On the number of neighborly simplices in R^d

Two $d$-dimensional simplices in $R^d$ are neighborly if its intersection is a $(d-1)$-dimensional set. A family of $d$-dimensional simplices in $R^d$ is called neighborly if every two simplices of the family are neighborly. Let $S_d$ be the maximal cardinality of a neighborly family of $d$-dimensional simplices in $R^d$. Based on the structure of some codes $V\subset \{0,1,\ast {\}}^n$ it is shown that ${\lim }_{d\to \mathrm{\infty }}(2^{d+1}-S_d)=\mathrm{\infty }$. Moreover, a result on the structure of codes $V\subset \{0,1,\ast {\}}^n$ is given.