Best Known (77−37, 77, s)-Nets in Base 27
(77−37, 77, 192)-Net over F27 — Constructive and digital
Digital (40, 77, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 29, 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, 48, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 29, 96)-net over F27, using
(77−37, 77, 370)-Net in Base 27 — Constructive
(40, 77, 370)-net in base 27, using
- 19 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 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, 72, 370)-net over F81, using
(77−37, 77, 670)-Net over F27 — Digital
Digital (40, 77, 670)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2777, 670, F27, 37) (dual of [670, 593, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2777, 752, F27, 37) (dual of [752, 675, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(2770, 729, F27, 37) (dual of [729, 659, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(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(36) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2777, 752, F27, 37) (dual of [752, 675, 38]-code), using
(77−37, 77, 321129)-Net in Base 27 — Upper bound on s
There is no (40, 77, 321130)-net in base 27, because
- 1 times m-reduction [i] would yield (40, 76, 321130)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 6 076574 617743 903347 996055 053060 593272 477355 388698 264892 555134 398972 712725 997154 109512 208699 884758 805640 089893 > 2776 [i]