Best Known (128, 128+33, s)-Nets in Base 3
(128, 128+33, 688)-Net over F3 — Constructive and digital
Digital (128, 161, 688)-net over F3, using
- 31 times duplication [i] based on digital (127, 160, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 40, 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, 40, 172)-net over F81, using
(128, 128+33, 1772)-Net over F3 — Digital
Digital (128, 161, 1772)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3161, 1772, F3, 33) (dual of [1772, 1611, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 2214, F3, 33) (dual of [2214, 2053, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [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]
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3161, 2214, F3, 33) (dual of [2214, 2053, 34]-code), using
(128, 128+33, 200764)-Net in Base 3 — Upper bound on s
There is no (128, 161, 200765)-net in base 3, because
- 1 times m-reduction [i] would yield (128, 160, 200765)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21847 654644 889627 449171 383959 164437 225115 625689 415982 544988 480143 620638 823361 > 3160 [i]