Best Known (18, 18+17, s)-Nets in Base 27
(18, 18+17, 140)-Net over F27 — Constructive and digital
Digital (18, 35, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (6, 23, 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, 12, 64)-net over F27, using
(18, 18+17, 182)-Net in Base 27 — Constructive
(18, 35, 182)-net in base 27, using
- (u, u+v)-construction [i] based on
- (3, 11, 82)-net in base 27, using
- 1 times m-reduction [i] based on (3, 12, 82)-net in base 27, using
- base change [i] based on digital (0, 9, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base change [i] based on digital (0, 9, 82)-net over F81, using
- 1 times m-reduction [i] based on (3, 12, 82)-net in base 27, using
- (7, 24, 100)-net in base 27, using
- base change [i] based on digital (1, 18, 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
- base change [i] based on digital (1, 18, 100)-net over F81, using
- (3, 11, 82)-net in base 27, using
(18, 18+17, 427)-Net over F27 — Digital
Digital (18, 35, 427)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2735, 427, F27, 17) (dual of [427, 392, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2735, 737, F27, 17) (dual of [737, 702, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(2733, 729, F27, 17) (dual of [729, 696, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2727, 729, F27, 14) (dual of [729, 702, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2735, 737, F27, 17) (dual of [737, 702, 18]-code), using
(18, 18+17, 175389)-Net in Base 27 — Upper bound on s
There is no (18, 35, 175390)-net in base 27, because
- 1 times m-reduction [i] would yield (18, 34, 175390)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 4 638529 378674 126456 352089 277473 614263 658084 633169 > 2734 [i]