Abstract
We investigate properties of two-weight codes over finite Frobenius rings, giving constructions for the modular case. A \(\delta \)-modular code (in: Honold T, Honold in Proceedings of the fifth international workshop on optimal codes and related topics, White Lagoon, Bulgaria, 2007) is characterized as having a generator matrix where each column \(g\) appears with multiplicity \(\delta |gR^\times |\) for some \(\delta \in \mathbb {Q}\). Generalizing (Delsarte in Discret Math 3:47–64, 1972) and (Byrne et al in Finite Fields Appl 18(4):711–727, 2012), we show that the additive group of a two-weight code satisfying certain constraint equations (and in particular a modular code) has a strongly regular Cayley graph and derive existence conditions on its parameters. We provide a construction for an infinite family of modular two-weight codes arising from unions of submodules with pairwise trivial intersection. The corresponding strongly regular graphs are isomorphic to graphs from orthogonal arrays.
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Notes
It is well-known that \(M_2(\mathbb {F}_q)\) determines a strongly regular graph by taking the ring elements as vertices and joining two vertices if their difference has rank 2 (cf. [9]); this is the same relation as induced by the homogeneous weight.
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Acknowledgments
The authors would like to thank the anonymous reviewers for their comments and suggestions, which has led to a great improvement in the presentation of this paper. Research supported by Science Foundation Ireland Grant 08/RFP/MTH1181.
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Communicated by W. H. Haemers.
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Byrne, E., Sneyd, A. Two-weight codes, graphs and orthogonal arrays. Des. Codes Cryptogr. 79, 201–217 (2016). https://doi.org/10.1007/s10623-015-0042-1
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DOI: https://doi.org/10.1007/s10623-015-0042-1
Keywords
- Ring-linear code
- Finite Frobenius ring
- Orthogonal array
- Strongly regular graph
- Homogeneous weight
- Two-weight code
- Modular code