Best Known (111, 111+36, s)-Nets in Base 3
(111, 111+36, 328)-Net over F3 — Constructive and digital
Digital (111, 147, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (111, 148, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 37, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 37, 82)-net over F81, using
(111, 111+36, 728)-Net over F3 — Digital
Digital (111, 147, 728)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3147, 728, F3, 36) (dual of [728, 581, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3147, 749, F3, 36) (dual of [749, 602, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- linear OA(3142, 729, F3, 37) (dual of [729, 587, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3124, 729, F3, 32) (dual of [729, 605, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3147, 749, F3, 36) (dual of [749, 602, 37]-code), using
(111, 111+36, 29739)-Net in Base 3 — Upper bound on s
There is no (111, 147, 29740)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13706 725995 292098 378478 798140 154113 834624 716691 444320 702372 786309 612841 > 3147 [i]