Best Known (186−32, 186, s)-Nets in Base 2
(186−32, 186, 380)-Net over F2 — Constructive and digital
Digital (154, 186, 380)-net over F2, using
- 21 times duplication [i] based on digital (153, 185, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 37, 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
- trace code for nets [i] based on digital (5, 37, 76)-net over F32, using
(186−32, 186, 907)-Net over F2 — Digital
Digital (154, 186, 907)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2186, 907, F2, 2, 32) (dual of [(907, 2), 1628, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2186, 1040, F2, 2, 32) (dual of [(1040, 2), 1894, 33]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2182, 1038, F2, 2, 32) (dual of [(1038, 2), 1894, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2182, 2076, F2, 32) (dual of [2076, 1894, 33]-code), using
- 1 times truncation [i] based on linear OA(2183, 2077, F2, 33) (dual of [2077, 1894, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2177, 2049, F2, 33) (dual of [2049, 1872, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2155, 2049, F2, 29) (dual of [2049, 1894, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [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 C([0,16]) ⊂ C([0,14]) [i] based on
- 1 times truncation [i] based on linear OA(2183, 2077, F2, 33) (dual of [2077, 1894, 34]-code), using
- OOA 2-folding [i] based on linear OA(2182, 2076, F2, 32) (dual of [2076, 1894, 33]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2182, 1038, F2, 2, 32) (dual of [(1038, 2), 1894, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2186, 1040, F2, 2, 32) (dual of [(1040, 2), 1894, 33]-NRT-code), using
(186−32, 186, 21455)-Net in Base 2 — Upper bound on s
There is no (154, 186, 21456)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 98 121305 091820 838262 681432 277219 292439 454104 252553 044060 > 2186 [i]