Best Known (119−20, 119, s)-Nets in Base 2
(119−20, 119, 320)-Net over F2 — Constructive and digital
Digital (99, 119, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (99, 120, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 24, 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, 24, 64)-net over F32, using
(119−20, 119, 823)-Net over F2 — Digital
Digital (99, 119, 823)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2119, 823, F2, 2, 20) (dual of [(823, 2), 1527, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2119, 1039, F2, 2, 20) (dual of [(1039, 2), 1959, 21]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2118, 1039, F2, 2, 20) (dual of [(1039, 2), 1960, 21]-NRT-code), using
- strength reduction [i] based on linear OOA(2118, 1039, F2, 2, 21) (dual of [(1039, 2), 1960, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2118, 2078, F2, 21) (dual of [2078, 1960, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2117, 2077, F2, 21) (dual of [2077, 1960, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2111, 2049, F2, 21) (dual of [2049, 1938, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(289, 2049, F2, 17) (dual of [2049, 1960, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,10]) ⊂ C([0,8]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2117, 2077, F2, 21) (dual of [2077, 1960, 22]-code), using
- OOA 2-folding [i] based on linear OA(2118, 2078, F2, 21) (dual of [2078, 1960, 22]-code), using
- strength reduction [i] based on linear OOA(2118, 1039, F2, 2, 21) (dual of [(1039, 2), 1960, 22]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2118, 1039, F2, 2, 20) (dual of [(1039, 2), 1960, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2119, 1039, F2, 2, 20) (dual of [(1039, 2), 1959, 21]-NRT-code), using
(119−20, 119, 17292)-Net in Base 2 — Upper bound on s
There is no (99, 119, 17293)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 664628 569701 556200 654815 509396 671774 > 2119 [i]