Best Known (137, 137+30, s)-Nets in Base 2
(137, 137+30, 320)-Net over F2 — Constructive and digital
Digital (137, 167, 320)-net over F2, using
- 22 times duplication [i] based on digital (135, 165, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 33, 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, 33, 64)-net over F32, using
(137, 137+30, 717)-Net over F2 — Digital
Digital (137, 167, 717)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2167, 717, F2, 2, 30) (dual of [(717, 2), 1267, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2167, 1030, F2, 2, 30) (dual of [(1030, 2), 1893, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2167, 2060, F2, 30) (dual of [2060, 1893, 31]-code), using
- strength reduction [i] based on linear OA(2167, 2060, F2, 31) (dual of [2060, 1893, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [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(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(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(30) ⊂ Ce(28) [i] based on
- strength reduction [i] based on linear OA(2167, 2060, F2, 31) (dual of [2060, 1893, 32]-code), using
- OOA 2-folding [i] based on linear OA(2167, 2060, F2, 30) (dual of [2060, 1893, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(2167, 1030, F2, 2, 30) (dual of [(1030, 2), 1893, 31]-NRT-code), using
(137, 137+30, 14406)-Net in Base 2 — Upper bound on s
There is no (137, 167, 14407)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 187 084166 359053 144309 118680 220352 044835 158861 257840 > 2167 [i]