Best Known (61−32, 61, s)-Nets in Base 9
(61−32, 61, 78)-Net over F9 — Constructive and digital
Digital (29, 61, 78)-net over F9, using
- t-expansion [i] based on digital (22, 61, 78)-net over F9, using
- net from sequence [i] based on digital (22, 77)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- F3 from the tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- net from sequence [i] based on digital (22, 77)-sequence over F9, using
(61−32, 61, 84)-Net in Base 9 — Constructive
(29, 61, 84)-net in base 9, using
- 2 times m-reduction [i] based on (29, 63, 84)-net in base 9, using
- base change [i] based on digital (8, 42, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- base change [i] based on digital (8, 42, 84)-net over F27, using
(61−32, 61, 112)-Net over F9 — Digital
Digital (29, 61, 112)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(961, 112, F9, 2, 32) (dual of [(112, 2), 163, 33]-NRT-code), using
- construction X applied to AG(2;F,185P) ⊂ AG(2;F,189P) [i] based on
- linear OOA(958, 109, F9, 2, 32) (dual of [(109, 2), 160, 33]-NRT-code), using algebraic-geometric NRT-code AG(2;F,185P) [i] based on function field F/F9 with g(F) = 26 and N(F) ≥ 110, using
- linear OOA(954, 109, F9, 2, 28) (dual of [(109, 2), 164, 29]-NRT-code), using algebraic-geometric NRT-code AG(2;F,189P) [i] based on function field F/F9 with g(F) = 26 and N(F) ≥ 110 (see above)
- linear OOA(93, 3, F9, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(93, 9, F9, 2, 3) (dual of [(9, 2), 15, 4]-NRT-code), using
- Reed–Solomon NRT-code RS(2;15,9) [i]
- discarding factors / shortening the dual code based on linear OOA(93, 9, F9, 2, 3) (dual of [(9, 2), 15, 4]-NRT-code), using
- construction X applied to AG(2;F,185P) ⊂ AG(2;F,189P) [i] based on
(61−32, 61, 3684)-Net in Base 9 — Upper bound on s
There is no (29, 61, 3685)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 16206 787805 959880 197692 496327 727861 630478 275647 838239 356033 > 961 [i]