Best Known (71−34, 71, s)-Nets in Base 81
(71−34, 71, 730)-Net over F81 — Constructive and digital
Digital (37, 71, 730)-net over F81, using
- t-expansion [i] based on digital (36, 71, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- the Hermitian function field over F81 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(71−34, 71, 2732)-Net over F81 — Digital
Digital (37, 71, 2732)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8171, 2732, F81, 2, 34) (dual of [(2732, 2), 5393, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8171, 3287, F81, 2, 34) (dual of [(3287, 2), 6503, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8171, 6574, F81, 34) (dual of [6574, 6503, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(8171, 6575, F81, 34) (dual of [6575, 6504, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(8171, 6575, F81, 34) (dual of [6575, 6504, 35]-code), using
- OOA 2-folding [i] based on linear OA(8171, 6574, F81, 34) (dual of [6574, 6503, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(8171, 3287, F81, 2, 34) (dual of [(3287, 2), 6503, 35]-NRT-code), using
(71−34, 71, 8386607)-Net in Base 81 — Upper bound on s
There is no (37, 71, 8386608)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 3180 067528 790705 285235 444160 720867 884107 085123 131777 749028 515946 102623 716328 233970 890261 148999 922227 610575 730804 012893 523840 611690 561281 > 8171 [i]