Best Known (52−9, 52, s)-Nets in Base 16
(52−9, 52, 4194301)-Net over F16 — Constructive and digital
Digital (43, 52, 4194301)-net over F16, using
- net defined by OOA [i] based on linear OOA(1652, 4194301, F16, 10, 9) (dual of [(4194301, 10), 41942958, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1652, large, F16, 2, 9), using
- 1 times NRT-code embedding in larger space [i] 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
- 1 times NRT-code embedding in larger space [i] 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(1652, large, F16, 2, 9), using
(52−9, 52, large)-Net over F16 — Digital
Digital (43, 52, large)-net over F16, using
- 163 times duplication [i] based on digital (40, 49, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1649, large, F16, 9) (dual of [large, large−49, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1649, large, F16, 9) (dual of [large, large−49, 10]-code), using
(52−9, 52, large)-Net in Base 16 — Upper bound on s
There is no (43, 52, large)-net in base 16, because
- 7 times m-reduction [i] would yield (43, 45, large)-net in base 16, but