Best Known (40, 40+36, s)-Nets in Base 27
(40, 40+36, 192)-Net over F27 — Constructive and digital
Digital (40, 76, 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, 47, 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
(40, 40+36, 370)-Net in Base 27 — Constructive
(40, 76, 370)-net in base 27, using
- 20 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
(40, 40+36, 732)-Net over F27 — Digital
Digital (40, 76, 732)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2776, 732, F27, 36) (dual of [732, 656, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(2776, 753, F27, 36) (dual of [753, 677, 37]-code), using
- construction XX applied to C1 = C([722,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([722,29]) [i] based on
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−6,−5,…,28}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2753, 728, F27, 27) (dual of [728, 675, 28]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {3,4,…,29}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−6,−5,…,29}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2751, 728, F27, 26) (dual of [728, 677, 27]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {3,4,…,28}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(278, 23, F27, 8) (dual of [23, 15, 9]-code or 23-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
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([722,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([722,29]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2776, 753, F27, 36) (dual of [753, 677, 37]-code), using
(40, 40+36, 321129)-Net in Base 27 — Upper bound on s
There is no (40, 76, 321130)-net in base 27, because
- 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]