Additive codes arising from hypergraphs
Abstract
We study the critical exponent of additive codes through an integer polymatroid associated with the code. We give a coding-theoretic proof of Whittle's Critical Theorem in this setting, a geometric description of the critical exponent in terms of h-projective systems, and general bounds, including an analogue of Kung's girth bound. We then study additive codes whose polymatroid is the hypergraphic polymatroid of a hypergraph H. For these codes the critical exponent turns to be determined by the weak chromatic number of H. If the code is faithful, then the minimum folded Hamming weight of the dual code is equal to the Berge girth of H. If H is connected, the minimum distance is equal to the edge-connectivity of H. As a consequence, for h≥2 we determine all such codes with connected H that attain the Singleton bound, that is, all faithful hypergraphic additive quasi-MDS codes. We specialise the Griesmer and linear programming bounds to hypergraphic codes, we derive a lower bound on the minimum distance from the Laplacian eigenvalues of the weighted $2$-section of H, and we compare all these bounds computationally.
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BibTeX
@misc{alfarano2026additive,
title = {{Additive codes arising from hypergraphs}},
author = {Gianira N. Alfarano},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39680},
eprint = {2609.39680},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05772906v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Additive codes arising from hypergraphs
AU - Gianira N. Alfarano
PY - 2026
DA - 2026/09/30
PB - HAL
DO - 10.48550/arxiv.2609.39680
UR - https://hal.science/hal-05772906v1
AB - We study the critical exponent of additive codes through an integer polymatroid associated with the code. We give a coding-theoretic proof of Whittle's Critical Theorem in this setting, a geometric description of the critical exponent in terms of h-projective systems, and general bounds, including an analogue of Kung's girth bound. We then study additive codes whose polymatroid is the hypergraphic polymatroid of a hypergraph H. For these codes the critical exponent turns to be determined by the weak chromatic number of H. If the code is faithful, then the minimum folded Hamming weight of the dual code is equal to the Berge girth of H. If H is connected, the minimum distance is equal to the edge-connectivity of H. As a consequence, for h≥2 we determine all such codes with connected H that attain the Singleton bound, that is, all faithful hypergraphic additive quasi-MDS codes. We specialise the Griesmer and linear programming bounds to hypergraphic codes, we derive a lower bound on the minimum distance from the Laplacian eigenvalues of the weighted $2$-section of H, and we compare all these bounds computationally.
ER -