Best Known (21−7, 21, s)-Nets in Base 3
(21−7, 21, 84)-Net over F3 — Constructive and digital
Digital (14, 21, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 7, 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
(21−7, 21, 122)-Net over F3 — Digital
Digital (14, 21, 122)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(321, 122, F3, 2, 7) (dual of [(122, 2), 223, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(321, 244, F3, 7) (dual of [244, 223, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(321, 244, F3, 7) (dual of [244, 223, 8]-code), using
(21−7, 21, 1375)-Net in Base 3 — Upper bound on s
There is no (14, 21, 1376)-net in base 3, because
- 1 times m-reduction [i] would yield (14, 20, 1376)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3492 667777 > 320 [i]