The Treewidth of MDS and Reed-Muller Codes

The constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization. The treewidth of a linear code is the least constraint complexity of any of its cycle-free graphical realizations. This notion provides a useful parametrization of the maximum-likelihood decoding complexity for linear codes. In this paper, we prove the surprising fact that for maximum distance separable codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code, is defined to be the least constraint complexity (or branch complexity) of any of its trellis realizations. From this, we obtain exact expressions for the treewidth of these codes, which constitute the only known explicit expressions for the treewidth of algebraic codes.

💡 Research Summary

The paper investigates the relationship between two graph‑theoretic complexity measures for linear codes: treewidth, defined as the minimum constraint complexity over all cycle‑free (tree) graphical realizations, and trelliswidth, the analogous quantity when the underlying graph is restricted to a path (a trellis). While it is always true that treewidth ≤ trelliswidth, it was previously unknown for which families of codes equality holds.

The authors develop a general proof strategy that combines (i) bounds derived from the generalized Hamming weights of a code and the associated maximal limited‑support subcode dimensions Uₛ(C), and (ii) separator theorems for cubic trees. For any tree decomposition (T, ω) of a code C, the local constraint dimension at an internal node v satisfies
κᵥ ≥ k – Σ_{e∈E(v)} U_{|I_{e,v}|}(C),
where I_{e,v} is the set of coordinates assigned to the leaf set of the component of T – e that does not contain v. This inequality provides a lower bound on the constraint complexity contributed by any node, expressed purely in terms of the number of leaves in the three sub‑trees incident to v.

The separator theorems guarantee that in any cubic tree with n leaves there exists an internal node v whose three incident sub‑trees have leaf counts n₁ ≤ n₂ ≤ n₃ satisfying either n₁, n₂, n₃ ≤ n/2 (Proposition 2) or n₃ ∈