Best Known (48, 48+8, s)-Nets in Base 16
(48, 48+8, 4259580)-Net over F16 — Constructive and digital
Digital (48, 56, 4259580)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 65280)-net over F16, using
- net defined by OOA [i] based on linear OOA(1612, 65280, F16, 4, 4) (dual of [(65280, 4), 261108, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1612, 130560, F16, 4) (dual of [130560, 130548, 5]-code), using
- trace code [i] based on linear OA(2566, 65280, F256, 4) (dual of [65280, 65274, 5]-code), using
- 1 times truncation [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- trace code [i] based on linear OA(2566, 65280, F256, 4) (dual of [65280, 65274, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(1612, 130560, F16, 4) (dual of [130560, 130548, 5]-code), using
- net defined by OOA [i] based on linear OOA(1612, 65280, F16, 4, 4) (dual of [(65280, 4), 261108, 5]-NRT-code), using
- digital (36, 44, 4194300)-net over F16, using
- net defined by OOA [i] based on linear OOA(1644, 4194300, F16, 10, 8) (dual of [(4194300, 10), 41942956, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1644, 8388601, F16, 2, 8) (dual of [(8388601, 2), 16777158, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1644, 8388602, F16, 2, 8) (dual of [(8388602, 2), 16777160, 9]-NRT-code), using
- trace code [i] based on linear OOA(25622, 4194301, F256, 2, 8) (dual of [(4194301, 2), 8388580, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25622, 8388602, F256, 8) (dual of [8388602, 8388580, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- OOA 2-folding [i] based on linear OA(25622, 8388602, F256, 8) (dual of [8388602, 8388580, 9]-code), using
- trace code [i] based on linear OOA(25622, 4194301, F256, 2, 8) (dual of [(4194301, 2), 8388580, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1644, 8388602, F16, 2, 8) (dual of [(8388602, 2), 16777160, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1644, 8388601, F16, 2, 8) (dual of [(8388601, 2), 16777158, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1644, 4194300, F16, 10, 8) (dual of [(4194300, 10), 41942956, 9]-NRT-code), using
- digital (8, 12, 65280)-net over F16, using
(48, 48+8, large)-Net over F16 — Digital
Digital (48, 56, large)-net over F16, using
- 161 times duplication [i] based on digital (47, 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
(48, 48+8, large)-Net in Base 16 — Upper bound on s
There is no (48, 56, large)-net in base 16, because
- 6 times m-reduction [i] would yield (48, 50, large)-net in base 16, but