Best Known (39−6, 39, s)-Nets in Base 16
(39−6, 39, 5658450)-Net over F16 — Constructive and digital
Digital (33, 39, 5658450)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 7, 66048)-net over F16, using
- net defined by OOA [i] based on linear OOA(167, 66048, F16, 3, 3) (dual of [(66048, 3), 198137, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(167, 66048, F16, 2, 3) (dual of [(66048, 2), 132089, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(167, 66048, F16, 3, 3) (dual of [(66048, 3), 198137, 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 (4, 7, 66048)-net over F16, using
(39−6, 39, large)-Net over F16 — Digital
Digital (33, 39, large)-net over F16, using
- 162 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
(39−6, 39, large)-Net in Base 16 — Upper bound on s
There is no (33, 39, large)-net in base 16, because
- 4 times m-reduction [i] would yield (33, 35, large)-net in base 16, but