Best Known (64−29, 64, s)-Nets in Base 81
(64−29, 64, 530)-Net over F81 — Constructive and digital
Digital (35, 64, 530)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- digital (16, 45, 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 (5, 19, 160)-net over F81, using
(64−29, 64, 3864)-Net over F81 — Digital
Digital (35, 64, 3864)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8164, 3864, F81, 29) (dual of [3864, 3800, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 6585, F81, 29) (dual of [6585, 6521, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,10]) [i] based on
- 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(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(817, 23, F81, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,81)), using
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- Reed–Solomon code RS(74,81) [i]
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- construction X applied to C([0,14]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8164, 6585, F81, 29) (dual of [6585, 6521, 30]-code), using
(64−29, 64, large)-Net in Base 81 — Upper bound on s
There is no (35, 64, large)-net in base 81, because
- 27 times m-reduction [i] would yield (35, 37, large)-net in base 81, but