Best Known (39−15, 39, s)-Nets in Base 27
(39−15, 39, 252)-Net over F27 — Constructive and digital
Digital (24, 39, 252)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- 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, 3, 28)-net 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, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 15, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 1, 28)-net over F27, using
(39−15, 39, 937)-Net in Base 27 — Constructive
(24, 39, 937)-net in base 27, using
- net defined by OOA [i] based on OOA(2739, 937, S27, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(2739, 6560, S27, 15), using
- discarding factors based on OA(2739, 6563, S27, 15), using
- discarding parts of the base [i] based on linear OA(8129, 6563, F81, 15) (dual of [6563, 6534, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(8129, 6561, F81, 15) (dual of [6561, 6532, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(8129, 6563, F81, 15) (dual of [6563, 6534, 16]-code), using
- discarding factors based on OA(2739, 6563, S27, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(2739, 6560, S27, 15), using
(39−15, 39, 2266)-Net over F27 — Digital
Digital (24, 39, 2266)-net over F27, using
(39−15, 39, 7659590)-Net in Base 27 — Upper bound on s
There is no (24, 39, 7659591)-net in base 27, because
- 1 times m-reduction [i] would yield (24, 38, 7659591)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 2 465035 954421 206094 250770 491556 105730 136048 981352 133675 > 2738 [i]