Best Known (82, 82+149, s)-Nets in Base 3
(82, 82+149, 57)-Net over F3 — Constructive and digital
Digital (82, 231, 57)-net over F3, using
- net from sequence [i] based on digital (82, 56)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 56)-sequence over F9, using
- s-reduction 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
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 56)-sequence over F9, using
(82, 82+149, 84)-Net over F3 — Digital
Digital (82, 231, 84)-net over F3, using
- t-expansion [i] based on digital (71, 231, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(82, 82+149, 354)-Net over F3 — Upper bound on s (digital)
There is no digital (82, 231, 355)-net over F3, because
- 2 times m-reduction [i] would yield digital (82, 229, 355)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3229, 355, F3, 147) (dual of [355, 126, 148]-code), but
- residual code [i] would yield linear OA(382, 207, F3, 49) (dual of [207, 125, 50]-code), but
- 1 times truncation [i] would yield linear OA(381, 206, F3, 48) (dual of [206, 125, 49]-code), but
- the Johnson bound shows that N ≤ 390067 621362 525166 675859 888729 736992 372634 079787 564789 713745 < 3125 [i]
- 1 times truncation [i] would yield linear OA(381, 206, F3, 48) (dual of [206, 125, 49]-code), but
- residual code [i] would yield linear OA(382, 207, F3, 49) (dual of [207, 125, 50]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3229, 355, F3, 147) (dual of [355, 126, 148]-code), but
(82, 82+149, 362)-Net in Base 3 — Upper bound on s
There is no (82, 231, 363)-net in base 3, because
- 1 times m-reduction [i] would yield (82, 230, 363)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 57 828635 889811 585598 260268 722342 331135 253177 118093 385288 242978 157257 925293 890529 959506 483376 485058 225021 705381 > 3230 [i]