A Note on the Existence of Special Laguerre i-Structures and Optimal Codes

A special Laguerre i-structure of order n (i ⩾1) is an incidence structure J = (P, B1 ⋃ B2, I) for which: (i) each element of P is incident with one element of B1, (ii) each i-residual space of J (with respect to B1) is a projective plane of order n minus one point, (iii) B2 ≠ ∅ and each element of B2 is incident with at least i elements of P. We prove some necessary conditions for the existence of special Laguerre i-structures, i≡2, of order n (resp. optimal (n + i + 1, i + 2) -codes of order n)