Best Known (143, 143+35, s)-Nets in Base 2
(143, 143+35, 260)-Net over F2 — Constructive and digital
Digital (143, 178, 260)-net over F2, using
- t-expansion [i] based on digital (141, 178, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (141, 180, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 45, 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, 45, 65)-net over F16, using
- 2 times m-reduction [i] based on digital (141, 180, 260)-net over F2, using
(143, 143+35, 530)-Net over F2 — Digital
Digital (143, 178, 530)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2178, 530, F2, 2, 35) (dual of [(530, 2), 882, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2178, 1060, F2, 35) (dual of [1060, 882, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 1061, F2, 35) (dual of [1061, 883, 36]-code), using
- adding a parity check bit [i] based on linear OA(2177, 1060, F2, 34) (dual of [1060, 883, 35]-code), using
- construction X applied to C([1,34]) ⊂ C([1,28]) [i] based on
- linear OA(2165, 1023, F2, 34) (dual of [1023, 858, 35]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2140, 1023, F2, 28) (dual of [1023, 883, 29]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(212, 37, F2, 5) (dual of [37, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 38, F2, 5) (dual of [38, 26, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(211, 32, F2, 5) (dual of [32, 21, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 31 = 25−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 31 = 25−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(21, 6, F2, 1) (dual of [6, 5, 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(4) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(212, 38, F2, 5) (dual of [38, 26, 6]-code), using
- construction X applied to C([1,34]) ⊂ C([1,28]) [i] based on
- adding a parity check bit [i] based on linear OA(2177, 1060, F2, 34) (dual of [1060, 883, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 1061, F2, 35) (dual of [1061, 883, 36]-code), using
- OOA 2-folding [i] based on linear OA(2178, 1060, F2, 35) (dual of [1060, 882, 36]-code), using
(143, 143+35, 9751)-Net in Base 2 — Upper bound on s
There is no (143, 178, 9752)-net in base 2, because
- 1 times m-reduction [i] would yield (143, 177, 9752)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 191604 206723 557279 042251 222868 903055 009892 296248 250909 > 2177 [i]