Best Known (116, 116+38, s)-Nets in Base 3
(116, 116+38, 328)-Net over F3 — Constructive and digital
Digital (116, 154, 328)-net over F3, using
- 32 times duplication [i] based on digital (114, 152, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 38, 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, 38, 82)-net over F81, using
(116, 116+38, 730)-Net over F3 — Digital
Digital (116, 154, 730)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3154, 730, F3, 38) (dual of [730, 576, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 753, F3, 38) (dual of [753, 599, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(3148, 729, F3, 38) (dual of [729, 581, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3130, 729, F3, 34) (dual of [729, 599, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(36, 24, F3, 3) (dual of [24, 18, 4]-code or 24-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(37) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3154, 753, F3, 38) (dual of [753, 599, 39]-code), using
(116, 116+38, 29181)-Net in Base 3 — Upper bound on s
There is no (116, 154, 29182)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 29 983987 030009 925079 419681 353610 677340 672056 336913 203955 385304 645688 138337 > 3154 [i]