Best Known (43, 84, s)-Nets in Base 27
(43, 84, 192)-Net over F27 — Constructive and digital
Digital (43, 84, 192)-net over F27, using
- 1 times m-reduction [i] based on digital (43, 85, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 32, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (11, 53, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 32, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(43, 84, 370)-Net in Base 27 — Constructive
(43, 84, 370)-net in base 27, using
- 24 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
(43, 84, 641)-Net over F27 — Digital
Digital (43, 84, 641)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2784, 641, F27, 41) (dual of [641, 557, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(2784, 741, F27, 41) (dual of [741, 657, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- linear OA(2781, 730, F27, 41) (dual of [730, 649, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(2773, 730, F27, 37) (dual of [730, 657, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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([0,20]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2784, 741, F27, 41) (dual of [741, 657, 42]-code), using
(43, 84, 278276)-Net in Base 27 — Upper bound on s
There is no (43, 84, 278277)-net in base 27, because
- 1 times m-reduction [i] would yield (43, 83, 278277)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 63563 693287 982487 686222 779119 877089 876019 157826 994719 657972 694189 830674 738917 020298 577890 375489 342514 345081 408889 174321 > 2783 [i]