Best Known (199, 199+43, s)-Nets in Base 2
(199, 199+43, 380)-Net over F2 — Constructive and digital
Digital (199, 242, 380)-net over F2, using
- 22 times duplication [i] based on digital (197, 240, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 48, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 48, 76)-net over F32, using
(199, 199+43, 952)-Net over F2 — Digital
Digital (199, 242, 952)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 952, F2, 2, 43) (dual of [(952, 2), 1662, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2242, 1040, F2, 2, 43) (dual of [(1040, 2), 1838, 44]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2238, 1038, F2, 2, 43) (dual of [(1038, 2), 1838, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2238, 2076, F2, 43) (dual of [2076, 1838, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(38) [i] based on
- linear OA(2232, 2048, F2, 43) (dual of [2048, 1816, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2210, 2048, F2, 39) (dual of [2048, 1838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-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
- OOA 2-folding [i] based on linear OA(2238, 2076, F2, 43) (dual of [2076, 1838, 44]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2238, 1038, F2, 2, 43) (dual of [(1038, 2), 1838, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2242, 1040, F2, 2, 43) (dual of [(1040, 2), 1838, 44]-NRT-code), using
(199, 199+43, 24695)-Net in Base 2 — Upper bound on s
There is no (199, 242, 24696)-net in base 2, because
- 1 times m-reduction [i] would yield (199, 241, 24696)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 533861 792392 264298 348578 042091 166700 914435 893308 029173 091901 394327 209337 > 2241 [i]