Best Known (78, 78+13, s)-Nets in Base 2
(78, 78+13, 5461)-Net over F2 — Constructive and digital
Digital (78, 91, 5461)-net over F2, using
- net defined by OOA [i] based on linear OOA(291, 5461, F2, 13, 13) (dual of [(5461, 13), 70902, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(291, 32767, F2, 13) (dual of [32767, 32676, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(291, 32767, F2, 13) (dual of [32767, 32676, 14]-code), using
(78, 78+13, 7059)-Net over F2 — Digital
Digital (78, 91, 7059)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(291, 7059, F2, 4, 13) (dual of [(7059, 4), 28145, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(291, 8192, F2, 4, 13) (dual of [(8192, 4), 32677, 14]-NRT-code), using
- OOA 4-folding [i] based on linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 4-folding [i] based on linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(291, 8192, F2, 4, 13) (dual of [(8192, 4), 32677, 14]-NRT-code), using
(78, 78+13, 98092)-Net in Base 2 — Upper bound on s
There is no (78, 91, 98093)-net in base 2, because
- 1 times m-reduction [i] would yield (78, 90, 98093)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1238 002103 382358 426798 455760 > 290 [i]