Best Known (147, 147+31, s)-Nets in Base 2
(147, 147+31, 320)-Net over F2 — Constructive and digital
Digital (147, 178, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (147, 180, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 36, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 36, 64)-net over F32, using
(147, 147+31, 841)-Net over F2 — Digital
Digital (147, 178, 841)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2178, 841, F2, 2, 31) (dual of [(841, 2), 1504, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2178, 1046, F2, 2, 31) (dual of [(1046, 2), 1914, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2178, 2092, F2, 31) (dual of [2092, 1914, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 2093, F2, 31) (dual of [2093, 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(212, 45, F2, 5) (dual of [45, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2178, 2093, F2, 31) (dual of [2093, 1915, 32]-code), using
- OOA 2-folding [i] based on linear OA(2178, 2092, F2, 31) (dual of [2092, 1914, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2178, 1046, F2, 2, 31) (dual of [(1046, 2), 1914, 32]-NRT-code), using
(147, 147+31, 22882)-Net in Base 2 — Upper bound on s
There is no (147, 178, 22883)-net in base 2, because
- 1 times m-reduction [i] would yield (147, 177, 22883)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 191624 973844 292686 588629 233042 700637 653593 552449 193040 > 2177 [i]