Best Known (180−31, 180, s)-Nets in Base 2
(180−31, 180, 380)-Net over F2 — Constructive and digital
Digital (149, 180, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 36, 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
(180−31, 180, 886)-Net over F2 — Digital
Digital (149, 180, 886)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 886, F2, 2, 31) (dual of [(886, 2), 1592, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2180, 1047, F2, 2, 31) (dual of [(1047, 2), 1914, 32]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2179, 1047, F2, 2, 31) (dual of [(1047, 2), 1915, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2179, 2094, F2, 31) (dual of [2094, 1915, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2166, 2048, F2, 31) (dual of [2048, 1882, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2133, 2048, F2, 25) (dual of [2048, 1915, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(24) [i] based on
- OOA 2-folding [i] based on linear OA(2179, 2094, F2, 31) (dual of [2094, 1915, 32]-code), using
- 21 times duplication [i] based on linear OOA(2179, 1047, F2, 2, 31) (dual of [(1047, 2), 1915, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2180, 1047, F2, 2, 31) (dual of [(1047, 2), 1914, 32]-NRT-code), using
(180−31, 180, 25100)-Net in Base 2 — Upper bound on s
There is no (149, 180, 25101)-net in base 2, because
- 1 times m-reduction [i] would yield (149, 179, 25101)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 766603 153458 353194 538711 838043 406523 292610 861561 900088 > 2179 [i]