Best Known (186−36, 186, s)-Nets in Base 3
(186−36, 186, 688)-Net over F3 — Constructive and digital
Digital (150, 186, 688)-net over F3, using
- t-expansion [i] based on digital (148, 186, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (148, 188, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (148, 188, 688)-net over F3, using
(186−36, 186, 2446)-Net over F3 — Digital
Digital (150, 186, 2446)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3186, 2446, F3, 36) (dual of [2446, 2260, 37]-code), using
- 241 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 21 times 0, 1, 27 times 0, 1, 34 times 0, 1, 42 times 0, 1, 51 times 0) [i] based on linear OA(3168, 2187, F3, 36) (dual of [2187, 2019, 37]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
- 241 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 21 times 0, 1, 27 times 0, 1, 34 times 0, 1, 42 times 0, 1, 51 times 0) [i] based on linear OA(3168, 2187, F3, 36) (dual of [2187, 2019, 37]-code), using
(186−36, 186, 321609)-Net in Base 3 — Upper bound on s
There is no (150, 186, 321610)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 55533 627364 043440 208299 905257 710819 745960 834114 640680 363492 349960 418134 555234 975407 055141 > 3186 [i]