Best Known (44, 44+17, s)-Nets in Base 16
(44, 44+17, 8191)-Net over F16 — Constructive and digital
Digital (44, 61, 8191)-net over F16, using
- net defined by OOA [i] based on linear OOA(1661, 8191, F16, 17, 17) (dual of [(8191, 17), 139186, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1661, 65529, F16, 17) (dual of [65529, 65468, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1661, 65529, F16, 17) (dual of [65529, 65468, 18]-code), using
(44, 44+17, 32768)-Net over F16 — Digital
Digital (44, 61, 32768)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1661, 32768, F16, 2, 17) (dual of [(32768, 2), 65475, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 2-folding [i] based on linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using
(44, 44+17, large)-Net in Base 16 — Upper bound on s
There is no (44, 61, large)-net in base 16, because
- 15 times m-reduction [i] would yield (44, 46, large)-net in base 16, but