Best Known (234−48, 234, s)-Nets in Base 3
(234−48, 234, 688)-Net over F3 — Constructive and digital
Digital (186, 234, 688)-net over F3, using
- t-expansion [i] based on digital (184, 234, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (184, 236, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 59, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 59, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (184, 236, 688)-net over F3, using
(234−48, 234, 2265)-Net over F3 — Digital
Digital (186, 234, 2265)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 2265, F3, 48) (dual of [2265, 2031, 49]-code), using
- 68 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 21 times 0) [i] based on linear OA(3224, 2187, F3, 48) (dual of [2187, 1963, 49]-code), using
- 1 times truncation [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
- 68 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 21 times 0) [i] based on linear OA(3224, 2187, F3, 48) (dual of [2187, 1963, 49]-code), using
(234−48, 234, 219896)-Net in Base 3 — Upper bound on s
There is no (186, 234, 219897)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4429 873426 532511 475048 926100 812621 429855 380512 324510 465145 812026 485108 003678 456917 875526 705531 551447 720235 468257 > 3234 [i]