Best Known (80−38, 80, s)-Nets in Base 27
(80−38, 80, 192)-Net over F27 — Constructive and digital
Digital (42, 80, 192)-net over F27, using
- 2 times m-reduction [i] based on digital (42, 82, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 31, 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, 51, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 31, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(80−38, 80, 370)-Net in Base 27 — Constructive
(42, 80, 370)-net in base 27, using
- 24 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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, 78, 370)-net over F81, using
(80−38, 80, 743)-Net over F27 — Digital
Digital (42, 80, 743)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2780, 743, F27, 38) (dual of [743, 663, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2780, 755, F27, 38) (dual of [755, 675, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- linear OA(2772, 729, F27, 38) (dual of [729, 657, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2754, 729, F27, 29) (dual of [729, 675, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(278, 26, F27, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,27)), using
- discarding factors / shortening the dual code based on linear OA(278, 27, F27, 8) (dual of [27, 19, 9]-code or 27-arc in PG(7,27)), using
- Reed–Solomon code RS(19,27) [i]
- discarding factors / shortening the dual code based on linear OA(278, 27, F27, 8) (dual of [27, 19, 9]-code or 27-arc in PG(7,27)), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2780, 755, F27, 38) (dual of [755, 675, 39]-code), using
(80−38, 80, 324357)-Net in Base 27 — Upper bound on s
There is no (42, 80, 324358)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 3 229343 161809 755365 480299 308471 052923 351736 547773 156269 037962 000324 127652 065537 731148 973746 706236 721965 458337 191505 > 2780 [i]