Best Known (51−10, 51, s)-Nets in Base 2
(51−10, 51, 207)-Net over F2 — Constructive and digital
Digital (41, 51, 207)-net over F2, using
- net defined by OOA [i] based on linear OOA(251, 207, F2, 10, 10) (dual of [(207, 10), 2019, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(251, 1035, F2, 10) (dual of [1035, 984, 11]-code), using
- 1 times truncation [i] based on linear OA(252, 1036, F2, 11) (dual of [1036, 984, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(251, 1024, F2, 11) (dual of [1024, 973, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(241, 1024, F2, 9) (dual of [1024, 983, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(211, 12, F2, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,2)), using
- dual of repetition code with length 12 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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 X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- 1 times truncation [i] based on linear OA(252, 1036, F2, 11) (dual of [1036, 984, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(251, 1035, F2, 10) (dual of [1035, 984, 11]-code), using
(51−10, 51, 423)-Net over F2 — Digital
Digital (41, 51, 423)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(251, 423, F2, 2, 10) (dual of [(423, 2), 795, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(251, 517, F2, 2, 10) (dual of [(517, 2), 983, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(251, 1034, F2, 10) (dual of [1034, 983, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(251, 1035, F2, 10) (dual of [1035, 984, 11]-code), using
- 1 times truncation [i] based on linear OA(252, 1036, F2, 11) (dual of [1036, 984, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(251, 1024, F2, 11) (dual of [1024, 973, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(241, 1024, F2, 9) (dual of [1024, 983, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(211, 12, F2, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,2)), using
- dual of repetition code with length 12 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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 X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- 1 times truncation [i] based on linear OA(252, 1036, F2, 11) (dual of [1036, 984, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(251, 1035, F2, 10) (dual of [1035, 984, 11]-code), using
- OOA 2-folding [i] based on linear OA(251, 1034, F2, 10) (dual of [1034, 983, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(251, 517, F2, 2, 10) (dual of [(517, 2), 983, 11]-NRT-code), using
(51−10, 51, 3057)-Net in Base 2 — Upper bound on s
There is no (41, 51, 3058)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2254 064289 121228 > 251 [i]