Best Known (40−6, 40, s)-Nets in Base 16
(40−6, 40, 5754054)-Net over F16 — Constructive and digital
Digital (34, 40, 5754054)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 8, 161652)-net over F16, using
- net defined by OOA [i] based on linear OOA(168, 161652, F16, 3, 3) (dual of [(161652, 3), 484948, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(168, 161652, F16, 2, 3) (dual of [(161652, 2), 323296, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(168, 161652, F16, 3, 3) (dual of [(161652, 3), 484948, 4]-NRT-code), using
- digital (26, 32, 5592402)-net over F16, using
- trace code for nets [i] based on digital (10, 16, 2796201)-net over F256, using
- net defined by OOA [i] based on linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- net defined by OOA [i] based on linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- trace code for nets [i] based on digital (10, 16, 2796201)-net over F256, using
- digital (5, 8, 161652)-net over F16, using
(40−6, 40, large)-Net over F16 — Digital
Digital (34, 40, large)-net over F16, using
- 163 times duplication [i] based on digital (31, 37, large)-net over F16, using
- t-expansion [i] based on digital (30, 37, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1637, large, F16, 7) (dual of [large, large−37, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1637, large, F16, 7) (dual of [large, large−37, 8]-code), using
- t-expansion [i] based on digital (30, 37, large)-net over F16, using
(40−6, 40, large)-Net in Base 16 — Upper bound on s
There is no (34, 40, large)-net in base 16, because
- 4 times m-reduction [i] would yield (34, 36, large)-net in base 16, but