Best Known (6, 6+9, s)-Nets in Base 27
(6, 6+9, 76)-Net over F27 — Constructive and digital
Digital (6, 15, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
(6, 6+9, 92)-Net over F27 — Digital
Digital (6, 15, 92)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2715, 92, F27, 9) (dual of [92, 77, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2715, 104, F27, 9) (dual of [104, 89, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 104 | 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(2715, 104, F27, 9) (dual of [104, 89, 10]-code), using
(6, 6+9, 116)-Net in Base 27 — Constructive
(6, 15, 116)-net in base 27, using
- 1 times m-reduction [i] based on (6, 16, 116)-net in base 27, using
- base change [i] based on digital (2, 12, 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, 12, 116)-net over F81, using
(6, 6+9, 118)-Net in Base 27
(6, 15, 118)-net in base 27, using
- 1 times m-reduction [i] based on (6, 16, 118)-net in base 27, using
- base change [i] based on digital (2, 12, 118)-net over F81, using
- net from sequence [i] based on digital (2, 117)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 118, using
- net from sequence [i] based on digital (2, 117)-sequence over F81, using
- base change [i] based on digital (2, 12, 118)-net over F81, using
(6, 6+9, 8705)-Net in Base 27 — Upper bound on s
There is no (6, 15, 8706)-net in base 27, because
- 1 times m-reduction [i] would yield (6, 14, 8706)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 109 467892 108799 580201 > 2714 [i]