Best Known (133, 168, s)-Nets in Base 3
(133, 168, 688)-Net over F3 — Constructive and digital
Digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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
(133, 168, 1679)-Net over F3 — Digital
Digital (133, 168, 1679)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3168, 1679, F3, 35) (dual of [1679, 1511, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3168, 2214, F3, 35) (dual of [2214, 2046, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [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(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3168, 2214, F3, 35) (dual of [2214, 2046, 36]-code), using
(133, 168, 174537)-Net in Base 3 — Upper bound on s
There is no (133, 168, 174538)-net in base 3, because
- 1 times m-reduction [i] would yield (133, 167, 174538)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 47 782538 981920 303810 337096 850355 065985 308160 525886 992174 568426 990320 666091 959797 > 3167 [i]