Best Known (21−12, 21, s)-Nets in Base 27
(21−12, 21, 88)-Net over F27 — Constructive and digital
Digital (9, 21, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
(21−12, 21, 109)-Net over F27 — Digital
Digital (9, 21, 109)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2721, 109, F27, 12) (dual of [109, 88, 13]-code), using
- 16 step Varšamov–Edel lengthening with (ri) = (1, 15 times 0) [i] based on linear OA(2720, 92, F27, 12) (dual of [92, 72, 13]-code), using
- extended algebraic-geometric code AGe(F,79P) [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 92, using
- 16 step Varšamov–Edel lengthening with (ri) = (1, 15 times 0) [i] based on linear OA(2720, 92, F27, 12) (dual of [92, 72, 13]-code), using
(21−12, 21, 116)-Net in Base 27 — Constructive
(9, 21, 116)-net in base 27, using
- 7 times m-reduction [i] based on (9, 28, 116)-net in base 27, using
- base change [i] based on digital (2, 21, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 21, 116)-net over F81, using
(21−12, 21, 136)-Net in Base 27
(9, 21, 136)-net in base 27, using
- 3 times m-reduction [i] based on (9, 24, 136)-net in base 27, using
- base change [i] based on digital (3, 18, 136)-net over F81, using
- net from sequence [i] based on digital (3, 135)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 3 and N(F) ≥ 136, using
- net from sequence [i] based on digital (3, 135)-sequence over F81, using
- base change [i] based on digital (3, 18, 136)-net over F81, using
(21−12, 21, 11773)-Net in Base 27 — Upper bound on s
There is no (9, 21, 11774)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 1 144606 038267 792705 679618 169445 > 2721 [i]