Best Known (68, 68+25, s)-Nets in Base 16
(68, 68+25, 5461)-Net over F16 — Constructive and digital
Digital (68, 93, 5461)-net over F16, using
- net defined by OOA [i] based on linear OOA(1693, 5461, F16, 25, 25) (dual of [(5461, 25), 136432, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1693, 65533, F16, 25) (dual of [65533, 65440, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(1693, 65536, F16, 25) (dual of [65536, 65443, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(1693, 65536, F16, 25) (dual of [65536, 65443, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1693, 65533, F16, 25) (dual of [65533, 65440, 26]-code), using
(68, 68+25, 41183)-Net over F16 — Digital
Digital (68, 93, 41183)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1693, 41183, F16, 25) (dual of [41183, 41090, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(1693, 65536, F16, 25) (dual of [65536, 65443, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(1693, 65536, F16, 25) (dual of [65536, 65443, 26]-code), using
(68, 68+25, large)-Net in Base 16 — Upper bound on s
There is no (68, 93, large)-net in base 16, because
- 23 times m-reduction [i] would yield (68, 70, large)-net in base 16, but