Best Known (68, 68+12, s)-Nets in Base 8
(68, 68+12, 349550)-Net over F8 — Constructive and digital
Digital (68, 80, 349550)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 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 (59, 71, 349526)-net over F8, using
- net defined by OOA [i] based on linear OOA(871, 349526, F8, 12, 12) (dual of [(349526, 12), 4194241, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(871, 2097156, F8, 12) (dual of [2097156, 2097085, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(871, 2097159, F8, 12) (dual of [2097159, 2097088, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(871, 2097152, F8, 12) (dual of [2097152, 2097081, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(871, 2097159, F8, 12) (dual of [2097159, 2097088, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(871, 2097156, F8, 12) (dual of [2097156, 2097085, 13]-code), using
- net defined by OOA [i] based on linear OOA(871, 349526, F8, 12, 12) (dual of [(349526, 12), 4194241, 13]-NRT-code), using
- digital (3, 9, 24)-net over F8, using
(68, 68+12, 699051)-Net in Base 8 — Constructive
(68, 80, 699051)-net in base 8, using
- net defined by OOA [i] based on OOA(880, 699051, S8, 12, 12), using
- OA 6-folding and stacking [i] based on OA(880, 4194306, S8, 12), using
- discarding factors based on OA(880, 4194310, S8, 12), using
- trace code [i] based on OA(6440, 2097155, S64, 12), using
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- trace code [i] based on OA(6440, 2097155, S64, 12), using
- discarding factors based on OA(880, 4194310, S8, 12), using
- OA 6-folding and stacking [i] based on OA(880, 4194306, S8, 12), using
(68, 68+12, 2593249)-Net over F8 — Digital
Digital (68, 80, 2593249)-net over F8, using
(68, 68+12, large)-Net in Base 8 — Upper bound on s
There is no (68, 80, large)-net in base 8, because
- 10 times m-reduction [i] would yield (68, 70, large)-net in base 8, but