Best Known (141−29, 141, s)-Nets in Base 3
(141−29, 141, 640)-Net over F3 — Constructive and digital
Digital (112, 141, 640)-net over F3, using
- 31 times duplication [i] based on digital (111, 140, 640)-net over F3, using
- t-expansion [i] based on digital (110, 140, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 35, 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, 35, 160)-net over F81, using
- t-expansion [i] based on digital (110, 140, 640)-net over F3, using
(141−29, 141, 1602)-Net over F3 — Digital
Digital (112, 141, 1602)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 1602, F3, 29) (dual of [1602, 1461, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 2215, F3, 29) (dual of [2215, 2074, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3140, 2214, F3, 29) (dual of [2214, 2074, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(28) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3140, 2214, F3, 29) (dual of [2214, 2074, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 2215, F3, 29) (dual of [2215, 2074, 30]-code), using
(141−29, 141, 178487)-Net in Base 3 — Upper bound on s
There is no (112, 141, 178488)-net in base 3, because
- 1 times m-reduction [i] would yield (112, 140, 178488)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 266106 214594 626688 628822 783866 297985 354270 074217 065218 437858 794193 > 3140 [i]