Best Known (214−39, 214, s)-Nets in Base 3
(214−39, 214, 896)-Net over F3 — Constructive and digital
Digital (175, 214, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (175, 216, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 54, 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, 54, 224)-net over F81, using
(214−39, 214, 4053)-Net over F3 — Digital
Digital (175, 214, 4053)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3214, 4053, F3, 39) (dual of [4053, 3839, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3214, 6585, F3, 39) (dual of [6585, 6371, 40]-code), using
- construction XX applied to Ce(39) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(31, 20, F3, 1) (dual of [20, 19, 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(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(39) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3214, 6585, F3, 39) (dual of [6585, 6371, 40]-code), using
(214−39, 214, 885027)-Net in Base 3 — Upper bound on s
There is no (175, 214, 885028)-net in base 3, because
- 1 times m-reduction [i] would yield (175, 213, 885028)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 423478 128539 884887 639594 571161 713506 967200 977589 874994 207427 018331 172492 973107 164078 385085 182966 001713 > 3213 [i]