Best Known (134, 134+29, s)-Nets in Base 2
(134, 134+29, 320)-Net over F2 — Constructive and digital
Digital (134, 163, 320)-net over F2, using
- 23 times duplication [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
(134, 134+29, 735)-Net over F2 — Digital
Digital (134, 163, 735)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2163, 735, F2, 2, 29) (dual of [(735, 2), 1307, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2163, 1039, F2, 2, 29) (dual of [(1039, 2), 1915, 30]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2162, 1039, F2, 2, 29) (dual of [(1039, 2), 1916, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2162, 2078, F2, 29) (dual of [2078, 1916, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2161, 2077, F2, 29) (dual of [2077, 1916, 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(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([0,14]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2161, 2077, F2, 29) (dual of [2077, 1916, 30]-code), using
- OOA 2-folding [i] based on linear OA(2162, 2078, F2, 29) (dual of [2078, 1916, 30]-code), using
- 21 times duplication [i] based on linear OOA(2162, 1039, F2, 2, 29) (dual of [(1039, 2), 1916, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2163, 1039, F2, 2, 29) (dual of [(1039, 2), 1915, 30]-NRT-code), using
(134, 134+29, 18378)-Net in Base 2 — Upper bound on s
There is no (134, 163, 18379)-net in base 2, because
- 1 times m-reduction [i] would yield (134, 162, 18379)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 846184 735121 334889 283035 697992 007594 854426 669728 > 2162 [i]