Best Known (40, 40+7, s)-Nets in Base 16
(40, 40+7, 6781282)-Net over F16 — Constructive and digital
Digital (40, 47, 6781282)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 9, 1188882)-net over F16, using
- net defined by OOA [i] based on linear OOA(169, 1188882, F16, 3, 3) (dual of [(1188882, 3), 3566637, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(169, 1188882, F16, 2, 3) (dual of [(1188882, 2), 2377755, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(169, 1188882, F16, 3, 3) (dual of [(1188882, 3), 3566637, 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 (6, 9, 1188882)-net over F16, using
(40, 40+7, large)-Net over F16 — Digital
Digital (40, 47, large)-net over F16, using
- 2 times m-reduction [i] based on digital (40, 49, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1649, large, F16, 9) (dual of [large, large−49, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1649, large, F16, 9) (dual of [large, large−49, 10]-code), using
(40, 40+7, large)-Net in Base 16 — Upper bound on s
There is no (40, 47, large)-net in base 16, because
- 5 times m-reduction [i] would yield (40, 42, large)-net in base 16, but