Best Known (131, 131+29, s)-Nets in Base 3
(131, 131+29, 696)-Net over F3 — Constructive and digital
Digital (131, 160, 696)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 16, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (115, 144, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 36, 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, 36, 172)-net over F81, using
- digital (2, 16, 8)-net over F3, using
(131, 131+29, 3499)-Net over F3 — Digital
Digital (131, 160, 3499)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3160, 3499, F3, 29) (dual of [3499, 3339, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3160, 6592, F3, 29) (dual of [6592, 6432, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3159, 6591, F3, 29) (dual of [6591, 6432, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(28) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3159, 6591, F3, 29) (dual of [6591, 6432, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3160, 6592, F3, 29) (dual of [6592, 6432, 30]-code), using
(131, 131+29, 792783)-Net in Base 3 — Upper bound on s
There is no (131, 160, 792784)-net in base 3, because
- 1 times m-reduction [i] would yield (131, 159, 792784)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7282 548644 129279 007991 966894 488146 667612 597404 682649 742077 631861 107747 884513 > 3159 [i]