On the Number of Neighborly Simplices in Rᵈ

Two d -dimensional simplices in Rᵈ R d are neighborly if its intersection is a $$(d-1)$$ ( d - 1 ) -dimensional set. A family of d -dimensional simplices in Rᵈ R d is called neighborly if every two simplices of the family are neighborly. Let Sd S d be the maximal cardinality of a neighborly family of d -dimensional simplices in Rᵈ R d . Based on the structure of some codes V⊂ \0,1,*\ⁿ V ⊂ { 0 , 1 , ∗ } n it is shown that lim d→ ∞(2ᵈ⁺¹-Sd)=∞ lim d → ∞ ( 2 d + 1 - S d ) = ∞ . Moreover, a result on the structure of codes V⊂ \0,1,*\ⁿ V ⊂ { 0 , 1 , ∗ } n is given.