Best Known (72−12, 72, s)-Nets in Base 2
(72−12, 72, 682)-Net over F2 — Constructive and digital
Digital (60, 72, 682)-net over F2, using
- net defined by OOA [i] based on linear OOA(272, 682, F2, 12, 12) (dual of [(682, 12), 8112, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(272, 4092, F2, 12) (dual of [4092, 4020, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(272, 4096, F2, 12) (dual of [4096, 4024, 13]-code), using
- 1 times truncation [i] based on linear OA(273, 4097, F2, 13) (dual of [4097, 4024, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(273, 4097, F2, 13) (dual of [4097, 4024, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(272, 4096, F2, 12) (dual of [4096, 4024, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(272, 4092, F2, 12) (dual of [4092, 4020, 13]-code), using
(72−12, 72, 1365)-Net over F2 — Digital
Digital (60, 72, 1365)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(272, 1365, F2, 3, 12) (dual of [(1365, 3), 4023, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(272, 4095, F2, 12) (dual of [4095, 4023, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(272, 4096, F2, 12) (dual of [4096, 4024, 13]-code), using
- 1 times truncation [i] based on linear OA(273, 4097, F2, 13) (dual of [4097, 4024, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(273, 4097, F2, 13) (dual of [4097, 4024, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(272, 4096, F2, 12) (dual of [4096, 4024, 13]-code), using
- OOA 3-folding [i] based on linear OA(272, 4095, F2, 12) (dual of [4095, 4023, 13]-code), using
(72−12, 72, 12254)-Net in Base 2 — Upper bound on s
There is no (60, 72, 12255)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 4724 475173 529438 898547 > 272 [i]