Best Known (40−8, 40, s)-Nets in Base 16
(40−8, 40, 262162)-Net over F16 — Constructive and digital
Digital (32, 40, 262162)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- 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
- digital (0, 4, 17)-net over F16, using
(40−8, 40, 524289)-Net in Base 16 — Constructive
(32, 40, 524289)-net in base 16, using
- net defined by OOA [i] based on OOA(1640, 524289, S16, 8, 8), using
- OA 4-folding and stacking [i] based on OA(1640, 2097156, S16, 8), using
- 1 times code embedding in larger space [i] based on OA(1639, 2097155, S16, 8), using
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- 1 times code embedding in larger space [i] based on OA(1639, 2097155, S16, 8), using
- OA 4-folding and stacking [i] based on OA(1640, 2097156, S16, 8), using
(40−8, 40, 1712040)-Net over F16 — Digital
Digital (32, 40, 1712040)-net over F16, using
(40−8, 40, large)-Net in Base 16 — Upper bound on s
There is no (32, 40, large)-net in base 16, because
- 6 times m-reduction [i] would yield (32, 34, large)-net in base 16, but