Best Known (30−8, 30, s)-Nets in Base 2
(30−8, 30, 67)-Net over F2 — Constructive and digital
Digital (22, 30, 67)-net over F2, using
- net defined by OOA [i] based on linear OOA(230, 67, F2, 8, 8) (dual of [(67, 8), 506, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(230, 67, F2, 7, 8) (dual of [(67, 7), 439, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 7, 4) (dual of [(32, 7), 214, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- linear OOA(220, 35, F2, 7, 8) (dual of [(35, 7), 225, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (12, 20, 35)-net over F2, using
- linear OOA(210, 32, F2, 7, 4) (dual of [(32, 7), 214, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(230, 67, F2, 7, 8) (dual of [(67, 7), 439, 9]-NRT-code), using
(30−8, 30, 78)-Net over F2 — Digital
Digital (22, 30, 78)-net over F2, using
- net defined by OOA [i] based on linear OOA(230, 78, F2, 8, 8) (dual of [(78, 8), 594, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(230, 78, F2, 7, 8) (dual of [(78, 7), 516, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(230, 78, F2, 8) (dual of [78, 48, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(230, 137, F2, 8) (dual of [137, 107, 9]-code), using
- construction X4 applied to C([0,8]) ⊂ C([1,6]) [i] based on
- linear OA(229, 127, F2, 9) (dual of [127, 98, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(221, 127, F2, 6) (dual of [127, 106, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(29, 10, F2, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,2)), using
- dual of repetition code with length 10 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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 C([0,8]) ⊂ C([1,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(230, 137, F2, 8) (dual of [137, 107, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(230, 78, F2, 8) (dual of [78, 48, 9]-code), using
- appending kth column [i] based on linear OOA(230, 78, F2, 7, 8) (dual of [(78, 7), 516, 9]-NRT-code), using
(30−8, 30, 395)-Net in Base 2 — Upper bound on s
There is no (22, 30, 396)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1082 419306 > 230 [i]