Best Known (65, 65+12, s)-Nets in Base 8
(65, 65+12, 349535)-Net over F8 — Constructive and digital
Digital (65, 77, 349535)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-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 (0, 6, 9)-net over F8, using
(65, 65+12, 2097186)-Net over F8 — Digital
Digital (65, 77, 2097186)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(877, 2097186, F8, 12) (dual of [2097186, 2097109, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [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(843, 2097152, F8, 7) (dual of [2097152, 2097109, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(86, 34, F8, 4) (dual of [34, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
(65, 65+12, large)-Net in Base 8 — Upper bound on s
There is no (65, 77, large)-net in base 8, because
- 10 times m-reduction [i] would yield (65, 67, large)-net in base 8, but