Best Known (73−13, 73, s)-Nets in Base 2
(73−13, 73, 682)-Net over F2 — Constructive and digital
Digital (60, 73, 682)-net over F2, using
- net defined by OOA [i] based on linear OOA(273, 682, F2, 13, 13) (dual of [(682, 13), 8793, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(273, 4093, F2, 13) (dual of [4093, 4020, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(273, 4096, F2, 13) (dual of [4096, 4023, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(273, 4096, F2, 13) (dual of [4096, 4023, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(273, 4093, F2, 13) (dual of [4093, 4020, 14]-code), using
(73−13, 73, 1024)-Net over F2 — Digital
Digital (60, 73, 1024)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(273, 1024, F2, 4, 13) (dual of [(1024, 4), 4023, 14]-NRT-code), using
- OOA 4-folding [i] based on linear OA(273, 4096, F2, 13) (dual of [4096, 4023, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 4-folding [i] based on linear OA(273, 4096, F2, 13) (dual of [4096, 4023, 14]-code), using
(73−13, 73, 12254)-Net in Base 2 — Upper bound on s
There is no (60, 73, 12255)-net in base 2, because
- 1 times m-reduction [i] would yield (60, 72, 12255)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4724 475173 529438 898547 > 272 [i]