Best Known (65, 81, s)-Nets in Base 8
(65, 81, 4120)-Net over F8 — Constructive and digital
Digital (65, 81, 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, 70, 4096)-net over F8, using
- net defined by OOA [i] based on linear OOA(870, 4096, F8, 16, 16) (dual of [(4096, 16), 65466, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(870, 32768, F8, 16) (dual of [32768, 32698, 17]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(870, 32768, F8, 16) (dual of [32768, 32698, 17]-code), using
- net defined by OOA [i] based on linear OOA(870, 4096, F8, 16, 16) (dual of [(4096, 16), 65466, 17]-NRT-code), using
- digital (3, 11, 24)-net over F8, using
(65, 81, 8192)-Net in Base 8 — Constructive
(65, 81, 8192)-net in base 8, using
- net defined by OOA [i] based on OOA(881, 8192, S8, 16, 16), using
- OA 8-folding and stacking [i] based on OA(881, 65536, S8, 16), using
- 1 times code embedding in larger space [i] based on OA(880, 65535, S8, 16), using
- discarding parts of the base [i] based on linear OA(1660, 65535, F16, 16) (dual of [65535, 65475, 17]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using
- discarding parts of the base [i] based on linear OA(1660, 65535, F16, 16) (dual of [65535, 65475, 17]-code), using
- 1 times code embedding in larger space [i] based on OA(880, 65535, S8, 16), using
- OA 8-folding and stacking [i] based on OA(881, 65536, S8, 16), using
(65, 81, 69088)-Net over F8 — Digital
Digital (65, 81, 69088)-net over F8, using
(65, 81, large)-Net in Base 8 — Upper bound on s
There is no (65, 81, large)-net in base 8, because
- 14 times m-reduction [i] would yield (65, 67, large)-net in base 8, but