Best Known (129, 129+28, s)-Nets in Base 2
(129, 129+28, 320)-Net over F2 — Constructive and digital
Digital (129, 157, 320)-net over F2, using
- 22 times duplication [i] based on digital (127, 155, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 31, 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, 31, 64)-net over F32, using
(129, 129+28, 713)-Net over F2 — Digital
Digital (129, 157, 713)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 713, F2, 2, 28) (dual of [(713, 2), 1269, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2157, 1030, F2, 2, 28) (dual of [(1030, 2), 1903, 29]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2156, 1030, F2, 2, 28) (dual of [(1030, 2), 1904, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2156, 2060, F2, 28) (dual of [2060, 1904, 29]-code), using
- strength reduction [i] based on linear OA(2156, 2060, F2, 29) (dual of [2060, 1904, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2155, 2048, F2, 29) (dual of [2048, 1893, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(21, 12, F2, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- strength reduction [i] based on linear OA(2156, 2060, F2, 29) (dual of [2060, 1904, 30]-code), using
- OOA 2-folding [i] based on linear OA(2156, 2060, F2, 28) (dual of [2060, 1904, 29]-code), using
- 21 times duplication [i] based on linear OOA(2156, 1030, F2, 2, 28) (dual of [(1030, 2), 1904, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2157, 1030, F2, 2, 28) (dual of [(1030, 2), 1903, 29]-NRT-code), using
(129, 129+28, 14344)-Net in Base 2 — Upper bound on s
There is no (129, 157, 14345)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 182842 226650 824949 069626 852439 940204 938017 121984 > 2157 [i]