Best Known (86−42, 86, s)-Nets in Base 27
(86−42, 86, 192)-Net over F27 — Constructive and digital
Digital (44, 86, 192)-net over F27, using
- 2 times m-reduction [i] based on digital (44, 88, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 33, 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, 55, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 33, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(86−42, 86, 370)-Net in Base 27 — Constructive
(44, 86, 370)-net in base 27, using
- t-expansion [i] based on (43, 86, 370)-net in base 27, using
- 22 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
- 22 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(86−42, 86, 649)-Net over F27 — Digital
Digital (44, 86, 649)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2786, 649, F27, 42) (dual of [649, 563, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(2786, 749, F27, 42) (dual of [749, 663, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(34) [i] based on
- linear OA(2780, 729, F27, 42) (dual of [729, 649, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(2766, 729, F27, 35) (dual of [729, 663, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(276, 20, F27, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,27)), using
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- Reed–Solomon code RS(21,27) [i]
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- construction X applied to Ce(41) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(2786, 749, F27, 42) (dual of [749, 663, 43]-code), using
(86−42, 86, 242814)-Net in Base 27 — Upper bound on s
There is no (44, 86, 242815)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 1251 088617 904042 052956 234193 355623 251995 539751 369146 628361 155784 858850 208193 539622 273814 358063 070979 557973 686012 843739 311839 > 2786 [i]