Best Known (17, 17+15, s)-Nets in Base 27
(17, 17+15, 142)-Net over F27 — Constructive and digital
Digital (17, 32, 142)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 66)-net over F27, using
- (u, u+v)-construction [i] based on
- 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 (1, 8, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- digital (0, 3, 28)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 21, 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
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (4, 11, 66)-net over F27, using
(17, 17+15, 200)-Net in Base 27 — Constructive
(17, 32, 200)-net in base 27, using
- base change [i] based on digital (9, 24, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (1, 16, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 8, 100)-net over F81, using
- (u, u+v)-construction [i] based on
(17, 17+15, 559)-Net over F27 — Digital
Digital (17, 32, 559)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2732, 559, F27, 15) (dual of [559, 527, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2732, 741, F27, 15) (dual of [741, 709, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(2729, 730, F27, 15) (dual of [730, 701, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2721, 730, F27, 11) (dual of [730, 709, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(273, 11, F27, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2732, 741, F27, 15) (dual of [741, 709, 16]-code), using
(17, 17+15, 283685)-Net in Base 27 — Upper bound on s
There is no (17, 32, 283686)-net in base 27, because
- 1 times m-reduction [i] would yield (17, 31, 283686)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 235 658581 542219 991097 909942 832779 739006 380841 > 2731 [i]