Best Known (190−37, 190, s)-Nets in Base 3
(190−37, 190, 688)-Net over F3 — Constructive and digital
Digital (153, 190, 688)-net over F3, using
- t-expansion [i] based on digital (151, 190, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (151, 192, 688)-net over F3, using
(190−37, 190, 2435)-Net over F3 — Digital
Digital (153, 190, 2435)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3190, 2435, F3, 37) (dual of [2435, 2245, 38]-code), using
- 226 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 30 times 0, 1, 37 times 0, 1, 44 times 0) [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 226 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 30 times 0, 1, 37 times 0, 1, 44 times 0) [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
(190−37, 190, 386236)-Net in Base 3 — Upper bound on s
There is no (153, 190, 386237)-net in base 3, because
- 1 times m-reduction [i] would yield (153, 189, 386237)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 499418 391808 781875 884256 233415 770114 841477 052848 128031 590659 717111 806500 491933 607928 281281 > 3189 [i]