Best Known (159, 159+22, s)-Nets in Base 2
(159, 159+22, 5959)-Net over F2 — Constructive and digital
Digital (159, 181, 5959)-net over F2, using
- 23 times duplication [i] based on digital (156, 178, 5959)-net over F2, using
- t-expansion [i] based on digital (155, 178, 5959)-net over F2, using
- net defined by OOA [i] based on linear OOA(2178, 5959, F2, 23, 23) (dual of [(5959, 23), 136879, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2178, 65550, F2, 23) (dual of [65550, 65372, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2161, 65536, F2, 21) (dual of [65536, 65375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2178, 65550, F2, 23) (dual of [65550, 65372, 24]-code), using
- net defined by OOA [i] based on linear OOA(2178, 5959, F2, 23, 23) (dual of [(5959, 23), 136879, 24]-NRT-code), using
- t-expansion [i] based on digital (155, 178, 5959)-net over F2, using
(159, 159+22, 13111)-Net over F2 — Digital
Digital (159, 181, 13111)-net over F2, using
- 21 times duplication [i] based on digital (158, 180, 13111)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 13111, F2, 5, 22) (dual of [(13111, 5), 65375, 23]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2180, 65555, F2, 22) (dual of [65555, 65375, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2177, 65552, F2, 22) (dual of [65552, 65375, 23]-code), using
- 1 times truncation [i] based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2161, 65536, F2, 21) (dual of [65536, 65375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2178, 65553, F2, 23) (dual of [65553, 65375, 24]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2177, 65552, F2, 22) (dual of [65552, 65375, 23]-code), using
- OOA 5-folding [i] based on linear OA(2180, 65555, F2, 22) (dual of [65555, 65375, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 13111, F2, 5, 22) (dual of [(13111, 5), 65375, 23]-NRT-code), using
(159, 159+22, 440869)-Net in Base 2 — Upper bound on s
There is no (159, 181, 440870)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3 065022 528630 288592 079321 620244 695890 361869 953953 404984 > 2181 [i]