Best Known (159, 159+33, s)-Nets in Base 2
(159, 159+33, 380)-Net over F2 — Constructive and digital
Digital (159, 192, 380)-net over F2, using
- 22 times duplication [i] based on digital (157, 190, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 38, 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, 38, 76)-net over F32, using
(159, 159+33, 928)-Net over F2 — Digital
Digital (159, 192, 928)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2192, 928, F2, 2, 33) (dual of [(928, 2), 1664, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2192, 1048, F2, 2, 33) (dual of [(1048, 2), 1904, 34]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2190, 1047, F2, 2, 33) (dual of [(1047, 2), 1904, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2190, 2094, F2, 33) (dual of [2094, 1904, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- linear OA(2177, 2048, F2, 33) (dual of [2048, 1871, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2144, 2048, F2, 27) (dual of [2048, 1904, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(213, 46, F2, 5) (dual of [46, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(2190, 2094, F2, 33) (dual of [2094, 1904, 34]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2190, 1047, F2, 2, 33) (dual of [(1047, 2), 1904, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2192, 1048, F2, 2, 33) (dual of [(1048, 2), 1904, 34]-NRT-code), using
(159, 159+33, 26650)-Net in Base 2 — Upper bound on s
There is no (159, 192, 26651)-net in base 2, because
- 1 times m-reduction [i] would yield (159, 191, 26651)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3139 919836 961217 255714 323057 395688 516967 551287 619377 812471 > 2191 [i]