Best Known (73−35, 73, s)-Nets in Base 27
(73−35, 73, 190)-Net over F27 — Constructive and digital
Digital (38, 73, 190)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 27, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (11, 46, 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 (10, 27, 94)-net over F27, using
(73−35, 73, 370)-Net in Base 27 — Constructive
(38, 73, 370)-net in base 27, using
- 15 times m-reduction [i] based on (38, 88, 370)-net in base 27, using
- base change [i] based on digital (16, 66, 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, 66, 370)-net over F81, using
(73−35, 73, 657)-Net over F27 — Digital
Digital (38, 73, 657)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2773, 657, F27, 35) (dual of [657, 584, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2773, 750, F27, 35) (dual of [750, 677, 36]-code), using
- construction XX applied to C1 = C([723,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([723,29]) [i] based on
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−5,−4,…,28}, and designed minimum distance d ≥ |I|+1 = 35 [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(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 = {−5,−4,…,29}, and designed minimum distance d ≥ |I|+1 = 36 [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(277, 20, F27, 7) (dual of [20, 13, 8]-code or 20-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
- 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([723,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([723,29]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2773, 750, F27, 35) (dual of [750, 677, 36]-code), using
(73−35, 73, 318571)-Net in Base 27 — Upper bound on s
There is no (38, 73, 318572)-net in base 27, because
- 1 times m-reduction [i] would yield (38, 72, 318572)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 11 434263 246503 520182 392509 114025 144203 500450 698893 149862 693492 558841 632601 870018 026421 765299 344216 669433 > 2772 [i]