Best Known (186, 186+41, s)-Nets in Base 3
(186, 186+41, 896)-Net over F3 — Constructive and digital
Digital (186, 227, 896)-net over F3, using
- t-expansion [i] based on digital (184, 227, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (184, 228, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 57, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 57, 224)-net over F81, using
- 1 times m-reduction [i] based on digital (184, 228, 896)-net over F3, using
(186, 186+41, 4443)-Net over F3 — Digital
Digital (186, 227, 4443)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 4443, F3, 41) (dual of [4443, 4216, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 6597, F3, 41) (dual of [6597, 6370, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3227, 6597, F3, 41) (dual of [6597, 6370, 42]-code), using
(186, 186+41, 1022676)-Net in Base 3 — Upper bound on s
There is no (186, 227, 1022677)-net in base 3, because
- 1 times m-reduction [i] would yield (186, 226, 1022677)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 675166 193070 156773 747794 550442 873557 883006 214683 909077 922152 845463 839432 696626 275726 689855 616352 282377 156241 > 3226 [i]