The Neighbor Graph of Linear Complementary Dual (LCD) Codes
Abstract
Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whenever they intersect in codimension one. We determine the number of neighbors of an LCD code that are also LCD, and we use this result to analyze the structure of the corresponding neighbor graph. In particular, we prove its regularity over arbitrary finite fields and establish further regularity properties for its main structural subgraphs in the binary and odd-characteristic cases. These results provide a graph-theoretic framework for the study of LCD codes and reveal a strong combinatorial regularity in their neighborhood structure.
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BibTeX
@misc{cruz2026neighbor,
title = {{The Neighbor Graph of Linear Complementary Dual (LCD) Codes}},
author = {Javier de la Cruz and Anna-Lena Horlemann and Marc Newman and Carlos Vela Cabello and Wolfgang Willems},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07580},
eprint = {2609.07580},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07580},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Neighbor Graph of Linear Complementary Dual (LCD) Codes
AU - Javier de la Cruz
AU - Anna-Lena Horlemann
AU - Marc Newman
AU - Carlos Vela Cabello
AU - Wolfgang Willems
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07580
UR - https://arxiv.org/abs/2609.07580
AB - Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whenever they intersect in codimension one. We determine the number of neighbors of an LCD code that are also LCD, and we use this result to analyze the structure of the corresponding neighbor graph. In particular, we prove its regularity over arbitrary finite fields and establish further regularity properties for its main structural subgraphs in the binary and odd-characteristic cases. These results provide a graph-theoretic framework for the study of LCD codes and reveal a strong combinatorial regularity in their neighborhood structure.
ER -