Best Known (56, 56+15, s)-Nets in Base 2
(56, 56+15, 146)-Net over F2 — Constructive and digital
Digital (56, 71, 146)-net over F2, using
- net defined by OOA [i] based on linear OOA(271, 146, F2, 15, 15) (dual of [(146, 15), 2119, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using
(56, 56+15, 341)-Net over F2 — Digital
Digital (56, 71, 341)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 341, F2, 3, 15) (dual of [(341, 3), 952, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OOA 3-folding [i] based on linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using
(56, 56+15, 3451)-Net in Base 2 — Upper bound on s
There is no (56, 71, 3452)-net in base 2, because
- 1 times m-reduction [i] would yield (56, 70, 3452)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1182 626927 139439 909146 > 270 [i]