Best Known (246−157, 246, s)-Nets in Base 3
(246−157, 246, 64)-Net over F3 — Constructive and digital
Digital (89, 246, 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
(246−157, 246, 96)-Net over F3 — Digital
Digital (89, 246, 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
(246−157, 246, 389)-Net over F3 — Upper bound on s (digital)
There is no digital (89, 246, 390)-net over F3, because
- 1 times m-reduction [i] would yield digital (89, 245, 390)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3245, 390, F3, 156) (dual of [390, 145, 157]-code), but
- residual code [i] would yield linear OA(389, 233, F3, 52) (dual of [233, 144, 53]-code), but
- the Johnson bound shows that N ≤ 495 041572 388827 382337 917134 411719 483784 605757 670798 500294 630099 278193 < 3144 [i]
- residual code [i] would yield linear OA(389, 233, F3, 52) (dual of [233, 144, 53]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3245, 390, F3, 156) (dual of [390, 145, 157]-code), but
(246−157, 246, 398)-Net in Base 3 — Upper bound on s
There is no (89, 246, 399)-net in base 3, because
- 1 times m-reduction [i] would yield (89, 245, 399)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 927 226067 258191 972335 912867 128033 465388 381282 952750 056386 045887 937289 803921 313487 574817 906963 822046 279965 053339 906565 > 3245 [i]