Best Known (56−9, 56, s)-Nets in Base 16
(56−9, 56, 4194420)-Net over F16 — Constructive and digital
Digital (47, 56, 4194420)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 120)-net over F16, using
- net defined by OOA [i] based on linear OOA(166, 120, F16, 4, 4) (dual of [(120, 4), 474, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- net defined by OOA [i] based on linear OOA(166, 120, F16, 4, 4) (dual of [(120, 4), 474, 5]-NRT-code), using
- digital (41, 50, 4194300)-net over F16, using
- net defined by OOA [i] based on linear OOA(1650, 4194300, F16, 10, 9) (dual of [(4194300, 10), 41942950, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1650, 8388601, F16, 2, 9) (dual of [(8388601, 2), 16777152, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1650, 8388602, F16, 2, 9) (dual of [(8388602, 2), 16777154, 10]-NRT-code), using
- trace code [i] based on linear OOA(25625, 4194301, F256, 2, 9) (dual of [(4194301, 2), 8388577, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25625, 8388602, F256, 9) (dual of [8388602, 8388577, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- OOA 2-folding [i] based on linear OA(25625, 8388602, F256, 9) (dual of [8388602, 8388577, 10]-code), using
- trace code [i] based on linear OOA(25625, 4194301, F256, 2, 9) (dual of [(4194301, 2), 8388577, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1650, 8388602, F16, 2, 9) (dual of [(8388602, 2), 16777154, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1650, 8388601, F16, 2, 9) (dual of [(8388601, 2), 16777152, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1650, 4194300, F16, 10, 9) (dual of [(4194300, 10), 41942950, 10]-NRT-code), using
- digital (2, 6, 120)-net over F16, using
(56−9, 56, large)-Net over F16 — Digital
Digital (47, 56, large)-net over F16, using
- 161 times duplication [i] based on digital (46, 55, large)-net over F16, using
- t-expansion [i] based on digital (45, 55, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1655, large, F16, 10) (dual of [large, large−55, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1655, large, F16, 10) (dual of [large, large−55, 11]-code), using
- t-expansion [i] based on digital (45, 55, large)-net over F16, using
(56−9, 56, large)-Net in Base 16 — Upper bound on s
There is no (47, 56, large)-net in base 16, because
- 7 times m-reduction [i] would yield (47, 49, large)-net in base 16, but