Best Known (140−28, 140, s)-Nets in Base 3
(140−28, 140, 688)-Net over F3 — Constructive and digital
Digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 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
(140−28, 140, 1851)-Net over F3 — Digital
Digital (112, 140, 1851)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3140, 1851, F3, 28) (dual of [1851, 1711, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3140, 2228, F3, 28) (dual of [2228, 2088, 29]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3137, 2225, F3, 28) (dual of [2225, 2088, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3137, 2225, F3, 28) (dual of [2225, 2088, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3140, 2228, F3, 28) (dual of [2228, 2088, 29]-code), using
(140−28, 140, 178487)-Net in Base 3 — Upper bound on s
There is no (112, 140, 178488)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 6 266106 214594 626688 628822 783866 297985 354270 074217 065218 437858 794193 > 3140 [i]