Best Known (89−63, 89, s)-Nets in Base 3
(89−63, 89, 36)-Net over F3 — Constructive and digital
Digital (26, 89, 36)-net over F3, using
- net from sequence [i] based on digital (26, 35)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 26 and N(F) ≥ 36, using
(89−63, 89, 37)-Net over F3 — Digital
Digital (26, 89, 37)-net over F3, using
- net from sequence [i] based on digital (26, 36)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 25, N(F) = 36, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 25 and N(F) ≥ 36, using algebraic function fields over ℤ3 by Niederreiter/Xing [i]
(89−63, 89, 85)-Net over F3 — Upper bound on s (digital)
There is no digital (26, 89, 86)-net over F3, because
- 9 times m-reduction [i] would yield digital (26, 80, 86)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(380, 86, F3, 54) (dual of [86, 6, 55]-code), but
- residual code [i] would yield linear OA(326, 31, F3, 18) (dual of [31, 5, 19]-code), but
- residual code [i] would yield linear OA(38, 12, F3, 6) (dual of [12, 4, 7]-code), but
- residual code [i] would yield linear OA(326, 31, F3, 18) (dual of [31, 5, 19]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(380, 86, F3, 54) (dual of [86, 6, 55]-code), but
(89−63, 89, 87)-Net in Base 3 — Upper bound on s
There is no (26, 89, 88)-net in base 3, because
- 9 times m-reduction [i] would yield (26, 80, 88)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(380, 88, S3, 54), but
- the linear programming bound shows that M ≥ 11972 515182 562019 788602 740026 717047 105681 / 70 > 380 [i]
- extracting embedded orthogonal array [i] would yield OA(380, 88, S3, 54), but