Best Known (156, 156+41, s)-Nets in Base 2
(156, 156+41, 260)-Net over F2 — Constructive and digital
Digital (156, 197, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (156, 200, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 50, 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, 50, 65)-net over F16, using
(156, 156+41, 474)-Net over F2 — Digital
Digital (156, 197, 474)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2197, 474, F2, 2, 41) (dual of [(474, 2), 751, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2197, 517, F2, 2, 41) (dual of [(517, 2), 837, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2197, 1034, F2, 41) (dual of [1034, 837, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(2197, 1035, F2, 41) (dual of [1035, 838, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- linear OA(2196, 1024, F2, 41) (dual of [1024, 828, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2186, 1024, F2, 39) (dual of [1024, 838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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(40) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(2197, 1035, F2, 41) (dual of [1035, 838, 42]-code), using
- OOA 2-folding [i] based on linear OA(2197, 1034, F2, 41) (dual of [1034, 837, 42]-code), using
- discarding factors / shortening the dual code based on linear OOA(2197, 517, F2, 2, 41) (dual of [(517, 2), 837, 42]-NRT-code), using
(156, 156+41, 7373)-Net in Base 2 — Upper bound on s
There is no (156, 197, 7374)-net in base 2, because
- 1 times m-reduction [i] would yield (156, 196, 7374)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 100589 880118 056752 469375 814130 916135 323927 015841 329239 978756 > 2196 [i]