Best Known (40, 40+8, s)-Nets in Base 8
(40, 40+8, 131082)-Net over F8 — Constructive and digital
Digital (40, 48, 131082)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (36, 44, 131073)-net over F8, using
- net defined by OOA [i] based on linear OOA(844, 131073, F8, 8, 8) (dual of [(131073, 8), 1048540, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(844, 524292, F8, 8) (dual of [524292, 524248, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 524294, F8, 8) (dual of [524294, 524250, 9]-code), using
- trace code [i] based on linear OA(6422, 262147, F64, 8) (dual of [262147, 262125, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(6422, 262147, F64, 8) (dual of [262147, 262125, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 524294, F8, 8) (dual of [524294, 524250, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(844, 524292, F8, 8) (dual of [524292, 524248, 9]-code), using
- net defined by OOA [i] based on linear OOA(844, 131073, F8, 8, 8) (dual of [(131073, 8), 1048540, 9]-NRT-code), using
- digital (0, 4, 9)-net over F8, using
(40, 40+8, 262145)-Net in Base 8 — Constructive
(40, 48, 262145)-net in base 8, using
- base change [i] based on digital (28, 36, 262145)-net over F16, using
- net defined by OOA [i] based on linear OOA(1636, 262145, F16, 8, 8) (dual of [(262145, 8), 2097124, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(1636, 1048580, F16, 8) (dual of [1048580, 1048544, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(1636, 1048581, F16, 8) (dual of [1048581, 1048545, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1631, 1048576, F16, 7) (dual of [1048576, 1048545, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(1636, 1048581, F16, 8) (dual of [1048581, 1048545, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(1636, 1048580, F16, 8) (dual of [1048580, 1048544, 9]-code), using
- net defined by OOA [i] based on linear OOA(1636, 262145, F16, 8, 8) (dual of [(262145, 8), 2097124, 9]-NRT-code), using
(40, 40+8, 752384)-Net over F8 — Digital
Digital (40, 48, 752384)-net over F8, using
(40, 40+8, 1048581)-Net in Base 8
(40, 48, 1048581)-net in base 8, using
- base change [i] based on digital (28, 36, 1048581)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1636, 1048581, F16, 8) (dual of [1048581, 1048545, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1631, 1048576, F16, 7) (dual of [1048576, 1048545, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1636, 1048581, F16, 8) (dual of [1048581, 1048545, 9]-code), using
(40, 40+8, large)-Net in Base 8 — Upper bound on s
There is no (40, 48, large)-net in base 8, because
- 6 times m-reduction [i] would yield (40, 42, large)-net in base 8, but