Best Known (79−13, 79, s)-Nets in Base 2
(79−13, 79, 1365)-Net over F2 — Constructive and digital
Digital (66, 79, 1365)-net over F2, using
- net defined by OOA [i] based on linear OOA(279, 1365, F2, 13, 13) (dual of [(1365, 13), 17666, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(279, 8191, F2, 13) (dual of [8191, 8112, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(279, 8191, F2, 13) (dual of [8191, 8112, 14]-code), using
(79−13, 79, 2048)-Net over F2 — Digital
Digital (66, 79, 2048)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(279, 2048, F2, 4, 13) (dual of [(2048, 4), 8113, 14]-NRT-code), using
- OOA 4-folding [i] based on linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 4-folding [i] based on linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using
(79−13, 79, 24516)-Net in Base 2 — Upper bound on s
There is no (66, 79, 24517)-net in base 2, because
- 1 times m-reduction [i] would yield (66, 78, 24517)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 302255 761358 084034 092860 > 278 [i]