Construction of 5-helix systems of index 2

Key Points

• The aim is to determine the spectrum of 5-helix S^{(3)} (1,5)-designs of index 2 with distinct blocks.
• Defined a 5-helix hypergraph system with specific properties.
• Analyzed the degree distribution of vertices and hyperedges.
• Conducted computations to derive the spectrum of designs.
• Identified the complete spectrum of S^{(3)} (1,5)-designs of index 2.
• Proved that all blocks are pairwise distinct.
• Established criteria for the degree structure within the hypergraph.