Best Known (83−16, 83, s)-Nets in Base 8
(83−16, 83, 4124)-Net over F8 — Constructive and digital
Digital (67, 83, 4124)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-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 (5, 13, 28)-net over F8, using
(83−16, 83, 16384)-Net in Base 8 — Constructive
(67, 83, 16384)-net in base 8, using
- net defined by OOA [i] based on OOA(883, 16384, S8, 16, 16), using
- OA 8-folding and stacking [i] based on OA(883, 131072, S8, 16), using
- discarding factors based on OA(883, 131076, S8, 16), using
- discarding parts of the base [i] based on linear OA(1662, 131076, F16, 16) (dual of [131076, 131014, 17]-code), using
- trace code [i] based on linear OA(25631, 65538, F256, 16) (dual of [65538, 65507, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- trace code [i] based on linear OA(25631, 65538, F256, 16) (dual of [65538, 65507, 17]-code), using
- discarding parts of the base [i] based on linear OA(1662, 131076, F16, 16) (dual of [131076, 131014, 17]-code), using
- discarding factors based on OA(883, 131076, S8, 16), using
- OA 8-folding and stacking [i] based on OA(883, 131072, S8, 16), using
(83−16, 83, 91159)-Net over F8 — Digital
Digital (67, 83, 91159)-net over F8, using
(83−16, 83, large)-Net in Base 8 — Upper bound on s
There is no (67, 83, large)-net in base 8, because
- 14 times m-reduction [i] would yield (67, 69, large)-net in base 8, but