Best Known (174, 174+38, s)-Nets in Base 3
(174, 174+38, 896)-Net over F3 — Constructive and digital
Digital (174, 212, 896)-net over F3, using
- t-expansion [i] based on digital (172, 212, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 53, 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, 53, 224)-net over F81, using
(174, 174+38, 4434)-Net over F3 — Digital
Digital (174, 212, 4434)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 4434, F3, 38) (dual of [4434, 4222, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 6602, F3, 38) (dual of [6602, 6390, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(311, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3212, 6602, F3, 38) (dual of [6602, 6390, 39]-code), using
(174, 174+38, 835304)-Net in Base 3 — Upper bound on s
There is no (174, 212, 835305)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 141160 954163 978256 154983 726002 307943 882019 035396 378885 151932 155120 327971 416092 262765 761356 646638 409147 > 3212 [i]