Best Known (133, 133+37, s)-Nets in Base 3
(133, 133+37, 640)-Net over F3 — Constructive and digital
Digital (133, 170, 640)-net over F3, using
- 32 times duplication [i] based on digital (131, 168, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 42, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 42, 160)-net over F81, using
(133, 133+37, 1368)-Net over F3 — Digital
Digital (133, 170, 1368)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3170, 1368, F3, 37) (dual of [1368, 1198, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 2195, F3, 37) (dual of [2195, 2025, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [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(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(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3170, 2195, F3, 37) (dual of [2195, 2025, 38]-code), using
(133, 133+37, 113938)-Net in Base 3 — Upper bound on s
There is no (133, 170, 113939)-net in base 3, because
- 1 times m-reduction [i] would yield (133, 169, 113939)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 430 033869 466418 047740 684683 370308 527223 073593 341292 186015 223806 966552 669654 103205 > 3169 [i]