Best Known (159−18, 159, s)-Nets in Base 2
(159−18, 159, 14567)-Net over F2 — Constructive and digital
Digital (141, 159, 14567)-net over F2, using
- net defined by OOA [i] based on linear OOA(2159, 14567, F2, 18, 18) (dual of [(14567, 18), 262047, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2159, 131103, F2, 18) (dual of [131103, 130944, 19]-code), using
- 1 times truncation [i] based on linear OA(2160, 131104, F2, 19) (dual of [131104, 130944, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2160, 131104, F2, 19) (dual of [131104, 130944, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(2159, 131103, F2, 18) (dual of [131103, 130944, 19]-code), using
(159−18, 159, 26220)-Net over F2 — Digital
Digital (141, 159, 26220)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2159, 26220, F2, 5, 18) (dual of [(26220, 5), 130941, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2159, 131100, F2, 18) (dual of [131100, 130941, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2159, 131103, F2, 18) (dual of [131103, 130944, 19]-code), using
- 1 times truncation [i] based on linear OA(2160, 131104, F2, 19) (dual of [131104, 130944, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2160, 131104, F2, 19) (dual of [131104, 130944, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2159, 131103, F2, 18) (dual of [131103, 130944, 19]-code), using
- OOA 5-folding [i] based on linear OA(2159, 131100, F2, 18) (dual of [131100, 130941, 19]-code), using
(159−18, 159, 862862)-Net in Base 2 — Upper bound on s
There is no (141, 159, 862863)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 730756 141671 314103 850077 317794 617992 017828 407256 > 2159 [i]