Binary LCD Codes and Their Graph Representations

We establish a bijection between binary even LCD codes and simple graphs whose adjacency matrices are idempotent over $\FF_2$. This bijection preserves equivalence: permutation equivalence of codes corresponds exactly to graph isomorphism. Based on this framework, we characterize which distance-regular graphs yield LCD codes via intersection array parameters, prove that non-isomorphic conference graphs yield inequivalent codes, and classify LCD-derived graphs of small orders.

💡 Research Summary

The paper establishes a precise correspondence between binary even linear complementary dual (LCD) codes and simple graphs whose adjacency matrices are idempotent over the field 𝔽₂. The core of the construction is the orthogonal projector Π_C associated with an LCD code C. Π_C is a symmetric n × n matrix satisfying Π_C² = Π_C and Im(Π_C) = C. The authors show that when C is even, every diagonal entry of Π_C is zero, so Π_C can be interpreted as the adjacency matrix A of a loop‑free simple graph on n vertices. Conversely, any symmetric idempotent (0,1)-matrix with zero diagonal is an orthogonal projector, and its row space is an LCD code. This establishes a bijection (Theorem 3.2) between binary even LCD codes of length n and simple graphs on n vertices with idempotent adjacency matrices. Moreover, permutation equivalence of codes corresponds exactly to graph isomorphism, because a permutation of coordinates translates into a simultaneous row‑column permutation of the projector matrix.

A combinatorial reformulation (Corollary 3.3) translates the idempotence condition into three elementary graph properties: (i) every vertex has even degree, (ii) any two adjacent vertices share an odd number of common neighbours, and (iii) any two non‑adjacent vertices share an even number of common neighbours. These conditions immediately rule out many familiar families (trees, bipartite graphs, etc.) and imply that any LCD‑producing graph must contain triangles.

The authors then focus on distance‑regular graphs (DRGs). Using the three‑term recurrence for the distance matrices, they derive a simple parity test (Theorem 4.1): a DRG yields a binary even LCD code iff (i) the valency b₀ is even, (ii) the parameter a₁ = k − b₁ − c₁ is odd, and (iii) the second intersection number c₂ is even. For strongly regular graphs (SRGs), these conditions reduce to k even, λ odd, μ even (Theorem 4.2). Thus the presence of an LCD code can be decided by checking only three parameters.

Applying this to conference graphs, the paper proves (Theorem 5.2) that non‑isomorphic conference graphs always give rise to inequivalent LCD codes. Since conference graphs have parameters (v,(v−1)/2,(v−5)/4,(v−1)/4) with v ≡ 1 (mod 4), the parity conditions are satisfied, and the distinct adjacency matrices lie in different permutation orbits, guaranteeing distinct codes. This provides a theoretical explanation for the empirical observations of Haemers et al. (2010).

Finally, leveraging known mass formulas for binary LCD codes, the authors perform an exhaustive enumeration for n ≤ 13. They list all simple graphs whose adjacency matrices are idempotent over 𝔽₂, thereby classifying every LCD‑derived graph up to order 13. The resulting catalogue serves as a concrete resource for both coding theorists and graph theorists.

In summary, the paper bridges coding theory and graph theory by showing that the code equivalence problem for binary even LCD codes is exactly the graph isomorphism problem for a well‑defined class of graphs. It supplies clean algebraic and combinatorial criteria for when a given graph family yields LCD codes, proves the uniqueness of codes from non‑isomorphic conference graphs, and provides a complete small‑order classification. These contributions open new avenues for transferring techniques between the two fields, such as using graph invariants to distinguish codes or applying coding‑theoretic mass formulas to enumerate special graph families.