Best Known (158−30, 158, s)-Nets in Base 3
(158−30, 158, 688)-Net over F3 — Constructive and digital
Digital (128, 158, 688)-net over F3, using
- t-expansion [i] based on digital (127, 158, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (127, 160, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 40, 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, 40, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (127, 160, 688)-net over F3, using
(158−30, 158, 2425)-Net over F3 — Digital
Digital (128, 158, 2425)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3158, 2425, F3, 30) (dual of [2425, 2267, 31]-code), using
- 220 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0, 1, 31 times 0, 1, 40 times 0, 1, 48 times 0) [i] based on linear OA(3140, 2187, F3, 30) (dual of [2187, 2047, 31]-code), using
- 1 times truncation [i] based on linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using
- 220 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0, 1, 31 times 0, 1, 40 times 0, 1, 48 times 0) [i] based on linear OA(3140, 2187, F3, 30) (dual of [2187, 2047, 31]-code), using
(158−30, 158, 340718)-Net in Base 3 — Upper bound on s
There is no (128, 158, 340719)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2427 575992 326134 060200 076392 995812 869536 778874 816049 810475 038467 172849 742755 > 3158 [i]