Best Known (18, 18+10, s)-Nets in Base 16
(18, 18+10, 819)-Net over F16 — Constructive and digital
Digital (18, 28, 819)-net over F16, using
- net defined by OOA [i] based on linear OOA(1628, 819, F16, 10, 10) (dual of [(819, 10), 8162, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1628, 4095, F16, 10) (dual of [4095, 4067, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1628, 4096, F16, 10) (dual of [4096, 4068, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(1628, 4096, F16, 10) (dual of [4096, 4068, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1628, 4095, F16, 10) (dual of [4095, 4067, 11]-code), using
(18, 18+10, 2904)-Net over F16 — Digital
Digital (18, 28, 2904)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1628, 2904, F16, 10) (dual of [2904, 2876, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1628, 4096, F16, 10) (dual of [4096, 4068, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(1628, 4096, F16, 10) (dual of [4096, 4068, 11]-code), using
(18, 18+10, 961204)-Net in Base 16 — Upper bound on s
There is no (18, 28, 961205)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 5192 307766 749913 117031 748052 305376 > 1628 [i]