Best Known (20, 20+5, s)-Nets in Base 4
(20, 20+5, 32767)-Net over F4 — Constructive and digital
Digital (20, 25, 32767)-net over F4, using
- net defined by OOA [i] based on linear OOA(425, 32767, F4, 5, 5) (dual of [(32767, 5), 163810, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(425, 65535, F4, 5) (dual of [65535, 65510, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(425, 65535, F4, 5) (dual of [65535, 65510, 6]-code), using
(20, 20+5, 39694)-Net over F4 — Digital
Digital (20, 25, 39694)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(425, 39694, F4, 5) (dual of [39694, 39669, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using
(20, 20+5, 7908854)-Net in Base 4 — Upper bound on s
There is no (20, 25, 7908855)-net in base 4, because
- 1 times m-reduction [i] would yield (20, 24, 7908855)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 281 475026 392591 > 424 [i]