Best Known (28, 28+13, s)-Nets in Base 3
(28, 28+13, 84)-Net over F3 — Constructive and digital
Digital (28, 41, 84)-net over F3, using
- 1 times m-reduction [i] based on digital (28, 42, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 28)-net over F27, using
(28, 28+13, 125)-Net over F3 — Digital
Digital (28, 41, 125)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(341, 125, F3, 13) (dual of [125, 84, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 242, F3, 13) (dual of [242, 201, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(341, 242, F3, 13) (dual of [242, 201, 14]-code), using
(28, 28+13, 2264)-Net in Base 3 — Upper bound on s
There is no (28, 41, 2265)-net in base 3, because
- 1 times m-reduction [i] would yield (28, 40, 2265)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 12 177835 352375 642697 > 340 [i]