Best Known (194, 194+41, s)-Nets in Base 2
(194, 194+41, 380)-Net over F2 — Constructive and digital
Digital (194, 235, 380)-net over F2, using
- t-expansion [i] based on digital (193, 235, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 47, 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, 47, 76)-net over F32, using
(194, 194+41, 999)-Net over F2 — Digital
Digital (194, 235, 999)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2235, 999, F2, 2, 41) (dual of [(999, 2), 1763, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2235, 1047, F2, 2, 41) (dual of [(1047, 2), 1859, 42]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2234, 1047, F2, 2, 41) (dual of [(1047, 2), 1860, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2234, 2094, F2, 41) (dual of [2094, 1860, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- linear OA(2221, 2048, F2, 41) (dual of [2048, 1827, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(40) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(2234, 2094, F2, 41) (dual of [2094, 1860, 42]-code), using
- 21 times duplication [i] based on linear OOA(2234, 1047, F2, 2, 41) (dual of [(1047, 2), 1860, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2235, 1047, F2, 2, 41) (dual of [(1047, 2), 1859, 42]-NRT-code), using
(194, 194+41, 27599)-Net in Base 2 — Upper bound on s
There is no (194, 235, 27600)-net in base 2, because
- 1 times m-reduction [i] would yield (194, 234, 27600)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 27626 851151 435502 170093 298811 289497 809305 343413 127172 532733 146138 873356 > 2234 [i]