Best Known (172−19, 172, s)-Nets in Base 2
(172−19, 172, 58254)-Net over F2 — Constructive and digital
Digital (153, 172, 58254)-net over F2, using
- net defined by OOA [i] based on linear OOA(2172, 58254, F2, 19, 19) (dual of [(58254, 19), 1106654, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2172, 524287, F2, 19) (dual of [524287, 524115, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2172, 524287, F2, 19) (dual of [524287, 524115, 20]-code), using
(172−19, 172, 77182)-Net over F2 — Digital
Digital (153, 172, 77182)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 77182, F2, 6, 19) (dual of [(77182, 6), 462920, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 87381, F2, 6, 19) (dual of [(87381, 6), 524114, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2172, 524286, F2, 19) (dual of [524286, 524114, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using
- OOA 6-folding [i] based on linear OA(2172, 524286, F2, 19) (dual of [524286, 524114, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 87381, F2, 6, 19) (dual of [(87381, 6), 524114, 20]-NRT-code), using
(172−19, 172, 2174296)-Net in Base 2 — Upper bound on s
There is no (153, 172, 2174297)-net in base 2, because
- 1 times m-reduction [i] would yield (153, 171, 2174297)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2993 161402 934852 181964 425254 780358 839342 200471 255278 > 2171 [i]