Best Known (217, 217+42, s)-Nets in Base 2
(217, 217+42, 520)-Net over F2 — Constructive and digital
Digital (217, 259, 520)-net over F2, using
- 1 times m-reduction [i] based on digital (217, 260, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 52, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 52, 104)-net over F32, using
(217, 217+42, 1427)-Net over F2 — Digital
Digital (217, 259, 1427)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2259, 1427, F2, 2, 42) (dual of [(1427, 2), 2595, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 2063, F2, 2, 42) (dual of [(2063, 2), 3867, 43]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2259, 4126, F2, 42) (dual of [4126, 3867, 43]-code), using
- strength reduction [i] based on linear OA(2259, 4126, F2, 43) (dual of [4126, 3867, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(38) [i] based on
- linear OA(2253, 4096, F2, 43) (dual of [4096, 3843, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2229, 4096, F2, 39) (dual of [4096, 3867, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on 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(42) ⊂ Ce(38) [i] based on
- strength reduction [i] based on linear OA(2259, 4126, F2, 43) (dual of [4126, 3867, 44]-code), using
- OOA 2-folding [i] based on linear OA(2259, 4126, F2, 42) (dual of [4126, 3867, 43]-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 2063, F2, 2, 42) (dual of [(2063, 2), 3867, 43]-NRT-code), using
(217, 217+42, 44760)-Net in Base 2 — Upper bound on s
There is no (217, 259, 44761)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 926538 864446 654423 037484 309610 548781 597441 449809 942186 912159 887970 472006 504428 > 2259 [i]