Best Known (3, 7, s)-Nets in Base 8
(3, 7, 65)-Net over F8 — Constructive and digital
Digital (3, 7, 65)-net over F8, using
- base reduction for projective spaces (embedding PG(3,64) in PG(6,8)) for nets [i] based on digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
(3, 7, 81)-Net over F8 — Digital
Digital (3, 7, 81)-net over F8, using
- net defined by OOA [i] based on linear OOA(87, 81, F8, 4, 4) (dual of [(81, 4), 317, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(87, 81, F8, 3, 4) (dual of [(81, 3), 236, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(87, 81, F8, 4) (dual of [81, 74, 5]-code), using
- a “Gra†code from Grassl’s database [i]
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(87, 81, F8, 4) (dual of [81, 74, 5]-code), using
- appending kth column [i] based on linear OOA(87, 81, F8, 3, 4) (dual of [(81, 3), 236, 5]-NRT-code), using
(3, 7, 291)-Net in Base 8 — Upper bound on s
There is no (3, 7, 292)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 2 100211 > 87 [i]
- extracting embedded orthogonal array [i] would yield OA(87, 292, S8, 4), but
- the linear programming bound shows that M ≥ 240314 880000 / 114211 > 87 [i]