Best Known (19, 36, s)-Nets in Base 27
(19, 36, 146)-Net over F27 — Constructive and digital
Digital (19, 36, 146)-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 (7, 24, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (4, 12, 64)-net over F27, using
(19, 36, 200)-Net in Base 27 — Constructive
(19, 36, 200)-net in base 27, using
- base change [i] based on digital (10, 27, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 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, 18, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 9, 100)-net over F81, using
- (u, u+v)-construction [i] based on
(19, 36, 534)-Net over F27 — Digital
Digital (19, 36, 534)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2736, 534, F27, 17) (dual of [534, 498, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2736, 741, F27, 17) (dual of [741, 705, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2733, 730, F27, 17) (dual of [730, 697, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2725, 730, F27, 13) (dual of [730, 705, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2736, 741, F27, 17) (dual of [741, 705, 18]-code), using
(19, 36, 264805)-Net in Base 27 — Upper bound on s
There is no (19, 36, 264806)-net in base 27, because
- 1 times m-reduction [i] would yield (19, 35, 264806)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 125 238768 220997 631397 666065 711151 709182 901535 458769 > 2735 [i]