Best Known (28, 28+29, s)-Nets in Base 27
(28, 28+29, 164)-Net over F27 — Constructive and digital
Digital (28, 57, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (7, 36, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 21, 82)-net over F27, using
(28, 28+29, 224)-Net in Base 27 — Constructive
(28, 57, 224)-net in base 27, using
- 3 times m-reduction [i] based on (28, 60, 224)-net in base 27, using
- base change [i] based on digital (13, 45, 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
- base change [i] based on digital (13, 45, 224)-net over F81, using
(28, 28+29, 379)-Net over F27 — Digital
Digital (28, 57, 379)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2757, 379, F27, 29) (dual of [379, 322, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2757, 728, F27, 29) (dual of [728, 671, 30]-code), using
(28, 28+29, 123570)-Net in Base 27 — Upper bound on s
There is no (28, 57, 123571)-net in base 27, because
- 1 times m-reduction [i] would yield (28, 56, 123571)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 143 349663 855173 955890 966620 110600 749912 000112 108429 554955 339732 451458 797715 207261 > 2756 [i]