Best Known (170−31, 170, s)-Nets in Base 2
(170−31, 170, 320)-Net over F2 — Constructive and digital
Digital (139, 170, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 34, 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
(170−31, 170, 687)-Net over F2 — Digital
Digital (139, 170, 687)-net over F2, using
- 22 times duplication [i] based on digital (137, 168, 687)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2168, 687, F2, 3, 31) (dual of [(687, 3), 1893, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2168, 2061, F2, 31) (dual of [2061, 1893, 32]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(2167, 2060, F2, 31) (dual of [2060, 1893, 32]-code), using
- OOA 3-folding [i] based on linear OA(2168, 2061, F2, 31) (dual of [2061, 1893, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2168, 687, F2, 3, 31) (dual of [(687, 3), 1893, 32]-NRT-code), using
(170−31, 170, 15804)-Net in Base 2 — Upper bound on s
There is no (139, 170, 15805)-net in base 2, because
- 1 times m-reduction [i] would yield (139, 169, 15805)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 748 986042 645740 178844 266208 527644 711108 921921 868168 > 2169 [i]