Best Known (169−19, 169, s)-Nets in Base 2
(169−19, 169, 29130)-Net over F2 — Constructive and digital
Digital (150, 169, 29130)-net over F2, using
- net defined by OOA [i] based on linear OOA(2169, 29130, F2, 19, 19) (dual of [(29130, 19), 553301, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2169, 262171, F2, 19) (dual of [262171, 262002, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2169, 262176, F2, 19) (dual of [262176, 262007, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2163, 262144, F2, 19) (dual of [262144, 261981, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2169, 262176, F2, 19) (dual of [262176, 262007, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2169, 262171, F2, 19) (dual of [262171, 262002, 20]-code), using
(169−19, 169, 43696)-Net over F2 — Digital
Digital (150, 169, 43696)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2169, 43696, F2, 6, 19) (dual of [(43696, 6), 262007, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2169, 262176, F2, 19) (dual of [262176, 262007, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2163, 262144, F2, 19) (dual of [262144, 261981, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- OOA 6-folding [i] based on linear OA(2169, 262176, F2, 19) (dual of [262176, 262007, 20]-code), using
(169−19, 169, 1725737)-Net in Base 2 — Upper bound on s
There is no (150, 169, 1725738)-net in base 2, because
- 1 times m-reduction [i] would yield (150, 168, 1725738)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 374 145193 574319 550466 034315 851877 858332 851409 355431 > 2168 [i]