Best Known (234−44, 234, s)-Nets in Base 3
(234−44, 234, 896)-Net over F3 — Constructive and digital
Digital (190, 234, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (190, 236, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 59, 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, 59, 224)-net over F81, using
(234−44, 234, 3622)-Net over F3 — Digital
Digital (190, 234, 3622)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 3622, F3, 44) (dual of [3622, 3388, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 6570, F3, 44) (dual of [6570, 6336, 45]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3233, 6569, F3, 44) (dual of [6569, 6336, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3233, 6561, F3, 44) (dual of [6561, 6328, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3225, 6561, F3, 43) (dual of [6561, 6336, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3233, 6569, F3, 44) (dual of [6569, 6336, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 6570, F3, 44) (dual of [6570, 6336, 45]-code), using
(234−44, 234, 537824)-Net in Base 3 — Upper bound on s
There is no (190, 234, 537825)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4429 726736 113161 141225 924737 100140 727167 660992 640676 382982 465549 698502 959692 540518 490575 252407 978130 594979 759321 > 3234 [i]