Best Known (173−34, 173, s)-Nets in Base 3
(173−34, 173, 688)-Net over F3 — Constructive and digital
Digital (139, 173, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (139, 176, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 172)-net over F81, using
(173−34, 173, 2310)-Net over F3 — Digital
Digital (139, 173, 2310)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3173, 2310, F3, 34) (dual of [2310, 2137, 35]-code), using
- 105 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 11 times 0, 1, 14 times 0, 1, 18 times 0, 1, 23 times 0) [i] based on linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 105 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 11 times 0, 1, 14 times 0, 1, 18 times 0, 1, 23 times 0) [i] based on linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using
(173−34, 173, 257215)-Net in Base 3 — Upper bound on s
There is no (139, 173, 257216)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 34832 179872 885459 961587 087981 814842 033952 499599 091863 739627 630131 605558 938421 125505 > 3173 [i]