Best Known (176−37, 176, s)-Nets in Base 3
(176−37, 176, 688)-Net over F3 — Constructive and digital
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
(176−37, 176, 1658)-Net over F3 — Digital
Digital (139, 176, 1658)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3176, 1658, F3, 37) (dual of [1658, 1482, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3176, 2214, F3, 37) (dual of [2214, 2038, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3148, 2187, F3, 32) (dual of [2187, 2039, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3176, 2214, F3, 37) (dual of [2214, 2038, 38]-code), using
(176−37, 176, 164335)-Net in Base 3 — Upper bound on s
There is no (139, 176, 164336)-net in base 3, because
- 1 times m-reduction [i] would yield (139, 175, 164336)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 313487 401274 417268 656866 413134 476882 916217 770779 254747 227483 108431 078479 597311 915937 > 3175 [i]