Best Known (130, 130+29, s)-Nets in Base 2
(130, 130+29, 269)-Net over F2 — Constructive and digital
Digital (130, 159, 269)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 9)-net over F2, using
- net from sequence [i] based on digital (5, 8)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 5 and N(F) ≥ 9, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
- net from sequence [i] based on digital (5, 8)-sequence over F2, using
- digital (111, 140, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 35, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 35, 65)-net over F16, using
- digital (5, 19, 9)-net over F2, using
(130, 130+29, 687)-Net over F2 — Digital
Digital (130, 159, 687)-net over F2, using
- 22 times duplication [i] based on digital (128, 157, 687)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 687, F2, 3, 29) (dual of [(687, 3), 1904, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2157, 2061, F2, 29) (dual of [2061, 1904, 30]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(2156, 2060, F2, 29) (dual of [2060, 1904, 30]-code), using
- OOA 3-folding [i] based on linear OA(2157, 2061, F2, 29) (dual of [2061, 1904, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 687, F2, 3, 29) (dual of [(687, 3), 1904, 30]-NRT-code), using
(130, 130+29, 15073)-Net in Base 2 — Upper bound on s
There is no (130, 159, 15074)-net in base 2, because
- 1 times m-reduction [i] would yield (130, 158, 15074)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 365635 893465 557741 783346 130228 287789 493140 165712 > 2158 [i]