Best Known (220, 220+39, s)-Nets in Base 2
(220, 220+39, 624)-Net over F2 — Constructive and digital
Digital (220, 259, 624)-net over F2, using
- 21 times duplication [i] based on digital (219, 258, 624)-net over F2, using
- t-expansion [i] based on digital (218, 258, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 43, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 43, 104)-net over F64, using
- t-expansion [i] based on digital (218, 258, 624)-net over F2, using
(220, 220+39, 2163)-Net over F2 — Digital
Digital (220, 259, 2163)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2259, 2163, F2, 3, 39) (dual of [(2163, 3), 6230, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 2743, F2, 3, 39) (dual of [(2743, 3), 7970, 40]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2256, 2742, F2, 3, 39) (dual of [(2742, 3), 7970, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2256, 8226, F2, 39) (dual of [8226, 7970, 40]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2254, 8224, F2, 39) (dual of [8224, 7970, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(34) [i] based on
- linear OA(2248, 8192, F2, 39) (dual of [8192, 7944, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(38) ⊂ Ce(34) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2254, 8224, F2, 39) (dual of [8224, 7970, 40]-code), using
- OOA 3-folding [i] based on linear OA(2256, 8226, F2, 39) (dual of [8226, 7970, 40]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2256, 2742, F2, 3, 39) (dual of [(2742, 3), 7970, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 2743, F2, 3, 39) (dual of [(2743, 3), 7970, 40]-NRT-code), using
(220, 220+39, 96997)-Net in Base 2 — Upper bound on s
There is no (220, 259, 96998)-net in base 2, because
- 1 times m-reduction [i] would yield (220, 258, 96998)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463211 665794 400310 399039 662249 567564 522824 731266 211389 959507 718060 335542 401284 > 2258 [i]