{t_1,\dots,t_s}\subset \mathbb{F}_p^{n=q-1}$ with generator $g(x)$ is $d=2$ if and only if a certain GCD condition holds, and then the number of weight-2 codewords is given by

$A_2 = \frac{(p-1)(q-1)(D-1)}{2},$

with $D$ as a function of the exponents $t_i$ . For binary two-zero codes and weight 3, there are further criteria utilizing the auxiliary polynomial $U_t(x)$ and recursive enumerative techniques connecting weight distributions to solution counts of diagonal systems (Coelho et al., 25 Jul 2024).

4. Algebraically Defined Codes: Reed–Muller, Schubert, Polar, BCH and Cartesian Codes

Projective Reed–Muller (PRM) Codes: The minimum weight in $\mathrm{PRM}_q(d,n)$ is $(q-s)q^{n-t-1}$ , where $d-1 = t(q-1)+s$ with $0\leq s < q-1$ . Minimum-weight codewords correspond to projective evaluations of products of $t+1$ independent linear forms (i.e., $t+1$ hyperplanes) in prescribed position, refined by a splitting in the last hyperplane. Their count is expressible via the Gaussian binomial coefficient and a parameter $N_s$ distinguishing whether the last term splits (Ghorpade et al., 2023).
Cartesian and Reed–Muller Codes: For affine Cartesian codes $AC_q(u,A)$ , with $A_i$ drawn from nested subfields, every minimum-weight codeword can be written (up to affine transformations) as a product of $(d_{j+1}-\ell)$ linear factors and specific normal forms in earlier coordinates, with $d_{j+1}$ the size of the next coordinate set and $\ell$ determined as in the decomposition $u=\sum_{i=1}^j(d_i-1)+\ell$ . The enumeration formula recovers as special cases the classical results for Reed–Solomon and Reed–Muller codes (Dang et al., 30 Apr 2025).
Schubert Codes: The minimal codewords are exactly the "Schubert-decomposable elements": decomposable tensors in an appropriate exterior power whose annihilators meet a prescribed flag in precise codimension. The number of such codewords, up to scalars, is given by a nested $q$ -binomial formula dictated by the geometry of the corresponding Schubert variety (Ghorpade et al., 2016).
Polar Codes and Variants: For classical polar codes of length $N=2^m$ and information set $I$ of maximal monomial degree $r$ , the minimum-weight is $d_{\min}=2^{m-r}$ , with orbits under the lower-triangular affine (LTA) group providing explicit formulae for the count. For pre-transformed polar codes (PTPCs), efficient enumeration exploits tree-intersection structures, yielding algorithms with cost $O(n A_{d_{\min}})$ and permitting per-code optimization (Zunker et al., 2023, Cheng et al., 2019). Higher-weight codewords (e.g., $1.5d$) are now tractable via explicit Minkowski-sum constructions, with generalized orbit-collision corrections (Rowshan et al., 2023).
Extended Binary BCH Codes: For $n=2^m$ and designed distance $d(m,s,i)$ , minimum-weight codewords have supports of the form $X+V$ , with $|X|=2^{2i-1}-2^{i-1}$ and $V$ a subspace of dimension $m-2i-s$ . Explicit deterministic and probabilistic polynomial-time algorithms exist for constructing such codewords in "hard" parameter regimes, relying on bilinear system solutions and Gold-function type polynomials (Berman et al., 2023).

5. Modern Themes: Enumeration, Algorithms, and Hardness

Recent progress has been made in both the efficient enumeration and explicit construction of minimum-weight codewords in broad classes of codes, as well as in identifying principal hardness barriers:

Enumeration Algorithms: Several papers provide explicit closed-form counts for minimum-weight codewords, leveraging orbit decompositions, geometric configurations, or recursive reduction (e.g., projective Reed–Muller, Cartesian, Schubert, and extended BCH codes). In polar-code families, combinatorial and group-theoretic structures permit enumeration in sublinear or nearly-linear time in the code length.
Algorithmic Constructions: For BCH, Reed–Muller-like codes and polar codes, recent work supplies randomized and deterministic polynomial-time algorithms for constructing minimum-weight codewords, often with support structures corresponding to highly symmetric algebraic varieties or orbits.
Complexity Barriers: NP-hardness and inapproximability results (notably for permutation codes under $\ell_\infty$ ) underscore the fact that these advances rely critically on the algebraic, geometric, or group-theoretic structure of the family in question. For arbitrary subgroup permutation codes, no general polynomial-time algorithm approximates the minimum weight to within any factor better than $7/6$ (Shieh et al., 2010).

6. Combinatorial, Geometric, and Dual Interpretations

The minimal supports often correspond to extremal combinatorial objects (e.g., blocking sets, partial spreads, secants), maximal value sets in algebraic varieties, and in dual codes to complements of hyperplane sections achieving maximal intersection. For projective and Schubert codes, geometric interpretations provide a dictionary between codeword weight and zero patterns on subspaces or algebraic curves, bringing insights from finite geometry and algebraic geometry to bear on code design and analysis.

7. Applications, Future Directions, and Open Problems

Applications of the structure and enumeration of minimum-weight codewords include:

Error Probability and Decoding Analysis: Enumeration feeds directly into union bounds and high-SNR error floor estimates.
Code Design: Ability to enumerate and minimize minimum-weight words enables optimization for both classical parameters and modern, capacity-approaching codes.
Geometric and Algebraic Insight: Explicit connections to blocking sets, spreads, and geometric objects spawn advances in both code theory and finite geometry.

Open questions include the detailed classification of minimal codewords beyond small weights (especially in non-prime and high-dimensional cases), classification of minimum-weight supports in duals of newer geometric codes, complexity aspects in partially-structured code families, and the extension of orbit-sum and intersection methods to higher weights and non-binary codes. For permutation codes, the identification of rich subclasses admitting efficient minimum-weight algorithms remains ongoing (Shieh et al., 2010).

In summary, minimum-weight codewords not only delimit the error correction limits of a code but, via their exact geometric and algebraic structure, anchor major advances in both theory and practice across the landscape of modern coding theory.