Best Known (33, 33+9, s)-Nets in Base 2
(33, 33+9, 258)-Net over F2 — Constructive and digital
Digital (33, 42, 258)-net over F2, using
- net defined by OOA [i] based on linear OOA(242, 258, F2, 9, 9) (dual of [(258, 9), 2280, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(242, 258, F2, 8, 9) (dual of [(258, 8), 2022, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(242, 1033, F2, 9) (dual of [1033, 991, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(242, 1035, F2, 9) (dual of [1035, 993, 10]-code), using
- construction X 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(21, 11, F2, 1) (dual of [11, 10, 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(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(242, 1035, F2, 9) (dual of [1035, 993, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(242, 1033, F2, 9) (dual of [1033, 991, 10]-code), using
- appending kth column [i] based on linear OOA(242, 258, F2, 8, 9) (dual of [(258, 8), 2022, 10]-NRT-code), using
(33, 33+9, 345)-Net over F2 — Digital
Digital (33, 42, 345)-net over F2, using
- net defined by OOA [i] based on linear OOA(242, 345, F2, 9, 9) (dual of [(345, 9), 3063, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(242, 345, F2, 8, 9) (dual of [(345, 8), 2718, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(242, 345, F2, 3, 9) (dual of [(345, 3), 993, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(242, 1035, F2, 9) (dual of [1035, 993, 10]-code), using
- construction X 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(21, 11, F2, 1) (dual of [11, 10, 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(8) ⊂ Ce(6) [i] based on
- OOA 3-folding [i] based on linear OA(242, 1035, F2, 9) (dual of [1035, 993, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(242, 345, F2, 3, 9) (dual of [(345, 3), 993, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(242, 345, F2, 8, 9) (dual of [(345, 8), 2718, 10]-NRT-code), using
(33, 33+9, 2689)-Net in Base 2 — Upper bound on s
There is no (33, 42, 2690)-net in base 2, because
- 1 times m-reduction [i] would yield (33, 41, 2690)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2 199596 837136 > 241 [i]