Best Known (59−47, 59, s)-Nets in Base 3
(59−47, 59, 20)-Net over F3 — Constructive and digital
Digital (12, 59, 20)-net over F3, using
- t-expansion [i] based on digital (11, 59, 20)-net over F3, using
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 9, N(F) = 19, and 1 place with degree 3 [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
(59−47, 59, 22)-Net over F3 — Digital
Digital (12, 59, 22)-net over F3, using
- net from sequence [i] based on digital (12, 21)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 12 and N(F) ≥ 22, using
(59−47, 59, 42)-Net over F3 — Upper bound on s (digital)
There is no digital (12, 59, 43)-net over F3, because
- 20 times m-reduction [i] would yield digital (12, 39, 43)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(339, 43, F3, 27) (dual of [43, 4, 28]-code), but
(59−47, 59, 45)-Net in Base 3 — Upper bound on s
There is no (12, 59, 46)-net in base 3, because
- 20 times m-reduction [i] would yield (12, 39, 46)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(339, 46, S3, 27), but
- the linear programming bound shows that M ≥ 17725 876239 305002 191858 / 4165 > 339 [i]
- extracting embedded orthogonal array [i] would yield OA(339, 46, S3, 27), but