Best Known (53, 53+9, s)-Nets in Base 16
(53, 53+9, 4259580)-Net over F16 — Constructive and digital
Digital (53, 62, 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 (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 (8, 12, 65280)-net over F16, using
(53, 53+9, large)-Net over F16 — Digital
Digital (53, 62, large)-net over F16, using
- 161 times duplication [i] based on digital (52, 61, large)-net over F16, using
- t-expansion [i] based on digital (50, 61, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1661, large, F16, 11) (dual of [large, large−61, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1661, large, F16, 11) (dual of [large, large−61, 12]-code), using
- t-expansion [i] based on digital (50, 61, large)-net over F16, using
(53, 53+9, large)-Net in Base 16 — Upper bound on s
There is no (53, 62, large)-net in base 16, because
- 7 times m-reduction [i] would yield (53, 55, large)-net in base 16, but