Best Known (66−32, 66, s)-Nets in Base 81
(66−32, 66, 486)-Net over F81 — Constructive and digital
Digital (34, 66, 486)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 18, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- digital (16, 48, 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
- digital (2, 18, 116)-net over F81, using
(66−32, 66, 2362)-Net over F81 — Digital
Digital (34, 66, 2362)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8166, 2362, F81, 2, 32) (dual of [(2362, 2), 4658, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8166, 3286, F81, 2, 32) (dual of [(3286, 2), 6506, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8166, 6572, F81, 32) (dual of [6572, 6506, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(8166, 6572, F81, 32) (dual of [6572, 6506, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(8166, 3286, F81, 2, 32) (dual of [(3286, 2), 6506, 33]-NRT-code), using
(66−32, 66, 6337951)-Net in Base 81 — Upper bound on s
There is no (34, 66, 6337952)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 912035 561576 044206 730954 162593 974462 088645 539447 202053 306127 443899 440081 966686 729445 168550 167029 118034 131597 251639 382260 162561 > 8166 [i]