Best Known (154, 154+27, s)-Nets in Base 2
(154, 154+27, 632)-Net over F2 — Constructive and digital
Digital (154, 181, 632)-net over F2, using
- 25 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
(154, 154+27, 2057)-Net over F2 — Digital
Digital (154, 181, 2057)-net over F2, using
- 21 times duplication [i] based on digital (153, 180, 2057)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 2057, F2, 4, 27) (dual of [(2057, 4), 8048, 28]-NRT-code), using
- 1 times NRT-code embedding in larger space [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
- 1 times NRT-code embedding in larger space [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(2180, 2057, F2, 4, 27) (dual of [(2057, 4), 8048, 28]-NRT-code), using
(154, 154+27, 83447)-Net in Base 2 — Upper bound on s
There is no (154, 181, 83448)-net in base 2, because
- 1 times m-reduction [i] would yield (154, 180, 83448)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 532540 662511 539079 995696 987662 627144 078981 327697 655778 > 2180 [i]