Best Known (161−30, 161, s)-Nets in Base 3
(161−30, 161, 688)-Net over F3 — Constructive and digital
Digital (131, 161, 688)-net over F3, using
- t-expansion [i] based on digital (130, 161, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (130, 164, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 41, 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, 41, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (130, 164, 688)-net over F3, using
(161−30, 161, 3284)-Net over F3 — Digital
Digital (131, 161, 3284)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3161, 3284, F3, 2, 30) (dual of [(3284, 2), 6407, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3161, 6568, F3, 30) (dual of [6568, 6407, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 6569, F3, 30) (dual of [6569, 6408, 31]-code), using
- 1 times truncation [i] based on linear OA(3162, 6570, F3, 31) (dual of [6570, 6408, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(31, 9, F3, 1) (dual of [9, 8, 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
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(3162, 6570, F3, 31) (dual of [6570, 6408, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 6569, F3, 30) (dual of [6569, 6408, 31]-code), using
- OOA 2-folding [i] based on linear OA(3161, 6568, F3, 30) (dual of [6568, 6407, 31]-code), using
(161−30, 161, 424446)-Net in Base 3 — Upper bound on s
There is no (131, 161, 424447)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 65542 784505 087889 342393 517918 136562 711099 542903 776960 871645 225283 116096 481251 > 3161 [i]