Best Known (193−34, 193, s)-Nets in Base 2
(193−34, 193, 320)-Net over F2 — Constructive and digital
Digital (159, 193, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (159, 195, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 39, 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, 39, 64)-net over F32, using
(193−34, 193, 844)-Net over F2 — Digital
Digital (159, 193, 844)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2193, 844, F2, 2, 34) (dual of [(844, 2), 1495, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 1037, F2, 2, 34) (dual of [(1037, 2), 1881, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2193, 2074, F2, 34) (dual of [2074, 1881, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2193, 2075, F2, 34) (dual of [2075, 1882, 35]-code), using
- 1 times truncation [i] based on linear OA(2194, 2076, F2, 35) (dual of [2076, 1882, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(2194, 2076, F2, 35) (dual of [2076, 1882, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2193, 2075, F2, 34) (dual of [2075, 1882, 35]-code), using
- OOA 2-folding [i] based on linear OA(2193, 2074, F2, 34) (dual of [2074, 1881, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 1037, F2, 2, 34) (dual of [(1037, 2), 1881, 35]-NRT-code), using
(193−34, 193, 18747)-Net in Base 2 — Upper bound on s
There is no (159, 193, 18748)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 12560 244756 369867 912022 308726 294601 245279 358435 787708 813853 > 2193 [i]