Best Known (42, 42+19, s)-Nets in Base 2
(42, 42+19, 54)-Net over F2 — Constructive and digital
Digital (42, 61, 54)-net over F2, using
- 3 times m-reduction [i] based on digital (42, 64, 54)-net over F2, using
- trace code for nets [i] based on digital (10, 32, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- trace code for nets [i] based on digital (10, 32, 27)-net over F4, using
(42, 42+19, 68)-Net over F2 — Digital
Digital (42, 61, 68)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(261, 68, F2, 2, 19) (dual of [(68, 2), 75, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(261, 136, F2, 19) (dual of [136, 75, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(257, 128, F2, 19) (dual of [128, 71, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(250, 128, F2, 15) (dual of [128, 78, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(24, 8, F2, 3) (dual of [8, 4, 4]-code or 8-cap in PG(3,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(261, 136, F2, 19) (dual of [136, 75, 20]-code), using
(42, 42+19, 408)-Net in Base 2 — Upper bound on s
There is no (42, 61, 409)-net in base 2, because
- 1 times m-reduction [i] would yield (42, 60, 409)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 165252 764250 823006 > 260 [i]