Best Known (54−20, 54, s)-Nets in Base 27
(54−20, 54, 252)-Net over F27 — Constructive and digital
Digital (34, 54, 252)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 3, 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
- digital (0, 4, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 5, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 10, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 20, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
(54−20, 54, 656)-Net in Base 27 — Constructive
(34, 54, 656)-net in base 27, using
- 1 times m-reduction [i] based on (34, 55, 656)-net in base 27, using
- net defined by OOA [i] based on OOA(2755, 656, S27, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(2755, 6561, S27, 21), using
- discarding factors based on OA(2755, 6563, S27, 21), using
- discarding parts of the base [i] based on linear OA(8141, 6563, F81, 21) (dual of [6563, 6522, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(8141, 6563, F81, 21) (dual of [6563, 6522, 22]-code), using
- discarding factors based on OA(2755, 6563, S27, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(2755, 6561, S27, 21), using
- net defined by OOA [i] based on OOA(2755, 656, S27, 21, 21), using
(54−20, 54, 3577)-Net over F27 — Digital
Digital (34, 54, 3577)-net over F27, using
(54−20, 54, large)-Net in Base 27 — Upper bound on s
There is no (34, 54, large)-net in base 27, because
- 18 times m-reduction [i] would yield (34, 36, large)-net in base 27, but