Best Known (87−42, 87, s)-Nets in Base 27
(87−42, 87, 192)-Net over F27 — Constructive and digital
Digital (45, 87, 192)-net over F27, using
- 4 times m-reduction [i] based on digital (45, 91, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 34, 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, 57, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 34, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(87−42, 87, 370)-Net in Base 27 — Constructive
(45, 87, 370)-net in base 27, using
- t-expansion [i] based on (43, 87, 370)-net in base 27, using
- 21 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
- 21 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(87−42, 87, 706)-Net over F27 — Digital
Digital (45, 87, 706)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2787, 706, F27, 42) (dual of [706, 619, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(2787, 752, F27, 42) (dual of [752, 665, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(33) [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(2764, 729, F27, 34) (dual of [729, 665, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(277, 23, F27, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,27)), using
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- Reed–Solomon code RS(20,27) [i]
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- construction X applied to Ce(41) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(2787, 752, F27, 42) (dual of [752, 665, 43]-code), using
(87−42, 87, 284078)-Net in Base 27 — Upper bound on s
There is no (45, 87, 284079)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 33780 424389 973835 829954 272568 845990 834545 496359 164121 333400 795222 012193 074482 840056 565570 905495 484897 937433 244263 025019 776575 > 2787 [i]