Best Known (67−33, 67, s)-Nets in Base 27
(67−33, 67, 178)-Net over F27 — Constructive and digital
Digital (34, 67, 178)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 24, 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 (10, 43, 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 (8, 24, 84)-net over F27, using
(67−33, 67, 370)-Net in Base 27 — Constructive
(34, 67, 370)-net in base 27, using
- 5 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
- base change [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
(67−33, 67, 519)-Net over F27 — Digital
Digital (34, 67, 519)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2767, 519, F27, 33) (dual of [519, 452, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2767, 744, F27, 33) (dual of [744, 677, 34]-code), using
- construction XX applied to C1 = C([725,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([725,29]) [i] based on
- linear OA(2760, 728, F27, 32) (dual of [728, 668, 33]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,…,28}, and designed minimum distance d ≥ |I|+1 = 33 [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(2762, 728, F27, 33) (dual of [728, 666, 34]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,…,29}, and designed minimum distance d ≥ |I|+1 = 34 [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(275, 14, F27, 5) (dual of [14, 9, 6]-code or 14-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
- 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([725,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([725,29]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2767, 744, F27, 33) (dual of [744, 677, 34]-code), using
(67−33, 67, 209857)-Net in Base 27 — Upper bound on s
There is no (34, 67, 209858)-net in base 27, because
- 1 times m-reduction [i] would yield (34, 66, 209858)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 29514 209274 711671 386778 700931 711649 950845 237863 148746 329487 742367 142240 810608 881245 985146 205345 > 2766 [i]