Best Known (151, 151+27, s)-Nets in Base 2
(151, 151+27, 632)-Net over F2 — Constructive and digital
Digital (151, 178, 632)-net over F2, using
- 22 times duplication [i] based on digital (149, 176, 632)-net over F2, using
- net defined by OOA [i] based on linear OOA(2176, 632, F2, 27, 27) (dual of [(632, 27), 16888, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2176, 8217, F2, 27) (dual of [8217, 8041, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2176, 8224, F2, 27) (dual of [8224, 8048, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2144, 8192, F2, 23) (dual of [8192, 8048, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2176, 8224, F2, 27) (dual of [8224, 8048, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2176, 8217, F2, 27) (dual of [8217, 8041, 28]-code), using
- net defined by OOA [i] based on linear OOA(2176, 632, F2, 27, 27) (dual of [(632, 27), 16888, 28]-NRT-code), using
(151, 151+27, 2056)-Net over F2 — Digital
Digital (151, 178, 2056)-net over F2, using
- 21 times duplication [i] based on digital (150, 177, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2177, 2056, F2, 4, 27) (dual of [(2056, 4), 8047, 28]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2176, 2056, F2, 4, 27) (dual of [(2056, 4), 8048, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2176, 8224, F2, 27) (dual of [8224, 8048, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2144, 8192, F2, 23) (dual of [8192, 8048, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- OOA 4-folding [i] based on linear OA(2176, 8224, F2, 27) (dual of [8224, 8048, 28]-code), using
- 21 times duplication [i] based on linear OOA(2176, 2056, F2, 4, 27) (dual of [(2056, 4), 8048, 28]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2177, 2056, F2, 4, 27) (dual of [(2056, 4), 8047, 28]-NRT-code), using
(151, 151+27, 71109)-Net in Base 2 — Upper bound on s
There is no (151, 178, 71110)-net in base 2, because
- 1 times m-reduction [i] would yield (151, 177, 71110)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 191569 621372 930305 776734 355594 049133 978955 579251 633768 > 2177 [i]