Error-Correcting Graph Codes

TL;DR

The paper defines and studies error-correcting graph codes under the graph distance $d_{graph}$, establishing a fundamental rate–distance tradeoff and providing both nonconstructive and explicit constructions. It proves a nonconstructive optimum with $R = (1-\delta)^2 - o(1)$ for constant $\delta$ via probabilistic methods, and then develops explicit schemes by combining symmetric tensor-zero-diagonal codes with Reed–Solomon outer codes through symmetric concatenation to achieve $R = (1-\sqrt{\delta})^4 - o(1)$, with stronger variants via triple concatenation. For very high distance near 1, it delivers explicit dual-BCH-type graph codes, including warmup constructions with $\dim = \Omega(\log n)$ and $d = 1 - O(n^{-1/2})$, and scalable generalizations achieving $\dim = \Omega(d\log n)$ and $d = 1 - O(d n^{-1/2})$, linking to Ramsey graphs. Collectively, the results deliver both foundational limits and practical, explicit constructions of large graph-code families with positive rate for any constant distance and near-perfect distance for large n, with implications for pseudorandom graphs and potential decoding algorithms.

Abstract

In this paper, we construct Error-Correcting Graph Codes. An error-correcting graph code of distance $δ$ is a family $C$ of graphs on a common vertex set of size $n$, such that if we start with any graph in $C$, we would have to modify the neighborhoods of at least $δn$ vertices in order to obtain some other graph in $C$. This is a natural graph generalization of the standard Hamming distance error-correcting codes for binary strings. Yohananov and Yaakobi were the first to construct codes in this metric, constructing good codes for $δ< 1/2$, and optimal codes for a large-alphabet analogue. We extend their work by showing 1. Combinatorial results determining the optimal rate vs. distance trade-off nonconstructively. 2. Graph code analogues of Reed-Solomon codes and code concatenation, leading to positive distance codes for all rates and positive rate codes for all distances. 3. Graph code analogues of dual-BCH codes, yielding large codes with distance $δ= 1-o(1)$. This gives an explicit ''graph code of Ramsey graphs''. Several recent works, starting with the paper of Alon, Gujgiczer, Körner, Milojević, and Simonyi, have studied more general graph codes; where the symmetric difference between any two graphs in the code is required to have some desired property. Error-correcting graph codes are a particularly interesting instantiation of this concept.

Figures (2)

• Figure 1: Example of symmetric concatenation. An outer codeword is shown on the left, with field elements represented as different colors. The concatenation with the inner code is shown to the right. Black squares represent $0$, and white squares represent $1$s.
• Figure 2: Inner code arrangement for $C_{\mathrm{Justensen}}$

Theorems & Definitions (39)

• Definition 2.1: Graph distance and relative graph distance
• Proposition 2.2: Alternate characterizations of $\mathop{\mathrm{d_{graph}}}\nolimits$
• Definition 2.3: Graph code
• Proposition 2.4
• proof
• Proposition 3.1
• proof
• Corollary 3.2
• Corollary 3.3
• Definition 4.1: Symmetric Tensor Code with Zeros on the Diagonal