Best Known (50−13, 50, s)-Nets in Base 16
(50−13, 50, 21845)-Net over F16 — Constructive and digital
Digital (37, 50, 21845)-net over F16, using
- net defined by OOA [i] based on linear OOA(1650, 21845, F16, 13, 13) (dual of [(21845, 13), 283935, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
(50−13, 50, 75631)-Net over F16 — Digital
Digital (37, 50, 75631)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1650, 75631, F16, 13) (dual of [75631, 75581, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
(50−13, 50, large)-Net in Base 16 — Upper bound on s
There is no (37, 50, large)-net in base 16, because
- 11 times m-reduction [i] would yield (37, 39, large)-net in base 16, but