Best Known (171, 171+31, s)-Nets in Base 2
(171, 171+31, 548)-Net over F2 — Constructive and digital
Digital (171, 202, 548)-net over F2, using
- net defined by OOA [i] based on linear OOA(2202, 548, F2, 31, 31) (dual of [(548, 31), 16786, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2202, 8221, F2, 31) (dual of [8221, 8019, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2202, 8224, F2, 31) (dual of [8224, 8022, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2202, 8224, F2, 31) (dual of [8224, 8022, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2202, 8221, F2, 31) (dual of [8221, 8019, 32]-code), using
(171, 171+31, 2031)-Net over F2 — Digital
Digital (171, 202, 2031)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2202, 2031, F2, 4, 31) (dual of [(2031, 4), 7922, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2202, 2056, F2, 4, 31) (dual of [(2056, 4), 8022, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2202, 8224, F2, 31) (dual of [8224, 8022, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- OOA 4-folding [i] based on linear OA(2202, 8224, F2, 31) (dual of [8224, 8022, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2202, 2056, F2, 4, 31) (dual of [(2056, 4), 8022, 32]-NRT-code), using
(171, 171+31, 69411)-Net in Base 2 — Upper bound on s
There is no (171, 202, 69412)-net in base 2, because
- 1 times m-reduction [i] would yield (171, 201, 69412)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 214273 805798 581948 804782 323351 182675 632337 439282 022304 736648 > 2201 [i]