Best Known (78−15, 78, s)-Nets in Base 2
(78−15, 78, 292)-Net over F2 — Constructive and digital
Digital (63, 78, 292)-net over F2, using
- net defined by OOA [i] based on linear OOA(278, 292, F2, 15, 15) (dual of [(292, 15), 4302, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(278, 2045, F2, 15) (dual of [2045, 1967, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(278, 2045, F2, 15) (dual of [2045, 1967, 16]-code), using
(78−15, 78, 539)-Net over F2 — Digital
Digital (63, 78, 539)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(278, 539, F2, 3, 15) (dual of [(539, 3), 1539, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(278, 682, F2, 3, 15) (dual of [(682, 3), 1968, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(278, 2046, F2, 15) (dual of [2046, 1968, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using
- OOA 3-folding [i] based on linear OA(278, 2046, F2, 15) (dual of [2046, 1968, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(278, 682, F2, 3, 15) (dual of [(682, 3), 1968, 16]-NRT-code), using
(78−15, 78, 6912)-Net in Base 2 — Upper bound on s
There is no (63, 78, 6913)-net in base 2, because
- 1 times m-reduction [i] would yield (63, 77, 6913)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 151226 184839 500083 124352 > 277 [i]