Best Known (37−12, 37, s)-Nets in Base 3
(37−12, 37, 84)-Net over F3 — Constructive and digital
Digital (25, 37, 84)-net over F3, using
- 31 times duplication [i] based on digital (24, 36, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 12, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- trace code for nets [i] based on digital (0, 12, 28)-net over F27, using
(37−12, 37, 111)-Net over F3 — Digital
Digital (25, 37, 111)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(337, 111, F3, 12) (dual of [111, 74, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(337, 113, F3, 12) (dual of [113, 76, 13]-code), using
- 7 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0) [i] based on linear OA(333, 102, F3, 12) (dual of [102, 69, 13]-code), using
- 1 times truncation [i] based on linear OA(334, 103, F3, 13) (dual of [103, 69, 14]-code), using
- a “Glo†code from Brouwer’s database [i]
- 1 times truncation [i] based on linear OA(334, 103, F3, 13) (dual of [103, 69, 14]-code), using
- 7 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0) [i] based on linear OA(333, 102, F3, 12) (dual of [102, 69, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(337, 113, F3, 12) (dual of [113, 76, 13]-code), using
(37−12, 37, 1305)-Net in Base 3 — Upper bound on s
There is no (25, 37, 1306)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 452311 915690 279245 > 337 [i]