Best Known (168−33, 168, s)-Nets in Base 3
(168−33, 168, 688)-Net over F3 — Constructive and digital
Digital (135, 168, 688)-net over F3, using
- t-expansion [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
(168−33, 168, 2275)-Net over F3 — Digital
Digital (135, 168, 2275)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3168, 2275, F3, 33) (dual of [2275, 2107, 34]-code), using
- 75 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 8 times 0, 1, 11 times 0, 1, 14 times 0, 1, 19 times 0) [i] based on linear OA(3154, 2186, F3, 33) (dual of [2186, 2032, 34]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using
- 75 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 8 times 0, 1, 11 times 0, 1, 14 times 0, 1, 19 times 0) [i] based on linear OA(3154, 2186, F3, 33) (dual of [2186, 2032, 34]-code), using
(168−33, 168, 324669)-Net in Base 3 — Upper bound on s
There is no (135, 168, 324670)-net in base 3, because
- 1 times m-reduction [i] would yield (135, 167, 324670)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 47 782254 312466 785568 059622 707067 559840 766159 223418 203858 102114 720456 800645 101985 > 3167 [i]