Best Known (82−17, 82, s)-Nets in Base 8
(82−17, 82, 4120)-Net over F8 — Constructive and digital
Digital (65, 82, 4120)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- digital (54, 71, 4096)-net over F8, using
- net defined by OOA [i] based on linear OOA(871, 4096, F8, 17, 17) (dual of [(4096, 17), 69561, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using
- net defined by OOA [i] based on linear OOA(871, 4096, F8, 17, 17) (dual of [(4096, 17), 69561, 18]-NRT-code), using
- digital (3, 11, 24)-net over F8, using
(82−17, 82, 8191)-Net in Base 8 — Constructive
(65, 82, 8191)-net in base 8, using
- net defined by OOA [i] based on OOA(882, 8191, S8, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(882, 65529, S8, 17), using
- discarding factors based on OA(882, 65536, S8, 17), using
- discarding parts of the base [i] based on linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding parts of the base [i] based on linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using
- discarding factors based on OA(882, 65536, S8, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(882, 65529, S8, 17), using
(82−17, 82, 41291)-Net over F8 — Digital
Digital (65, 82, 41291)-net over F8, using
(82−17, 82, large)-Net in Base 8 — Upper bound on s
There is no (65, 82, large)-net in base 8, because
- 15 times m-reduction [i] would yield (65, 67, large)-net in base 8, but