Best Known (131, 131+28, s)-Nets in Base 2
(131, 131+28, 320)-Net over F2 — Constructive and digital
Digital (131, 159, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (131, 160, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 32, 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, 32, 64)-net over F32, using
(131, 131+28, 755)-Net over F2 — Digital
Digital (131, 159, 755)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2159, 755, F2, 2, 28) (dual of [(755, 2), 1351, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2159, 1032, F2, 2, 28) (dual of [(1032, 2), 1905, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2159, 2064, F2, 28) (dual of [2064, 1905, 29]-code), using
- 1 times truncation [i] based on linear OA(2160, 2065, F2, 29) (dual of [2065, 1905, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 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(2133, 2049, F2, 25) (dual of [2049, 1916, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(25, 16, F2, 3) (dual of [16, 11, 4]-code or 16-cap in PG(4,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 1 times truncation [i] based on linear OA(2160, 2065, F2, 29) (dual of [2065, 1905, 30]-code), using
- OOA 2-folding [i] based on linear OA(2159, 2064, F2, 28) (dual of [2064, 1905, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(2159, 1032, F2, 2, 28) (dual of [(1032, 2), 1905, 29]-NRT-code), using
(131, 131+28, 15839)-Net in Base 2 — Upper bound on s
There is no (131, 159, 15840)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 731178 303036 495603 590904 374030 704779 159577 533573 > 2159 [i]