Best Known (70−34, 70, s)-Nets in Base 81
(70−34, 70, 730)-Net over F81 — Constructive and digital
Digital (36, 70, 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
(70−34, 70, 2369)-Net over F81 — Digital
Digital (36, 70, 2369)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8170, 2369, F81, 2, 34) (dual of [(2369, 2), 4668, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8170, 3286, F81, 2, 34) (dual of [(3286, 2), 6502, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8170, 6572, F81, 34) (dual of [6572, 6502, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(29) [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(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(33) ⊂ Ce(29) [i] based on
- OOA 2-folding [i] based on linear OA(8170, 6572, F81, 34) (dual of [6572, 6502, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(8170, 3286, F81, 2, 34) (dual of [(3286, 2), 6502, 35]-NRT-code), using
(70−34, 70, 6476231)-Net in Base 81 — Upper bound on s
There is no (36, 70, 6476232)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 39 260099 491629 020535 443257 268896 720848 624422 274954 956978 491639 804742 564828 030456 448134 671120 463808 978977 976455 474434 330552 013655 086721 > 8170 [i]