Best Known (33, 41, s)-Nets in Base 2
(33, 41, 258)-Net over F2 — Constructive and digital
Digital (33, 41, 258)-net over F2, using
- net defined by OOA [i] based on linear OOA(241, 258, F2, 8, 8) (dual of [(258, 8), 2023, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(241, 1032, F2, 8) (dual of [1032, 991, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(241, 1035, F2, 8) (dual of [1035, 994, 9]-code), using
- 1 times truncation [i] based on linear OA(242, 1036, F2, 9) (dual of [1036, 994, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 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(231, 1024, F2, 7) (dual of [1024, 993, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(242, 1036, F2, 9) (dual of [1036, 994, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(241, 1035, F2, 8) (dual of [1035, 994, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(241, 1032, F2, 8) (dual of [1032, 991, 9]-code), using
(33, 41, 517)-Net over F2 — Digital
Digital (33, 41, 517)-net over F2, using
- net defined by OOA [i] based on linear OOA(241, 517, F2, 8, 8) (dual of [(517, 8), 4095, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(241, 517, F2, 7, 8) (dual of [(517, 7), 3578, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(241, 517, F2, 2, 8) (dual of [(517, 2), 993, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(241, 1034, F2, 8) (dual of [1034, 993, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(241, 1035, F2, 8) (dual of [1035, 994, 9]-code), using
- 1 times truncation [i] based on linear OA(242, 1036, F2, 9) (dual of [1036, 994, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 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(231, 1024, F2, 7) (dual of [1024, 993, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(242, 1036, F2, 9) (dual of [1036, 994, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(241, 1035, F2, 8) (dual of [1035, 994, 9]-code), using
- OOA 2-folding [i] based on linear OA(241, 1034, F2, 8) (dual of [1034, 993, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(241, 517, F2, 2, 8) (dual of [(517, 2), 993, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(241, 517, F2, 7, 8) (dual of [(517, 7), 3578, 9]-NRT-code), using
(33, 41, 2689)-Net in Base 2 — Upper bound on s
There is no (33, 41, 2690)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 199596 837136 > 241 [i]