Best Known (89, 89+161, s)-Nets in Base 3
(89, 89+161, 64)-Net over F3 — Constructive and digital
Digital (89, 250, 64)-net over F3, using
- net from sequence [i] based on digital (89, 63)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- base reduction for sequences [i] based on digital (13, 63)-sequence over F9, using
(89, 89+161, 96)-Net over F3 — Digital
Digital (89, 250, 96)-net over F3, using
- net from sequence [i] based on digital (89, 95)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 89 and N(F) ≥ 96, using
(89, 89+161, 384)-Net over F3 — Upper bound on s (digital)
There is no digital (89, 250, 385)-net over F3, because
- 2 times m-reduction [i] would yield digital (89, 248, 385)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3248, 385, F3, 159) (dual of [385, 137, 160]-code), but
- residual code [i] would yield linear OA(389, 225, F3, 53) (dual of [225, 136, 54]-code), but
- 1 times truncation [i] would yield linear OA(388, 224, F3, 52) (dual of [224, 136, 53]-code), but
- the Johnson bound shows that N ≤ 73138 750163 909178 998184 764057 381403 209399 446777 305834 354572 535302 < 3136 [i]
- 1 times truncation [i] would yield linear OA(388, 224, F3, 52) (dual of [224, 136, 53]-code), but
- residual code [i] would yield linear OA(389, 225, F3, 53) (dual of [225, 136, 54]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3248, 385, F3, 159) (dual of [385, 137, 160]-code), but
(89, 89+161, 393)-Net in Base 3 — Upper bound on s
There is no (89, 250, 394)-net in base 3, because
- 1 times m-reduction [i] would yield (89, 249, 394)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 74729 985957 878958 453997 690700 677279 863726 895274 798552 788757 460368 201821 904052 026808 609368 739551 920606 813436 351095 764513 > 3249 [i]