Best Known (29, 69, s)-Nets in Base 81
(29, 69, 370)-Net over F81 — Constructive and digital
Digital (29, 69, 370)-net over F81, using
- t-expansion [i] based on digital (16, 69, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(29, 69, 502)-Net over F81 — Digital
Digital (29, 69, 502)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8169, 502, F81, 2, 40) (dual of [(502, 2), 935, 41]-NRT-code), using
- construction X applied to AG(2;F,957P) ⊂ AG(2;F,961P) [i] based on
- linear OOA(8166, 499, F81, 2, 40) (dual of [(499, 2), 932, 41]-NRT-code), using algebraic-geometric NRT-code AG(2;F,957P) [i] based on function field F/F81 with g(F) = 26 and N(F) ≥ 500, using
- linear OOA(8162, 499, F81, 2, 36) (dual of [(499, 2), 936, 37]-NRT-code), using algebraic-geometric NRT-code AG(2;F,961P) [i] based on function field F/F81 with g(F) = 26 and N(F) ≥ 500 (see above)
- linear OOA(813, 3, F81, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(813, 81, F81, 2, 3) (dual of [(81, 2), 159, 4]-NRT-code), using
- Reed–Solomon NRT-code RS(2;159,81) [i]
- discarding factors / shortening the dual code based on linear OOA(813, 81, F81, 2, 3) (dual of [(81, 2), 159, 4]-NRT-code), using
- construction X applied to AG(2;F,957P) ⊂ AG(2;F,961P) [i] based on
(29, 69, 398546)-Net in Base 81 — Upper bound on s
There is no (29, 69, 398547)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 484706 113747 021993 934441 285354 313960 055259 706357 607572 369078 827338 281473 970700 050857 012410 651258 720418 234764 153995 752608 709601 643201 > 8169 [i]