Best Known (114, 114+28, s)-Nets in Base 3
(114, 114+28, 688)-Net over F3 — Constructive and digital
Digital (114, 142, 688)-net over F3, using
- 32 times duplication [i] based on digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 172)-net over F81, using
(114, 114+28, 2016)-Net over F3 — Digital
Digital (114, 142, 2016)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3142, 2016, F3, 28) (dual of [2016, 1874, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3142, 2231, F3, 28) (dual of [2231, 2089, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- 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(392, 2187, F3, 20) (dual of [2187, 2095, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(315, 44, F3, 7) (dual of [44, 29, 8]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3142, 2231, F3, 28) (dual of [2231, 2089, 29]-code), using
(114, 114+28, 208820)-Net in Base 3 — Upper bound on s
There is no (114, 142, 208821)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 56 395315 703433 107875 377109 730475 691910 878881 786894 710623 243860 342081 > 3142 [i]