Best Known (180−30, 180, s)-Nets in Base 2
(180−30, 180, 390)-Net over F2 — Constructive and digital
Digital (150, 180, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 30, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
(180−30, 180, 1145)-Net over F2 — Digital
Digital (150, 180, 1145)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 1145, F2, 3, 30) (dual of [(1145, 3), 3255, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2180, 1365, F2, 3, 30) (dual of [(1365, 3), 3915, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2180, 4095, F2, 30) (dual of [4095, 3915, 31]-code), using
- 1 times truncation [i] based on linear OA(2181, 4096, F2, 31) (dual of [4096, 3915, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 1 times truncation [i] based on linear OA(2181, 4096, F2, 31) (dual of [4096, 3915, 32]-code), using
- OOA 3-folding [i] based on linear OA(2180, 4095, F2, 30) (dual of [4095, 3915, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(2180, 1365, F2, 3, 30) (dual of [(1365, 3), 3915, 31]-NRT-code), using
(180−30, 180, 26288)-Net in Base 2 — Upper bound on s
There is no (150, 180, 26289)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 533058 208277 459894 978640 042697 257450 569867 035049 566208 > 2180 [i]