Best Known (18, 18+22, s)-Nets in Base 8
(18, 18+22, 65)-Net over F8 — Constructive and digital
Digital (18, 40, 65)-net over F8, using
- t-expansion [i] based on digital (14, 40, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(18, 18+22, 67)-Net over F8 — Digital
Digital (18, 40, 67)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(840, 67, F8, 3, 22) (dual of [(67, 3), 161, 23]-NRT-code), using
- construction X applied to AG(3;F,169P) ⊂ AG(3;F,174P) [i] based on
- linear OOA(836, 64, F8, 3, 22) (dual of [(64, 3), 156, 23]-NRT-code), using algebraic-geometric NRT-code AG(3;F,169P) [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- linear OOA(831, 64, F8, 3, 17) (dual of [(64, 3), 161, 18]-NRT-code), using algebraic-geometric NRT-code AG(3;F,174P) [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65 (see above)
- linear OOA(84, 3, F8, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(84, 8, F8, 3, 4) (dual of [(8, 3), 20, 5]-NRT-code), using
- Reed–Solomon NRT-code RS(3;20,8) [i]
- discarding factors / shortening the dual code based on linear OOA(84, 8, F8, 3, 4) (dual of [(8, 3), 20, 5]-NRT-code), using
- linear OOA(836, 64, F8, 3, 22) (dual of [(64, 3), 156, 23]-NRT-code), using algebraic-geometric NRT-code AG(3;F,169P) [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- construction X applied to AG(3;F,169P) ⊂ AG(3;F,174P) [i] based on
(18, 18+22, 1342)-Net in Base 8 — Upper bound on s
There is no (18, 40, 1343)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1 339802 396106 502807 112293 725097 048724 > 840 [i]