Best Known (119−21, 119, s)-Nets in Base 3
(119−21, 119, 688)-Net over F3 — Constructive and digital
Digital (98, 119, 688)-net over F3, using
- t-expansion [i] based on digital (97, 119, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (97, 120, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 30, 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, 30, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (97, 120, 688)-net over F3, using
(119−21, 119, 3624)-Net over F3 — Digital
Digital (98, 119, 3624)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3119, 3624, F3, 21) (dual of [3624, 3505, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3119, 6591, F3, 21) (dual of [6591, 6472, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-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(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3119, 6591, F3, 21) (dual of [6591, 6472, 22]-code), using
(119−21, 119, 965990)-Net in Base 3 — Upper bound on s
There is no (98, 119, 965991)-net in base 3, because
- 1 times m-reduction [i] would yield (98, 118, 965991)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 199 668904 727082 944166 622476 608128 543485 179755 639397 241917 > 3118 [i]