Best Known (25, 46, s)-Nets in Base 27
(25, 46, 166)-Net over F27 — Constructive and digital
Digital (25, 46, 166)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (8, 29, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (7, 17, 82)-net over F27, using
(25, 46, 224)-Net in Base 27 — Constructive
(25, 46, 224)-net in base 27, using
- 2 times m-reduction [i] based on (25, 48, 224)-net in base 27, using
- base change [i] based on digital (13, 36, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 36, 224)-net over F81, using
(25, 46, 740)-Net over F27 — Digital
Digital (25, 46, 740)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2746, 740, F27, 21) (dual of [740, 694, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2746, 747, F27, 21) (dual of [747, 701, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(2741, 730, F27, 21) (dual of [730, 689, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2729, 730, F27, 15) (dual of [730, 701, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(275, 17, F27, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2746, 747, F27, 21) (dual of [747, 701, 22]-code), using
(25, 46, 480989)-Net in Base 27 — Upper bound on s
There is no (25, 46, 480990)-net in base 27, because
- 1 times m-reduction [i] would yield (25, 45, 480990)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 25785 468314 426527 688374 386126 115002 961621 634004 286857 073069 576973 > 2745 [i]