Best Known (65, 65+24, s)-Nets in Base 16
(65, 65+24, 5461)-Net over F16 — Constructive and digital
Digital (65, 89, 5461)-net over F16, using
- net defined by OOA [i] based on linear OOA(1689, 5461, F16, 24, 24) (dual of [(5461, 24), 130975, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(1689, 65532, F16, 24) (dual of [65532, 65443, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(1689, 65536, F16, 24) (dual of [65536, 65447, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(1689, 65536, F16, 24) (dual of [65536, 65447, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(1689, 65532, F16, 24) (dual of [65532, 65443, 25]-code), using
(65, 65+24, 39547)-Net over F16 — Digital
Digital (65, 89, 39547)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1689, 39547, F16, 24) (dual of [39547, 39458, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(1689, 65536, F16, 24) (dual of [65536, 65447, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(1689, 65536, F16, 24) (dual of [65536, 65447, 25]-code), using
(65, 65+24, large)-Net in Base 16 — Upper bound on s
There is no (65, 89, large)-net in base 16, because
- 22 times m-reduction [i] would yield (65, 67, large)-net in base 16, but