Best Known (176−35, 176, s)-Nets in Base 3
(176−35, 176, 688)-Net over F3 — Constructive and digital
Digital (141, 176, 688)-net over F3, using
- t-expansion [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
(176−35, 176, 2200)-Net over F3 — Digital
Digital (141, 176, 2200)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3176, 2200, F3, 35) (dual of [2200, 2024, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3176, 2236, F3, 35) (dual of [2236, 2060, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(314, 49, F3, 6) (dual of [49, 35, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3176, 2236, F3, 35) (dual of [2236, 2060, 36]-code), using
(176−35, 176, 292706)-Net in Base 3 — Upper bound on s
There is no (141, 176, 292707)-net in base 3, because
- 1 times m-reduction [i] would yield (141, 175, 292707)-net in base 3, but
- 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]