Best Known (85−18, 85, s)-Nets in Base 16
(85−18, 85, 116510)-Net over F16 — Constructive and digital
Digital (67, 85, 116510)-net over F16, using
- 161 times duplication [i] based on digital (66, 84, 116510)-net over F16, using
- net defined by OOA [i] based on linear OOA(1684, 116510, F16, 18, 18) (dual of [(116510, 18), 2097096, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(1684, 1048590, F16, 18) (dual of [1048590, 1048506, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1684, 1048594, F16, 18) (dual of [1048594, 1048510, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(163, 18, F16, 3) (dual of [18, 15, 4]-code or 18-arc in PG(2,16) or 18-cap in PG(2,16)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(1684, 1048594, F16, 18) (dual of [1048594, 1048510, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(1684, 1048590, F16, 18) (dual of [1048590, 1048506, 19]-code), using
- net defined by OOA [i] based on linear OOA(1684, 116510, F16, 18, 18) (dual of [(116510, 18), 2097096, 19]-NRT-code), using
(85−18, 85, 950766)-Net over F16 — Digital
Digital (67, 85, 950766)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1685, 950766, F16, 18) (dual of [950766, 950681, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1685, 1048595, F16, 18) (dual of [1048595, 1048510, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1684, 1048594, F16, 18) (dual of [1048594, 1048510, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(163, 18, F16, 3) (dual of [18, 15, 4]-code or 18-arc in PG(2,16) or 18-cap in PG(2,16)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(1684, 1048594, F16, 18) (dual of [1048594, 1048510, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1685, 1048595, F16, 18) (dual of [1048595, 1048510, 19]-code), using
(85−18, 85, large)-Net in Base 16 — Upper bound on s
There is no (67, 85, large)-net in base 16, because
- 16 times m-reduction [i] would yield (67, 69, large)-net in base 16, but