Best Known (199−36, 199, s)-Nets in Base 2
(199−36, 199, 320)-Net over F2 — Constructive and digital
Digital (163, 199, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (163, 200, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 40, 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, 40, 64)-net over F32, using
(199−36, 199, 778)-Net over F2 — Digital
Digital (163, 199, 778)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2199, 778, F2, 2, 36) (dual of [(778, 2), 1357, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 1029, F2, 2, 36) (dual of [(1029, 2), 1859, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2199, 2058, F2, 36) (dual of [2058, 1859, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(2199, 2059, F2, 36) (dual of [2059, 1860, 37]-code), using
- 1 times truncation [i] based on linear OA(2200, 2060, F2, 37) (dual of [2060, 1860, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [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(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(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(36) ⊂ Ce(34) [i] based on
- 1 times truncation [i] based on linear OA(2200, 2060, F2, 37) (dual of [2060, 1860, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2199, 2059, F2, 36) (dual of [2059, 1860, 37]-code), using
- OOA 2-folding [i] based on linear OA(2199, 2058, F2, 36) (dual of [2058, 1859, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 1029, F2, 2, 36) (dual of [(1029, 2), 1859, 37]-NRT-code), using
(199−36, 199, 16049)-Net in Base 2 — Upper bound on s
There is no (163, 199, 16050)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 803704 525538 407168 566272 410974 775539 108463 547654 151004 245416 > 2199 [i]