Best Known (69−10, 69, s)-Nets in Base 16
(69−10, 69, 3388080)-Net over F16 — Constructive and digital
Digital (59, 69, 3388080)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 32640)-net over F16, using
- net defined by OOA [i] based on linear OOA(1613, 32640, F16, 5, 5) (dual of [(32640, 5), 163187, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1613, 65281, F16, 5) (dual of [65281, 65268, 6]-code), using
- net defined by OOA [i] based on linear OOA(1613, 32640, F16, 5, 5) (dual of [(32640, 5), 163187, 6]-NRT-code), using
- digital (46, 56, 3355440)-net over F16, using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- digital (8, 13, 32640)-net over F16, using
(69−10, 69, large)-Net over F16 — Digital
Digital (59, 69, large)-net over F16, using
- 162 times duplication [i] based on digital (57, 67, large)-net over F16, using
- t-expansion [i] based on digital (55, 67, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1667, large, F16, 12) (dual of [large, large−67, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1667, large, F16, 12) (dual of [large, large−67, 13]-code), using
- t-expansion [i] based on digital (55, 67, large)-net over F16, using
(69−10, 69, large)-Net in Base 16 — Upper bound on s
There is no (59, 69, large)-net in base 16, because
- 8 times m-reduction [i] would yield (59, 61, large)-net in base 16, but