Best Known (74−35, 74, s)-Nets in Base 81
(74−35, 74, 730)-Net over F81 — Constructive and digital
Digital (39, 74, 730)-net over F81, using
- t-expansion [i] based on digital (36, 74, 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
(74−35, 74, 3132)-Net over F81 — Digital
Digital (39, 74, 3132)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8174, 3132, F81, 2, 35) (dual of [(3132, 2), 6190, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8174, 3289, F81, 2, 35) (dual of [(3289, 2), 6504, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8174, 6578, F81, 35) (dual of [6578, 6504, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(8174, 6579, F81, 35) (dual of [6579, 6505, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- linear OA(8169, 6562, F81, 35) (dual of [6562, 6493, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(8157, 6562, F81, 29) (dual of [6562, 6505, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8174, 6579, F81, 35) (dual of [6579, 6505, 36]-code), using
- OOA 2-folding [i] based on linear OA(8174, 6578, F81, 35) (dual of [6578, 6504, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(8174, 3289, F81, 2, 35) (dual of [(3289, 2), 6504, 36]-NRT-code), using
(74−35, 74, large)-Net in Base 81 — Upper bound on s
There is no (39, 74, large)-net in base 81, because
- 33 times m-reduction [i] would yield (39, 41, large)-net in base 81, but