Best Known (163, 163+44, s)-Nets in Base 3
(163, 163+44, 688)-Net over F3 — Constructive and digital
Digital (163, 207, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (163, 208, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 52, 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, 52, 172)-net over F81, using
(163, 163+44, 1768)-Net over F3 — Digital
Digital (163, 207, 1768)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3207, 1768, F3, 44) (dual of [1768, 1561, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3207, 2200, F3, 44) (dual of [2200, 1993, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(40) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(43) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3207, 2200, F3, 44) (dual of [2200, 1993, 45]-code), using
(163, 163+44, 139647)-Net in Base 3 — Upper bound on s
There is no (163, 207, 139648)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 580 935535 540405 664647 370360 289459 849513 077572 622943 694778 025915 898617 526115 323976 086624 122695 158017 > 3207 [i]