Best Known (175−34, 175, s)-Nets in Base 3
(175−34, 175, 688)-Net over F3 — Constructive and digital
Digital (141, 175, 688)-net over F3, using
- t-expansion [i] based on digital (139, 175, 688)-net over F3, using
- 1 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
- 1 times m-reduction [i] based on digital (139, 176, 688)-net over F3, using
(175−34, 175, 2377)-Net over F3 — Digital
Digital (141, 175, 2377)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3175, 2377, F3, 34) (dual of [2377, 2202, 35]-code), using
- 170 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, 1, 29 times 0, 1, 34 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]
- 170 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, 1, 29 times 0, 1, 34 times 0) [i] based on linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using
(175−34, 175, 292706)-Net in Base 3 — Upper bound on s
There is no (141, 175, 292707)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 313487 636951 804523 148453 381291 586230 811636 676935 244700 801291 769387 690614 185048 370471 > 3175 [i]