Best Known (106−20, 106, s)-Nets in Base 2
(106−20, 106, 260)-Net over F2 — Constructive and digital
Digital (86, 106, 260)-net over F2, using
- 22 times duplication [i] based on digital (84, 104, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 26, 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, 26, 65)-net over F16, using
(106−20, 106, 475)-Net over F2 — Digital
Digital (86, 106, 475)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2106, 475, F2, 2, 20) (dual of [(475, 2), 844, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2106, 525, F2, 2, 20) (dual of [(525, 2), 944, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2106, 1050, F2, 20) (dual of [1050, 944, 21]-code), using
- 1 times truncation [i] based on linear OA(2107, 1051, F2, 21) (dual of [1051, 944, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2101, 1025, F2, 21) (dual of [1025, 924, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 220−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(281, 1025, F2, 17) (dual of [1025, 944, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 220−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-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([0,10]) ⊂ C([0,8]) [i] based on
- 1 times truncation [i] based on linear OA(2107, 1051, F2, 21) (dual of [1051, 944, 22]-code), using
- OOA 2-folding [i] based on linear OA(2106, 1050, F2, 20) (dual of [1050, 944, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(2106, 525, F2, 2, 20) (dual of [(525, 2), 944, 21]-NRT-code), using
(106−20, 106, 7014)-Net in Base 2 — Upper bound on s
There is no (86, 106, 7015)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 81 184488 016987 635143 113827 035588 > 2106 [i]