Best Known (42, 57, s)-Nets in Base 16
(42, 57, 9362)-Net over F16 — Constructive and digital
Digital (42, 57, 9362)-net over F16, using
- net defined by OOA [i] based on linear OOA(1657, 9362, F16, 15, 15) (dual of [(9362, 15), 140373, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1657, 65535, F16, 15) (dual of [65535, 65478, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1657, 65535, F16, 15) (dual of [65535, 65478, 16]-code), using
(42, 57, 58109)-Net over F16 — Digital
Digital (42, 57, 58109)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1657, 58109, F16, 15) (dual of [58109, 58052, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using
(42, 57, large)-Net in Base 16 — Upper bound on s
There is no (42, 57, large)-net in base 16, because
- 13 times m-reduction [i] would yield (42, 44, large)-net in base 16, but