Best Known (210−37, 210, s)-Nets in Base 2
(210−37, 210, 380)-Net over F2 — Constructive and digital
Digital (173, 210, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 42, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
(210−37, 210, 891)-Net over F2 — Digital
Digital (173, 210, 891)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2210, 891, F2, 2, 37) (dual of [(891, 2), 1572, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2210, 1042, F2, 2, 37) (dual of [(1042, 2), 1874, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2210, 2084, F2, 37) (dual of [2084, 1874, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2210, 2085, F2, 37) (dual of [2085, 1875, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2199, 2048, F2, 37) (dual of [2048, 1849, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2177, 2048, F2, 33) (dual of [2048, 1871, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(21, 5, F2, 1) (dual of [5, 4, 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 XX applied to Ce(36) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2210, 2085, F2, 37) (dual of [2085, 1875, 38]-code), using
- OOA 2-folding [i] based on linear OA(2210, 2084, F2, 37) (dual of [2084, 1874, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2210, 1042, F2, 2, 37) (dual of [(1042, 2), 1874, 38]-NRT-code), using
(210−37, 210, 23601)-Net in Base 2 — Upper bound on s
There is no (173, 210, 23602)-net in base 2, because
- 1 times m-reduction [i] would yield (173, 209, 23602)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 823 199791 466908 468924 962681 520969 727813 903907 360640 491677 116840 > 2209 [i]