Linear codes associated with the Desarguesian ovoids in Q^+(7,q)
share
The Desarguesian ovoids in the orthogonal polar space Q^+(7,q) with q even have first been introduced by Kantor by examining the 8-dimensional absolutely irreducible modular representations of PGL(2,q^3). We investigate this module for all prime power values of q. The shortest PGL(2,q^3)-orbit O gives the Desarguesian ovoid in Q^+(7,q) for even q and it is known to give a complete partial ovoid of the symplectic polar space W(7,q) for odd q. We determine the hyperplane sections of O. As a corollary, we obtain the parameters [q^3+1,8,q^3-q^2-q]_q and the weight distribution of the associated 𝔽_q-linear code C_O and the parameters [q^3+1,q^3-7,5]_q of the dual code C_O^⊥ for q ≥ 4. We also show that both codes C_O and C_O^⊥ are length-optimal for all prime power values of q.
READ FULL TEXT