Best Known (78−12, 78, s)-Nets in Base 8
(78−12, 78, 349540)-Net over F8 — Constructive and digital
Digital (66, 78, 349540)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 14)-net over F8, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 1 and N(F) ≥ 14, using
- net from sequence [i] based on digital (1, 13)-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 (1, 7, 14)-net over F8, using
(78−12, 78, 2097194)-Net over F8 — Digital
Digital (66, 78, 2097194)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(878, 2097194, F8, 12) (dual of [2097194, 2097116, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [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(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
(78−12, 78, large)-Net in Base 8 — Upper bound on s
There is no (66, 78, large)-net in base 8, because
- 10 times m-reduction [i] would yield (66, 68, large)-net in base 8, but