Best Known (46, 89, s)-Nets in Base 27
(46, 89, 192)-Net over F27 — Constructive and digital
Digital (46, 89, 192)-net over F27, using
- 5 times m-reduction [i] based on digital (46, 94, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 35, 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, 59, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 35, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(46, 89, 370)-Net in Base 27 — Constructive
(46, 89, 370)-net in base 27, using
- t-expansion [i] based on (43, 89, 370)-net in base 27, using
- 19 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
- 19 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(46, 89, 714)-Net over F27 — Digital
Digital (46, 89, 714)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2789, 714, F27, 43) (dual of [714, 625, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(2789, 752, F27, 43) (dual of [752, 663, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(34) [i] based on
- linear OA(2782, 729, F27, 43) (dual of [729, 647, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(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(42) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(2789, 752, F27, 43) (dual of [752, 663, 44]-code), using
(46, 89, 332353)-Net in Base 27 — Upper bound on s
There is no (46, 89, 332354)-net in base 27, because
- 1 times m-reduction [i] would yield (46, 88, 332354)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 912036 197397 608983 102887 961863 380435 564541 246074 196606 921433 914437 272166 987891 765464 898854 759357 473297 322793 016826 901763 511725 > 2788 [i]