Best Known (239−42, 239, s)-Nets in Base 2
(239−42, 239, 380)-Net over F2 — Constructive and digital
Digital (197, 239, 380)-net over F2, using
- 1 times m-reduction [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
(239−42, 239, 984)-Net over F2 — Digital
Digital (197, 239, 984)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2239, 984, F2, 2, 42) (dual of [(984, 2), 1729, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2239, 1038, F2, 2, 42) (dual of [(1038, 2), 1837, 43]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2238, 1038, F2, 2, 42) (dual of [(1038, 2), 1838, 43]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2238, 2076, F2, 42) (dual of [2076, 1838, 43]-code), using
- strength reduction [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
- strength reduction [i] based on linear OA(2238, 2076, F2, 43) (dual of [2076, 1838, 44]-code), using
- OOA 2-folding [i] based on linear OA(2238, 2076, F2, 42) (dual of [2076, 1838, 43]-code), using
- 21 times duplication [i] based on linear OOA(2238, 1038, F2, 2, 42) (dual of [(1038, 2), 1838, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2239, 1038, F2, 2, 42) (dual of [(1038, 2), 1837, 43]-NRT-code), using
(239−42, 239, 23116)-Net in Base 2 — Upper bound on s
There is no (197, 239, 23117)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 883957 899068 401043 646306 535357 952839 757699 236430 883644 020949 174944 585656 > 2239 [i]