Codeword Distance Matrices in Coding Theory

• Codeword Distance Matrices are symmetric arrays that capture pairwise distances among codewords to reveal the geometric structure of discrete codes.
• They utilize methods like brute-force, Gröbner basis computation, and rank reduction to efficiently compute distances in both Hamming and subspace codes.
• Their spectral invariants, determinant properties, and SDP applications offer crucial insights for optimizing code design and analyzing code structure.

A codeword distance matrix is the symmetric array which records all pairwise metric distances among the codewords of a discrete code. Such matrices play a central role in coding theory, combinatorial design, and the semidefinite programming bounds central to the analysis of code size and structure. Their explicit computation, invertibility properties, and spectral invariants provide essential insight into the geometry and optimality of codes in both linear and nonlinear settings.

1. Definitions and Fundamental Structures

Given a code $\mathcal{C}$ with codewords $c_1,\ldots,c_N$ in a metric space $(X, d)$, the codeword distance matrix $D = (D_{ij})$ is defined as

$D_{ij} = d(c_i, c_j),$

where $d$ is typically the Hamming metric on $X = \mathbb{F}_q^n$ or, in the case of subspace codes, the subspace distance on $X$ the Grassmannian of subspaces of $\mathbb{F}_q^n$. The matrix is symmetric with $D_{ii} = 0$.

In Hamming space, for $c_i, c_j \in \mathbb{F}_q^n$,

$$d_H(c_i, c_j) = |\{ \ell \mid (c_i)_\ell \neq (c_j)_\ell \}|.$$

For constant dimension subspace codes, the distance is

$$d_S(U, V) = \dim U + \dim V - 2\dim(U \cap V) = 2\,\mathrm{rank}\begin{pmatrix}\mathrm{RE}(U)\\mathrm{RE}(V)\end{pmatrix} - \dim U - \dim V,$$

where $\mathrm{RE}(U)$ is the reduced row echelon form of a generator matrix for $U$ (Silberstein et al., 2010). The distance matrix thus encodes a complete pairwise geometric profile of the code.

2. Explicit Computation: Algorithmic Approaches

Brute-Force Methods

The standard computational method is to enumerate all $N(N-1)/2$ off-diagonal pairs, evaluating $d(c_i, c_j)$ for each. Brute-force Hamming computation for systematic codes of size $S=q^k$ entails $\Theta(n S^2)$ total complexity ($\Theta(k 2^{2k})$ for $q=2$) (0909.1626).

Gröbner Basis Method for Nonlinear Systematic Codes

To overcome the brute-force barrier and to enable symbolic computation on parametric families, the Gröbner basis technique of Guerrini–Orsini–Sala encodes distance constraints into polynomial ideals. Specifically, one constructs, for each threshold $t$, the ideal

$$J_t = I_C + \langle x_i - x_i' \rangle_{i=1}^k + \langle M^{(n,k)}_{i_1 \dots i_t}(x,x') \rangle,$$

where $I_C$ encodes the systematic code structure, $M^{(n,k)}_{i_1 \dots i_t}$ are binomials vanishing on pairs at Hamming distance $\le t-1$, and the diagonal cut ensures $c_i \neq c_j$ (0909.1626). A Gröbner basis is computed for $J_t$, and roots are counted to identify all codeword pairs at or below a given distance. The full distance distribution (and hence $D$) is efficiently assembled from these counts using telescoping differences. This technique can be extended to entire code families, not just specific codes.

Distance Matrices for Subspace Codes

For codes whose elements are subspaces, one uses the rank-based distance formula. Given codewords $X_i, X_j$ with RREF generators $A, B$,

$$D_{ij} = 2\,\mathrm{rank}\begin{pmatrix}A \ B\end{pmatrix} - \mathrm{rank}(A) - \mathrm{rank}(B).$$

Practical computation is greatly accelerated by Hamming distance screening on “identifying vectors” (support sets of pivots), and by pruning via lexicode/Ferrers-diagram classes, so that only genuinely necessary row-reductions are performed (Silberstein et al., 2010).

Method	Metric	Complexity
Brute-force	Hamming	$\Theta(n S^2)$
Gröbner basis	Hamming	$O(2^{3k})$
Rank-reduction	Subspace	$O(N^2 k_\max^2 n)$
Hamming screening	Subspace	Reduces rank computations

3. Invariants: Determinant, Invertibility, and Type

Determinant Formulae in Hamming Space

For codewords $x_0, ..., x_m \in H_n$, the determinant of $D$ satisfies the generalized Graham–Winkler formula (Doust et al., 2020):

$\det D = (-1)^{m-1} 2^{m-1} \det(G) (G^{-1}u, u),$

where $G$ is the Gram matrix of the translated codeword vectors, $u$ encodes their squared norms, and $V^2 = \det G$ is the squared $m$-volume of the parallelotope spanned by the codewords translated to $x_0 = 0$. In the full-dimensional case ($m=n$),

$\det D = (-1)^n 2^{n-1} V^2.$

Vanishing of $\det D$ characterizes affine dependence; $\det D \neq 0$ if and only if the codewords are affinely independent (Doust et al., 2020).

Spectral Invariants and 1-Negative Type

For a finite metric space $(X,d)$, the “strict 1-negative type” criterion is satisfied if, for all real weightings $\sigma_i$ summing to zero and not all zero,

$\sum_{i,j} d(x_i, x_j) \sigma_i \sigma_j < 0.$

By the work of Murugan and others, in Hamming space this is equivalent to the invertibility of $D$ and the nonvanishing of $(D^{-1} 1, 1) > 0$ (Doust et al., 2020). For unweighted trees embedded in Hamming cubes, $(D^{-1} 1, 1) = 2/n$, independent of tree structure.

4. Applications: Bounds, Design, and Analysis

Semidefinite Programming Bounds

Higher-order distance matrices, notably quadruple-distance matrices $M_S(x)$ indexed by subsets of up to four codewords, are central to contemporary semidefinite programming (SDP) bounds on $A(n,d)$—the maximal code size in Hamming space with minimum distance $d$ (Gijswijt et al., 2010). The positivity of $M_S(x)$ for all $|S| \le 2$ is imposed as an SDP constraint, block-diagonalized under the Hamming automorphism group for tractability.

Structural Analysis and Classification

The distance matrix encodes the full geometric configuration of a code, allowing the analysis of isometric embeddings, diameter, distance distributions, and other combinatorial invariants. Its rank and spectrum provide quick tests for affine independence and code regularity, and the entries can be used to reconstruct properties such as covering radius and minimum distance via direct inspection.

5. Optimization and Computational Techniques

For large codes or high dimensions, direct computation of all pairwise distances becomes intractable. Key algorithmic strategies include:

• Hamming distance screening: For subspaces, if the Hamming distance of identifying vectors exceeds threshold, explicit rank computation is skipped (Silberstein et al., 2010).
• Lexicode/Ferrers pruning: Only compare candidate subspaces within relevant Ferrers classes, cutting the number of rank evaluations from quadratic to essentially linear in code size.
• Gröbner basis elimination: Systematic codes permit elimination of dependent variables "for free," reducing basis computations before applying F4/F5 or Buchberger algorithms.

Computational experiments confirm that these optimizations yield orders-of-magnitude improvements in constructing $D$ for large codes (0909.1626, Silberstein et al., 2010).

6. Representative Examples and Explicit Matrices

Explicit distance matrices provide concrete insight into code structure. For instance, the systematic $(4,2,2)$ binary code

$$C = \{ (x_1, x_2, x_1, x_1x_2) \mid x_1, x_2 \in \mathbb{F}_2 \}$$

produces the Hamming distance matrix

$$D = \begin{pmatrix} 0 & 2 & 2 & 2 \ 2 & 0 & 2 & 2 \ 2 & 2 & 0 & 3 \ 2 & 2 & 3 & 0 \ \end{pmatrix}$$

(0909.1626). For subspaces of $\mathbb{F}_2^4$, the corresponding matrix:

$$D = \begin{pmatrix} 0 & 2 & 2 & 4 \ 2 & 0 & 2 & 2 \ 2 & 2 & 0 & 2 \ 4 & 2 & 2 & 0 \ \end{pmatrix}$$

(Silberstein et al., 2010). These examples underscore the geometric diversity encoded by $D$ and its straightforward assembly from Gröbner or rank computations.

7. Connections, Generalizations, and Open Directions

Distance matrices are central in the theory of association schemes, eigenvalue methods, and their use in optimization via semidefinite programming. The transition from pairwise distance matrices to higher-order matrices (e.g., quadruple or higher) enables increasingly tight code bounds and reveals structural symmetries exploitable by group action block-diagonalization (Gijswijt et al., 2010).

For systematic nonlinear codes, Gröbner basis-based methods extend to parametric family analysis and provide a symbolic approach to minimum distance and weight spectrum bounds, which remains infeasible for brute-force approaches (0909.1626). In the context of constant dimension codes and network coding, distance matrices built via rank and identifying-vector methods continue to be a vital computational and analytical tool (Silberstein et al., 2010).

A plausible implication is that further advances in computational algebra and symmetry exploitation may yield more efficient methods for evaluating or bounding distance matrix spectra, automorphism groups, and code isomorphism classes at scale.