Best Known (42−7, 42, s)-Nets in Base 16
(42−7, 42, 5592657)-Net over F16 — Constructive and digital
Digital (35, 42, 5592657)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 257)-net over F16, using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(164, 257, F16, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- digital (31, 38, 5592400)-net over F16, using
- trace code for nets [i] based on digital (12, 19, 2796200)-net over F256, using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- trace code for nets [i] based on digital (12, 19, 2796200)-net over F256, using
- digital (1, 4, 257)-net over F16, using
(42−7, 42, large)-Net over F16 — Digital
Digital (35, 42, large)-net over F16, using
- 1 times m-reduction [i] based on digital (35, 43, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1643, large, F16, 8) (dual of [large, large−43, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1643, large, F16, 8) (dual of [large, large−43, 9]-code), using
(42−7, 42, large)-Net in Base 16 — Upper bound on s
There is no (35, 42, large)-net in base 16, because
- 5 times m-reduction [i] would yield (35, 37, large)-net in base 16, but