Best Known (172, 212, s)-Nets in Base 3
(172, 212, 896)-Net over F3 — Constructive and digital
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
(172, 212, 3314)-Net over F3 — Digital
Digital (172, 212, 3314)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 3314, F3, 40) (dual of [3314, 3102, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 6574, F3, 40) (dual of [6574, 6362, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(36) [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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(39) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3212, 6574, F3, 40) (dual of [6574, 6362, 41]-code), using
(172, 212, 473962)-Net in Base 3 — Upper bound on s
There is no (172, 212, 473963)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 141163 180984 825756 704888 049320 306891 163076 012227 500662 585534 136651 684257 503443 546434 027191 285342 799289 > 3212 [i]