Generalized graph codes and thier minimum distances

TL;DR

The article extends graph codes to ℓ-partite graphs, deriving a general lower bound on the minimum distance $D$ in terms of the second-largest eigenvalue $\\lambda_2$ and the component-code parameters: $D \ge \frac{dm(d-\\lambda_2)}{(\\ell-1)n-\\lambda_2}$. It provides a detailed proof using spectral techniques and a Rayleigh-quotient argument and demonstrates the construction with concrete examples on $K_{7,7,7}$ and $K_{3,3,3}$, achieving $[147,48,9]$ and $[9,3,3]$-types respectively. The work confirms the bound in specific cases but acknowledges that no ℓ≥3 example currently reaches the bound and highlights open questions about tightness and potential refinements. Overall, the paper contributes a theoretical framework for minimum-distance guarantees in generalized (ℓ-partite) graph codes and illustrates the potential gains with explicit constructions.

Abstract

Graph code is a linear code obtained from linear codes $C$ and a certain bipartite graph G. In this paper, I propose an expansion of the definition of graph code to general $l$-partite, and give its lower bound of minimum distance. I also give an example of generalized graph code and calculate its parameters $[n, k, d]$.

Theorems & Definitions (9)

• Definition 1.1: Hoholdt--Justesen
• Proposition 1.1: Hoholdt--Justesen
• Theorem 1.1
• Definition 2.1
• Remark 2.1
• Theorem \ref{main-theorem}: Recall
• proof : Proof of Theorem \ref{['main-theorem']}
• Lemma 3.1
• proof