Best Known (100−18, 100, s)-Nets in Base 8
(100−18, 100, 29136)-Net over F8 — Constructive and digital
Digital (82, 100, 29136)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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]
- a shift-net [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (73, 91, 29127)-net over F8, using
- net defined by OOA [i] based on linear OOA(891, 29127, F8, 18, 18) (dual of [(29127, 18), 524195, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(891, 262143, F8, 18) (dual of [262143, 262052, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(891, 262144, F8, 18) (dual of [262144, 262053, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(891, 262144, F8, 18) (dual of [262144, 262053, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(891, 262143, F8, 18) (dual of [262143, 262052, 19]-code), using
- net defined by OOA [i] based on linear OOA(891, 29127, F8, 18, 18) (dual of [(29127, 18), 524195, 19]-NRT-code), using
- digital (0, 9, 9)-net over F8, using
(100−18, 100, 262185)-Net over F8 — Digital
Digital (82, 100, 262185)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8100, 262185, F8, 18) (dual of [262185, 262085, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(898, 262181, F8, 18) (dual of [262181, 262083, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(891, 262144, F8, 18) (dual of [262144, 262053, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(861, 262144, F8, 12) (dual of [262144, 262083, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 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(17) ⊂ Ce(11) [i] based on
- linear OA(898, 262183, F8, 17) (dual of [262183, 262085, 18]-code), using Gilbert–Varšamov bound and bm = 898 > Vbs−1(k−1) = 791 407217 362850 835796 347714 135112 637659 979146 508560 156727 162742 064388 612959 266747 424462 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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
- linear OA(898, 262181, F8, 18) (dual of [262181, 262083, 19]-code), using
- construction X with Varšamov bound [i] based on
(100−18, 100, large)-Net in Base 8 — Upper bound on s
There is no (82, 100, large)-net in base 8, because
- 16 times m-reduction [i] would yield (82, 84, large)-net in base 8, but