The edge code of hypergraphs

TL;DR

This work introduces edge codes $C_{ H}$, a novel class of evaluation (toric) codes derived from hypergraphs by parameterizing polynomials with hypergraph edges and evaluating on the affine torus. It establishes fundamental parameters, provides sharp minimum-distance formulas for $d$-uniform clutters, proves self-orthogonality, and analyzes weight distributions for edge codes on trees, including exact computations for all connected graphs on five vertices. The results connect combinatorial hypergraph structures to algebraic-geometry–driven code properties, enabling both code construction and analysis of duals and potential quantum-code applications. Collectively, the paper extends toric and squarefree evaluation code frameworks to hypergraphs, offering new bounds, exact distances, and structural insights with practical implications for code design over finite fields.

Abstract

Given a hypergraph $\mathcal{H}$, we introduce a new class of evaluation toric codes called edge codes derived from $\mathcal{H}$. We analyze these codes, focusing on determining their basic parameters. We provide estimations for the minimum distance, particularly in scenarios involving $d$-uniform clutters. Additionally, we demonstrate that these codes exhibit self-orthogonality. Furthermore, we compute the minimum distances of edge codes for all graphs with five vertices.

Figures (3)

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Theorems & Definitions (35)

• Definition 1
• Theorem 2: Theorem \ref{['clutter_theorem']}
• Theorem 3: Theorem \ref{['relative_code']}
• Theorem 4: Theorem \ref{['theo-weight-dist']}
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Remark 2.5
• Proposition 3.1